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Page 1: Studies of Infinite Two-Dimensional Quantum Lattice …Studies of Infinite Two-Dimensional Quantum Lattice Systems with Projected Entangled Pair States By Jacob Jordan B.Eng. (Hons

Studies of Infinite Two-Dimensional Quantum Lattice

Systems with Projected Entangled Pair States

By

Jacob Jordan

B.Eng. (Hons 1st, 2003), UQ

A thesis submitted for the degree of Doctor of Philosophy at

The University of Queensland in January 2011

School of Physical Sciences

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c! Jacob Jordan, 2011.

Produced in LATEX2!.

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This thesis is composed of my original work, and contains no

material previously published or written by another person

except where due reference has been made in the text. I have

clearly stated the contribution by others to jointly-authored

works that I have included in my thesis.

I have clearly stated the contribution of others to my thesis as

a whole, including statistical assistance, survey design, data

analysis, significant technical procedures, professional edito-

rial advice, and any other original research work used or re-

ported in my thesis. The content of my thesis is the result

of work I have carried out since the commencement of my

research higher degree candidature and does not include a

substantial part of work that has been submitted to qualify

for the award of any other degree or diploma in any univer-

sity or other tertiary institution. I have clearly stated which

parts of my thesis, if any, have been submitted to qualify for

another award.

I acknowledge that an electronic copy of my thesis must be

lodged with the University Library and, subject to the Gen-

eral Award Rules of The University of Queensland, imme-

diately made available for research and study in accordance

with the Copyright Act 1968.

I acknowledge that copyright of all material contained in my

thesis resides with the copyright holder(s) of that material.

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Statements of Contributions

Statement of Contributions to Jointly Authored Works Contained in this The-

sis

• J. Jordan, R. Orus, G. Vidal, F. Verstraete and J. I. Cirac, Classical Simulation of

Infinite-Size Quantum Lattice Systems in Two Spatial Dimensions, Physical Review

Letters, 101, 250602, (2007). This paper outlined the first iPEPS algorithm for

computing the ground state of infinite 2D quantum lattice systems. The main ideas

behind this algorithm were devised by GV, FV and JIC. Most of the implementation

of the algorithm was performed by JJ and RO. The paper was written by RO and

GV.

• J. Jordan, R. Orus and G. Vidal, Numerical study of the hard-core Bose-Hubbard

model on an infinite square lattice, Physical Review B, 79, 174515, (2009). This

paper details the application of the iPEPS algorithm to the hard-core Bose-Hubbard

model. The simulations were performed by JJ. The paper was written by GV with

assistance from JJ and RO.

Statement of Contributions by Others to the Thesis as a Whole

This thesis was completed under the supervision of Prof. Guifre Vidal and Dr. Roman

Orus. The results of Chapter 6 are based on an algorithm conceived by RO and GV

(Ref. [OV08]). Chapter 8 is heavily based on a paper by JJ, GV and RO (Ref. [JOV09]).

Chapter 11 extends upon work in 1D by RO. The structure of this chapter was suggested

by RO. The thesis was otherwise entirely written by the author.

Statement of Parts of the Thesis Submitted to Qualify for the Award of An-

other Degree

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None.

Published Works by the Author Incorporated into the Thesis

• J. Jordan, R. Orus, G. Vidal, F. Verstraete and J. I. Cirac, Classical Simulation of

Infinite-Size Quantum Lattice Systems in Two Spatial Dimensions, Physical Review

Letters, 101, 250602, (2007). This publication is the basis for Chapters 5 & 7.

• J. Jordan, R. Orus and G. Vidal, Numerical study of the hard-core Bose-Hubbard

model on an infinite square lattice, Physical Review B, 79, 174515, (2009). This

publication is the basis for Chapter 8.

Additional Published Works by the Author Relevant to the Thesis but not

Forming Part of it

• P. Corboz, J. Jordan and G. Vidal Simulation of 2D fermionic lattice models with

Projected Entangled-Pair States: Next-nearest neighbor Hamiltonians, Physical Re-

view B, 82, 245119, (2010).

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Acknowledgments

I would firstly like to thank my supervisor Prof. Guifre Vidal for all of his support and

guidance throughout my PhD. It has been a great privilege to work alongside such an

accomplished scientist. I would also like to thank my co-supervisor Dr. Roman Orus.

Roman and I worked very closely on this particular project and his support, patience,

encouragement and positive frame of mind were exceedingly important.

I would also like thank my colleagues in the University of Queensland Physics Department

for their comradeship over the last few years. A PhD is by nature a lonely path and their

friendship has been important.

I would like to acknowledge the financial support provided by the Australian Research

Council, Prof. Vidal, the University of Queensland School of Mathematics and Physics

and the University of Queensland Graduate School.

In December of 2008 I visited the research group of Prof. Immanuel Bloch in Mainz as a

guest of Dr. Belen Paredes. I would like to thank Belen for her help in arranging a most

enjoyable visit.

Outside of UQ, I have enjoyed an association with Oxley United Football Club for many

years as a player, coach and committee member. This has not only been a place to

unwind, but has taught me the importance of community and volunteer spirit - values

that can easily dim in the singular vision of the research scientist. I thank the many close

friends of mine I have met through the club for the good times we have had.

Most of all, I would like to thank my parents, Peter and Linda Jordan, my brothers Luke

and Will, my sister Hannah and all of my extended family for their inspiration, love and

support. It is for you that I dedicate this thesis.

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Abstract

Determining the properties of quantum many-body systems is a central challenge in mod-

ern physics. Being able to determine the macroscopic properties of a system from its

microscopic description would hasten progress in many fields of science and technology.

However, we currently lack the tools to solve such problems generally, or even to develop a

theoretical intuition about how many systems might behave. From a simple Hamiltonian

description of the system, one may obtain complex, highly correlated collective behaviour.

Computational techniques have played a major part in the e!ort to determine properties

of quantum many-body systems. However, as the total degrees of freedom in the sys-

tem scales exponentially in the system size, numerical diagonalization of the Hamiltonian

quickly becomes computationally intractable and one must develop more e"cient approx-

imate techniques to explore the system. Present numerical methods such as quantum

Monte Carlo and series expansion have provided insight into many systems of interest,

but are also held back by fundamental di"culties.

In this thesis, we focus on tensor networks, a relatively new ansatz for representing quan-

tum many-body states. Tensor networks are motivated by two ideas from quantum in-

formation: firstly, that quantum entanglement is the source of the immense di"culty of

simulating quantum systems classically, and secondly that the ground states of certain

Hamiltonians exist in a low-entanglement region of the entire Hilbert space. The strength

of tensor networks is that they provide a systematic way of representing this class of low-

entanglement quantum states. In particular, this thesis describes the iPEPS algorithm for

computing the ground states of infinite, two-dimensional quantum lattice systems based

on the Projected Entangled Pair States (PEPS) ansatz. We then benchmark the algo-

rithm by computing the phase diagrams of several systems that have been studied with

other techniques. Lastly, we apply our algorithm to problems that are not well solved by

current approaches, such as frustrated spin systems.

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Keywords: Projected entangled pair states, quantum many-body systems, simulation

algorithms, tensor networks, quantum entanglement, quantum information.

Australian and New Zealand Standard Research Classifications (ANZRC):

020401 Condensed Matter Characterisation Technique Development (50%), 020603 Quan-

tum Information, Computation and Communication (50%).

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List of Abbreviations and Symbols

TN Tensor network

MPS Matrix product state

PEPS Projected entangled pair state

TPS Tensor product state

CTM Corner transfer matrix

CTMRG Corner transfer matrix renormalization group

DMRG Density matrix renormalization group

QMC Quantum Monte Carlo

MF Mean-field

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Contents

List of Tables xvii

List of Figures xix

1 Introduction 1

2 The Quantum-Classical Correspondence 5

3 Foundations of Tensor Networks 7

3.1 Basic tensor operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Tensor Network Descriptions of Quantum States . . . . . . . . . . . . . . . 14

3.3 E"cient Representations of Quantum States . . . . . . . . . . . . . . . . . 21

4 Dimension and Computational Complexity 25

4.1 D = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 D = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3 D = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Projected Entangled Pair States 37

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5.1 Projected Entangled Pair States . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Problem Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3 The iPEPS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.4 Computing Physical Properties of PEPS States . . . . . . . . . . . . . . . 46

5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6 The 2D Classical Ising Model 57

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7 The 2D Quantum Ising Model 65

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8 The Hard-Core Bose-Hubbard Model 73

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

9 The Quantum Potts Model 87

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

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9.2 The quantum Potts Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

9.3 q = 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

9.4 q = 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

10 The J1-J2 Model 107

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

10.2 The J1-J2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

10.3 Columnar and Plaquette Ordered Ground States . . . . . . . . . . . . . . . 112

10.4 Algorithmic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 112

10.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

10.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

11 Geometric Entanglement 125

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

11.2 Computing the Closest Product State . . . . . . . . . . . . . . . . . . . . . 127

11.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

11.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

12 Conclusion 137

12.1 Thesis Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

12.2 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

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A Infinite MPS Methods for Computing the Environment 141

A.1 Problem Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A.2 Evolution of an Infinite MPS by Non-unitary Operators . . . . . . . . . . . 151

A.3 MPS-based Contraction Schemes for PEPS . . . . . . . . . . . . . . . . . . 155

B Corner Transfer Matrix Renormalization Group Algorithms for the Square

Lattice 165

B.1 Problem Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

B.2 The Corner Transfer Matrix Renormalization Group for iPEPS . . . . . . . 166

B.3 Coarse-Graining Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 168

B.4 The Directional CTMRG Approach . . . . . . . . . . . . . . . . . . . . . . 169

B.5 An Improved Directional Algorithm . . . . . . . . . . . . . . . . . . . . . . 170

B.6 Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

C Update Schemes for PEPS tensors 175

C.1 Problem Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

C.2 A Variational Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

C.3 Conjugate Gradient Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 178

C.4 Comparison of the Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 181

References 185

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List of Tables

5.1 Leading order computational complexity of various iPEPS algorithms for

the square lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.1 Critical point and exponent " as a function of D. . . . . . . . . . . . . . . 70

9.1 The location of the phase transition for various versions of the algorithm

and di!erent D. The ’lowest energy’ solution is taken from the lowest energy

ground states on either side of the phase transition. . . . . . . . . . . . . . 92

9.2 A summary of results for the quantum Ising model and q = 3 and q = 4

quantum Potts models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

10.1 Leading order computational cost of four PEPS schemes for the J1-J2 model.116

11.1 Density of global geometric entanglement at the critical point of the quan-

tum Ising model for various values of dimensionality . . . . . . . . . . . . . 134

A.1 The leading order computational costs for the square, hexagonal, Kagome

and Triangular lattice infinite-MPS based contraction schemes. . . . . . . . 163

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List of Figures

3.1 Example of a simple three-index tensor. . . . . . . . . . . . . . . . . . . . . 7

3.2 Examples of common tensors and their equivalent mathematical form . . . 8

3.3 Example of a tensor permutation . . . . . . . . . . . . . . . . . . . . . . . 9

3.4 Example of a tensor reshape . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.5 The process for contracting a simple three-tensor tensor network. . . . . . 11

3.6 An example of the process splitting a tensor . . . . . . . . . . . . . . . . . 13

3.7 Some Common TN Quantum State Operations . . . . . . . . . . . . . . . 15

3.8 The area law of quantum entanglement for ground states of local Hamilto-

nians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.9 The representation of a quantum state with coe"cients Ci1i2...iN in an MPS

form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.10 The representation of a quantum state with coe"cients Ci1i2...iN in a PEPS

form. Here we label a D-dimensional bond index and a d-dimensional

physical index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.11 The refinement parameter and the accessibility of the Hilbert space with

TNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 The computation of a local observable of a 1D classical system . . . . . . . 27

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4.2 The imaginary-time evolution of a point particle . . . . . . . . . . . . . . . 28

4.3 A 4" 4 2D tensor network that might represent, for example, a partition

function of a 2D classical system at finite temperature . . . . . . . . . . . 29

4.4 An infinite, translationally invariant 2D network defined by the four-legged

tensor a. The tensor # describes the boundary state. . . . . . . . . . . . . 29

4.5 The iTEBD algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.6 The computation of a local observable quantity, M, of an MPS . . . . . . . 31

4.7 The contraction of a 2D tensor network, using the MPS to describe the

boundary state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.8 Corner Transfer Matrix basics . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.9 The CTMRG approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.1 Representing 2D systems with a 1D ansatz . . . . . . . . . . . . . . . . . . 39

5.2 The infinite PEPS for the square lattice . . . . . . . . . . . . . . . . . . . 41

5.3 The four basic stages of the PEPS algorithm . . . . . . . . . . . . . . . . 42

5.4 The environment around a tensor link . . . . . . . . . . . . . . . . . . . . . 44

5.5 The computation of a local observable as a 2D tensor network contraction 49

5.6 The calculation of spatial correlation functions with PEPS. . . . . . . . . . 51

5.7 The expression of the fidelity as a 2D tensor network. Here, the states

|$0 (%)# and |$0 (%!)# are translationally invariant and contain tensors A,B

and C,D respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.8 The tensor contraction that gives the four-legged tensor containing the

coe"cients of the two-site reduced density matrix. . . . . . . . . . . . . . . 54

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6.1 The process for expressing the partition function as a 2D square tensor

network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.2 A plot of the magnetization and magnetization error (below) of the 2D

classical Ising model for various &, along with the exact solution . . . . . . 62

6.3 A plot of the magnetization of the 2D classical Ising model for various

numbers of boundary state iterations. Note that as the number of iterations

increases, we more closely track the exact solution. . . . . . . . . . . . . . 63

6.4 A plot showing the two-point correlation function of the classical Ising

model for various & along with the exact solution . . . . . . . . . . . . . . 64

7.1 Transverse magnetization mx and energy per site e of the quantum Ising

model as a function of the transverse magnetic field h . . . . . . . . . . . . 68

7.2 Magnetization mz(%) of the quantum Ising model as a function of the trans-

verse magnetic field % . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.3 A comparison of the order parameter for iMPS, CTMRG and the simplified

update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.4 Two-point correlator Sxx(l) of the quantum Ising model near the critical

point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.5 Fidelity diagram of the quantum Ising model . . . . . . . . . . . . . . . . . 71

8.1 Particle density '(µ), energy per lattice site ((') and condensate fraction

'0(') of the HCBH model for a PEPS/TPS with D = 2, 3. . . . . . . . . . 78

8.2 Purity r and entanglement entropy SL as a function of the chemical poten-

tial µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.3 Two-point correlation function C(s) of the HCBH model, versus distance

s (measured in lattice sites), along a horizontal direction of the lattice . . . 81

8.4 Fidelity per lattice site f(µ1, µ2) for the ground states of the HCBH model 82

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8.5 Evolution of the energies $H0# and $H#, the density ', and condensate

fraction '0 after a translation invariant perturbation V is suddenly added

to the Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

9.1 q = 3 Potts model energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

9.2 q = 3 Potts model first derivative of the energy . . . . . . . . . . . . . . . 94

9.3 Plot showing the order parameter of the q = 3 Potts model as a function

of external field, %Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

9.4 Plot showing the magnetization of the q = 3 Potts model in the direction

of the magnetic field, as a function of %Z . . . . . . . . . . . . . . . . . . . . 96

9.5 Fidelity diagram for the q = 3 Potts model, computed from D = 3 ground

states evolved with the simplified update. . . . . . . . . . . . . . . . . . . . 97

9.6 Two point correlation function of the q = 3 Potts model for various values

of the external field, %Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

9.7 Entanglement entropy of the q = 3 Potts model . . . . . . . . . . . . . . . 99

9.8 q = 4 Potts model energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

9.9 The first derivative of the energy-peer-site of the q = 4 Potts model with

respect to the magnetic field, %Z , as calculated by a finite di!erence method101

9.10 A plot of the order parameter of the q = 4 Potts model . . . . . . . . . . . 102

9.11 A plot of the magnetization in the direction of the magnetic field for the

q = 4 Potts model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

9.12 The fidelity diagram for the q = 4 Potts model, using the simplified update

D = 4 ground states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

9.13 Two point correlation function of the q = 4 Potts model. . . . . . . . . . . 103

9.14 Entanglement entropy for the q = 4 Potts model . . . . . . . . . . . . . . . 104

xxii

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10.1 A simple example of frustration . . . . . . . . . . . . . . . . . . . . . . . . 108

10.2 Neel and Collinear ground states. . . . . . . . . . . . . . . . . . . . . . . . 110

10.3 The generally accepted phase diagram for theJ1-J2 model. . . . . . . . . . 111

10.4 The columnar dimer and plaquette RVB states . . . . . . . . . . . . . . . . 113

10.5 PEPS algorithm variants for the J1-J2 model . . . . . . . . . . . . . . . . . 114

10.6 A plot comparing the energies given by the four PEPS algorithm variants . 117

10.7 A plot of the energy per lattice site vs J2J1

for various values of D. (inset)

Convergence of the energy per lattice site with the bond dimension, D, atJ2J1

= 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

10.8 A plot of the Neel order parameter vs J2J1

for various values of D . . . . . . 119

10.9 A plot of the co-linear order parameter vs J2J1

for various values of D . . . . 120

10.10A plot of the nearest-neighbour spin-spin expectation values and plaquette

order parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

10.11A plot of the entanglement entropy for the J1-J2 model. . . . . . . . . . . . 124

11.1 The determination of the state µk, expressed as a 2D tensor contraction. . 128

11.2 Hexagonal lattice quantum Ising model: (a) expectation value of the Hamil-

tonian, and (b) order parameter. . . . . . . . . . . . . . . . . . . . . . . . . 131

11.3 Square lattice quantum Ising model: (a) expectation value of the Hamilto-

nian, and (b) order parameter . . . . . . . . . . . . . . . . . . . . . . . . . 132

11.4 Square lattice quantum 3-Potts model: (a) expectation value of the Hamil-

tonian, and (b) order parameter. . . . . . . . . . . . . . . . . . . . . . . . . 133

11.5 (a) Geometric entanglement for the hexagonal lattice quantum Ising model.

(b) Geometric entanglement for the square lattice quantum Ising model . . 135

xxiii

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A.1 The operation of the two-body gates a and b on the boundary MPS!

!)[0]"

. 142

A.2 The exact evolution of the boundary state with an MPS may lead to an

increase in the MPS bond dimension. . . . . . . . . . . . . . . . . . . . . . 144

A.3 Overview of Lemma 2, part 1 . . . . . . . . . . . . . . . . . . . . . . . . . 145

A.4 Overview of Lemma 2, part 2 . . . . . . . . . . . . . . . . . . . . . . . . . 146

A.5 Overview of Lemma 2, part 3. . . . . . . . . . . . . . . . . . . . . . . . . . 147

A.6 Overview of Lemma 3, part 1. . . . . . . . . . . . . . . . . . . . . . . . . . 148

A.7 Overview of Lemma 3, part 2. . . . . . . . . . . . . . . . . . . . . . . . . . 149

A.8 Overview of Lemma 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

A.9 Overview of Lemma 5, part 1 . . . . . . . . . . . . . . . . . . . . . . . . . 152

A.10 Overview of Lemma 5, part 2. . . . . . . . . . . . . . . . . . . . . . . . . . 153

A.11 The evolution of an MPS under non-unitary gates . . . . . . . . . . . . . . 154

A.12 The computation of the left and right scalar product matrices . . . . . . . 155

A.13 The gates ’a’ and ’b’, formed by contracting the PEPS tensors with their

conjugate versions along the physical index. . . . . . . . . . . . . . . . . . 156

A.14 The diagonal contraction scheme for the square lattice . . . . . . . . . . . 157

A.15 The parallel contraction scheme for the square lattice . . . . . . . . . . . . 158

A.16 The computation of the scalar product matrices in the parallel scheme. . . 159

A.17 A contraction scheme for the hexagonal lattice . . . . . . . . . . . . . . . . 160

A.18 A contraction scheme for the Kagome lattice . . . . . . . . . . . . . . . . . 161

A.19 A contraction scheme for the triangular lattice . . . . . . . . . . . . . . . . 162

xxiv

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B.1 The basic CTM structure for a PEPS with 2" 2 periodicity . . . . . . . . 166

B.2 Insertion of a 2 " 2 block of sites i) to the left of the existing unit-cell ii)

to the right of the existing unit-cell iii) in the middle of the existing unit-cell167

B.3 The absorption process in a horizontal direction . . . . . . . . . . . . . . . 168

B.4 Renormalization of the vertical bonds by the renormalization operators Q

and W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

B.5 The stages of the first directional CTMRG algorithm. . . . . . . . . . . . . 171

B.6 The stages of a more stable, but computationally more expensive direc-

tional CTMRG algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

C.1 The expression of the distance metric in terms of tensor contractions . . . 176

C.2 The tensor network representation of equation C.2. . . . . . . . . . . . . . 177

C.3 The matrices NA and MA . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

C.4 The splitting of A and B into W, X, Y and Z . . . . . . . . . . . . . . . . 179

xxv

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xxvi

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Chapter 1

Introduction

Determining the properties of quantum many-body systems is a central challenge in mod-

ern physics. Improving our understanding of such systems would catalyze progress in

many fields of physics and technology, such as condensed matter physics and quantum

field theory, the search for a quantum theory of gravity and the engineering of systems that

harness quantum phenomena as a basic resource. However, the complexity of unraveling

the properties of quantum many-body systems belies the simple mathematical elegance

of the postulates of quantum mechanics. From a very simple microscopic description, one

can observe complex, highly correlated collective behaviour. Indeed, it is still a point

of contention as to whether such physics can be explained entirely from a microscopic

description, or whether emergent physical laws play a part [And72]. Nonetheless, we do

not currently have the tools to fully verify the reductionist view of Nature. Even a very

fundamental task, such as determining the zero-temperature ground state of a quantum

many-body system typically requires classical resources that scale exponentially in the

system size.

Historically, there have been two major avenues for addressing this problem. In a paper

in 1982[Fey82], Richard Feynman suggested that the peculiarities of quantum mechan-

ics could be exploited to simulate quantum phenomena exponentially faster than with

a classical Turing machine. Since then, a great deal of e!ort has been directed at de-

veloping a scalable quantum computer to e"ciently solve such problems among many

others. At present however, a universal quantum simulation device seems on the distant

horizon, dogged by the experimental fragility of quantum mechanics and its tendency to

decohere[NC00]. In parallel, classical computer technology was advancing exponentially

year upon year in accordance with the much fabled Moore’s law. Whilst this did not

1

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2 Introduction

alleviate the computational intractability of exactly determining (for example) ground

states of macroscopically large systems, it did allow the exact diagonalization of progres-

sively larger systems in time. More importantly, it stimulated interest in a variety of

methods for approximately calculating ground states of many-body systems, automating

the computational work of existing techniques such as perturbation theory and leading to

genuinely novel methods such as the density matrix normalization group (DMRG)[Whi92]

and quantum Monte Carlo (QMC)[CA80].

Entanglement - the property whereby a measurement on one part of a quantum system

a!ects the measurement outcome of another - was initially a mysterious and controversial

aspect of quantum mechanics. In 1935, Einstein, Podolsky and Rosen could not accept

that the universe may have a fundamentally non-local interpretation and expressed as

much in the so-called EPR paradox[EPR35]. Here, they proposed that quantum mechan-

ics was incomplete, and that the missing physics must be accounted for by a local hidden

variable. The theoretical work of Bell[Bel64], carried forward by Clauser, Horne, Shimony

and Holt[CHSH69] suggested some simple inequalities that, when tested experimentally,

would verify if quantum mechanics obeyed such a local realist picture. The experiments of

Aspect[AGR82, ADR82] showed that quantum mechanics violates such inequalities, and

as such entanglement - an unmistakably non-classical physical property - was confirmed

as a very real part of Nature.

Entanglement is at the heart of the complexity of describing quantum many-body physics.

Interesting quantum many-body phenomena such as quantum phase transitions[Sac99] are

driven by the possibility for many-body states to exhibit strongly correlated behaviour

at absolute zero temperature. Entanglement is also of great consequence in the classical

simulation of quantum many-body systems for a couple of reasons. On one hand, it can

be shown that the presence of entanglement is key to the exponential scaling of the cost of

the exact classical simulation of quantum systems[Vid03]. On the other hand, it is known

that for a certain class of Hamiltonians, the amount of entanglement in the ground state

is limited by well established laws[VLRK03]. The ground state for these systems lives in a

low entanglement region of the entire Hilbert space. This means that we can approximate

the ground states of these systems by considering a subset of all the degrees of freedom.

In quantum Monte Carlo, for example, this is exploited by sampling the Hilbert space.

The question naturally arises - is there a representation of quantum states that is system-

atically confined to a low entanglement region of the entire Hilbert space? In this thesis

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3

we examine tensor networks (TNs), a recently developed ansatz for many-body states

that is limited in such a way. The earliest example of a tensor network representation of

a quantum state is in the A#eck-Kennedy-Lieb-Tasaki (AKLT) ground state [AKLT88].

The authors here identified a spin-1 Hamiltonian chain, where the ground state had many

unique properties and which decomposed in such a way that it could be exactly written in

terms of symmetrization operators on entangled spin 1/2 pairs (“lattice bonds”). Later,

Fannes, Nachtergale and Werner introduced finitely correlated states, a description for 1D

quantum systems in which many-body correlations are modeled with a finite dimensional

vector space at each lattice bond [FNW92]. This work formalised a representation, later

to become known as the matrix product state (MPS), which could potentially capture the

low energy properties of one-dimensional (1D) quantum systems, but did not present a

general method for obtaining the ground state of a particular Hamiltonian.

In parallel, White [Whi92] devised a variational method for obtaining the ground state of

the Heisenberg antiferromagnetic spin chain based on using renormalization group[Wil75]

methods. This procedure, known as the density matrix renormalization group (DMRG)

formulated a method of reducing the degrees of freedom of a quantum system in such a

way as to keep the most relevant many-body correlations.

The potential for the MPS to be used to simulate quantum dynamics was highlighted

when Ostlund and Rommer[OR95] demonstrated that the ground states determined by

DMRG had an equivalent MPS representation. On this basis, they suggested that these

ground states could be determined by a variational treatment of the family of MPS states.

Later, whilst investigating the e"cient classical simulation of quantum computation, Vi-

dal [Vid03] showed, using the MPS description of a pure quantum state and the circuit

model of quantum computation, that entanglement is a necessary resource for the ex-

ponential computational speedup of a quantum computer. Elsewhere, Vidal described

the time-evolving block decimation (TEBD) algorithm for finding the ground state of a

one-dimensional quantum system represented by an MPS and governed by a Hamiltonian

with nearest neighbour interactions[Vid04].

In this thesis, we focus on algorithms for finding the ground state of two-dimensional

(2D) lattice systems. Here, there is much need for novel e"cient computational methods,

as analytic techniques are more di"cult to devise, and DMRG has struggled to deal

with the correlations in systems in higher-than-one dimensions. Moreover, some of the

more interesting examples of exotic behaviour in many-body systems, such as topological

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4 Introduction

order, are only known to exist in 2D systems. The basis for our algorithm is the projected

entangled pair states (PEPS) ansatz, introduced in [VC04]. This natural extension of

the MPS to quantum systems in two and higher dimensions had also been introduced in

the context of 3D classical systems as the tensor product states (TPS) ansatz[NOH+00,

NHO+01, GMN03]. For this reason the literature may use the two terms interchangeably.

Our approach here is to firstly develop an understanding of the physical problem in

Chapter 2, by defining a standard technique for finding the ground states of quantum

systems known as imaginary-time evolution. This method, which lies at the heart of

our algorithm, also draws a well-known correspondence between computing the ground

state of D-dimensional quantum systems and finding the statistical properties of (D+1)-

dimensional classical systems at finite temperature. In Chapter 3, we review the founda-

tional ideas and notations of tensor network theory and motivate their consideration for

studying ground states of many-body Hamiltonians. In Chapter 4, we suggest a hierarchi-

cal picture of tensor networks in terms of their spatial dimensionality. We show that the

problem of finding the ground state in zero spatial dimensions (e.g. a single quantum spin)

is trivial for a classical computer, and that from this we can deduce e"cient algorithms

for approximating the ground state in higher dimensions. In Chapter 5, we describe the

properties of the PEPS ansatz, and introduce our algorithm for finding the ground states

of infinite, translationally invariant two-dimensional systems. We demonstrate the power

and versatility of the ansatz for computing physical quantities of significance. In Chapters

6, 7, 8 and 9 we benchmark the algorithm by applying it to systems for which there is

either an exact solution, or there has been extensive study with other numerical methods.

In Chapters 10 and 11 we show that the algorithm can be used to tackle problems that

are di"cult to study with existing algorithms.

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Chapter 2

The Quantum-Classical Correspondence

Consider a classical system made of N sites, where each site can be in one of d di!erent

states. Then at any given time, the system is in one of the dN possible configurations.

Each configuration * has an energy E!. In thermal equilibrium at temperature T , the

probability of finding the system in configuration * is given by e""E!/Z, where

Z =#

!

e""E! . (2.1)

is the partition function and " = 1/kT.

Thus, computing the partition function involves a summation over a number of configu-

rations, each labeled by a di!erent value of *, that scales exponentially in the system size.

Manipulating the partition function can give us a wealth of information about the statis-

tical properties of the ensemble, such as the average energy of the system or the average

magnetization of a magnetic system. So, for a given Hamiltonian, our partition function

is a function of temperature, T , and we may monitor how the statistical properties of the

system change with varying temperature.

In a quantum mechanical setting, quantum fluctuations play an analogous role to those

of temperature, such that even at zero temperature the description of the system can be

very rich. For a quantum Hamiltonian, HQ, the ground state, |$0#, is the eigenstate of

HQ for which the following equation holds:

HQ |$0# = E0 |$0# , (2.2)

5

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6 The Quantum-Classical Correspondence

where E0 is the lowest eigenvalue of H0.

In a straightforward choice of basis, |$0# may be described by a quantum superposition in

this basis. Here, we are interested in observable properties of the ground state. A quantum

phase transition (QPT) occurs at zero-temperature where upon changing some Hamilto-

nian parameter, the qualitative properties of the ground state undergo some change.

Computing observables for ground states corresponding to di!erent values of the Hamil-

tonian parameter, we can build a phase diagram of the system. In particular, for many

of the phase transitions we are interested in, the change in phase is captured by a local

order parameter that is zero in one phase and non-zero in the other.

In practice, we cannot compute the properties of the ground state until we have deter-

mined the ground state, and for most Hamiltonians of interest this is a highly non-trivial

task. Exact diagonalization becomes intractable for relatively small systems, so we must

consider other means of finding the lowest energy eigenstate of the Hamiltonian. A well-

known technique for doing so is imaginary-time evolution. Consider some initial state

|$i#, with non-zero overlap with the actual ground state |$g#. Then, it is possible to show

that evolving |$i# with the imaginary time evolution operator, e"H# , leads to the ground

state (up to some normalisation constant) in the limit of infinite imaginary-time, + , i.e.

|$g# % lim##$

e"H# |$i# (2.3)

One can treat this imaginary-time as an extra dimension in the quantum ground-state

problem. From this reasoning, a well-known correspondence[Sac99, Hen99] arises. Finding

the ground-state of an infinite quantum system in D dimensions is equivalent to calculating

the local observable properties of an infinite (D + 1)-dimensional classical system.

In this document we are concerned with lattice systems, where the spatial dimensions have

been discretized. To complete the above discussion, consider dividing the imaginary-time

evolution into steps of size ,+ . Now, both the space and time dimensions are discrete,

and so the correspondence between finding the ground state of D-dimensional quantum

systems and the partition function of (D)-dimensional classical systems also holds for

lattice systems. In Chapter 4 we will discuss this at greater length and describe the

consequences of this from a computational perspective.

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Chapter 3

Foundations of Tensor Networks

A tensor is a multi-dimensional array of complex coe"cients. A collection of such tensors,

with legs connected according to some network pattern is called a tensor network (TN).

Graphically, we usually denote each tensor by a closed shape with protruding legs (see

fig. 3.1). Each leg specifies an index of the array, and the total number of legs determines

the order of the tensor.

A generic coe"cient of the tensor A in fig. 3.1 is A$"% , where -, " and . run from 1

to &$, &" and &%. We say that the dimensions of the indices are &$, &" and &% and A

contains &$&"&% complex coe"cients.

Figure 3.1: Example of a simple three-index tensor.

7

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8 Foundations of Tensor Networks

Figure 3.2: Examples of common tensors and their equivalent mathematical form, (i) atensor with no legs is a complex number A. (ii) A tensor with one leg is a vector ofcoe"cients A$. (iii) A tensor with two legs is a matrix with elements A$".

Some special cases of tensors are shown in figure 3.2. One can see that a tensor with no

legs is a complex number, a tensor with a single leg is a vector and a tensor with two

legs is a matrix. Since such structures are easily dealt with in mathematical software we

will see that being able to manipulate parts of our tensor network into such structures is

extremely important.

3.1 Basic tensor operations

3.1.1 Permutation

A permutation of a tensor reorders the coe"cients of a tensor according to some reordering

of the tensor legs. In fig. 3.3 we depict the tensor A permuted into the tensor A, where

each coe"cient of A is determined according to the assignment A"$% = A$"% .

3.1.2 Reshaping

A reshape operation takes two or more legs of a tensor and joins them into a single

index, thus reducing the order of the tensor whilst keeping the number of coe"cients

unchanged. This means that the index of the new leg runs over the product of the indices

of the constituent legs. A simple example is shown in figure 3.4. Here, we have joined

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3.1 Basic tensor operations 9

Figure 3.3: Example of a tensor permutation, taking the tensor A with coe"cients A$"%and returning the tensor A with coe"cients A"$%.

Figure 3.4: Example of a tensor reshape, taking the tensor A with coe"cients A$"% andreturning the tensor A with coe"cients A$&.

two legs of a tensor A to create a new index , of dimension &"&%. The leg corresponding

to the index - is left untouched. The result is a new tensor A with only two legs, and

coe"cients A$&. Often, we wish to join indices together and later on recover the original

indices. A useful notation then is to express the coe"cients as A$("%) where the brackets

indicate that the index has been joined.

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10 Foundations of Tensor Networks

3.1.3 Tensor Multiplication

A tensor multiplication is defined as an inner product over an index that is shared be-

tween two tensors. That is, for two tensors A and B with coe"cients A$"% and B"', a

tensor multiplication obtains a new tensor C with coe"cients C$%' =$

"A$"%B"'. Such

a process may also be called the contraction of the shared index ". In practice, mathe-

matical software supports matrix multiplication and so we perform tensor multiplication

by succession of permutation, reshape and matrix multiplication operations. Often, we

want to contract many indices in a tensor network shared by many tensors. We do this

by performing a series of tensor multiplications. As a shorthand, we will usually call the

contraction of all the shared indices in a network the contraction of the tensor network.

We show in figure 3.5 a simple three-body tensor network contraction broken down into

its various stages. Here, the subscript indices for each tensor specify the ordering of the

indices.

The stages in this network contraction can be described as follows. Firstly (i), we select

a pair of connected tensors to multiply. Here, we choose A and C, which share the index

labeled .. The choice of order of multiplication does not a!ect the result, but is normally

made on consideration of computational cost. Next (ii), we permute the tensor A & A

such that the shared index, . appears last in A’s its list of indices. Since . appears first

in C’s index order, no permutation of C is required. Then (iii), we reshape A and C into

matrices A and C by joining the indices -, " and ,, and the indices ( and '. Then we

contract the tensors by means of matrix multiplication, returning a new tensor O. i.e.

O($"&)((') =$

%A($"&)%C%((')

O = AC(3.1)

Then (iv), we reshape O so that the non-contracted indices re-appear. Steps vi-ix closely

mirror steps ii-v.

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3.1 Basic tensor operations 11

Figure 3.5: The process for contracting a simple three-tensor tensor network.

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12 Foundations of Tensor Networks

3.1.4 Computational Cost Considerations of Tensor Network

Contractions

From the above description, it should be obvious that a generic N -body network con-

traction can be broken down into N ' 1 two-body tensor multiplications, each of which

contain permutation, reshape and matrix multiplication operations. It should also be

noted that there are two computational considerations. The computational cost of each

step is the number of computational cycles required to multiply two tensors. This is the

sum total of cycles needed for the permute, reshape and matrix multiplication stages and

is usually dominated by the last stage. The cost of multiplying a matrix A$" and B"%

scales as &$&"&% . The second computational consideration is the memory requirement,

which is the maximum amount of computer memory required to store the current state of

the tensor network at any stage of the contraction. For the same multiplication of A and

B, the memory requirement is &$&"+&"&%+&$&%. For the contraction of a many-tensor

network, it can also be seen that both of these costs depend on the order of contraction. In

general, the order of contraction that optimises the computational cost may not optimise

the memory requirement and vice versa. Since it is usually preferable to minimise the

total time required for a contraction, the contraction with minimal computational cost

is usually chosen. However, there are instances when the memory requirement becomes

so large that the computer’s memory resources are exhausted and the operating system

needs to store and retrieve information from the hard disk. Hard disk access is orders of

magnitude slower than RAM access and as such this can severely a!ect the time involved

in contracting the network. So the best contraction order is the one that minimises the

computational cost without exceeding a particular machine’s RAM resources.

3.1.5 Splitting

Splitting of a tensor involves making a decomposition into an equivalent many-tensor

form. We can choose whichever matrix decomposition we wish to split a tensor, so long

as the original tensor is recovered when we contract the resulting split tensors. Typically

however, we will make use of either a singular value decomposition (SVD) or an eigenvalue

decomposition (ED). For instance, in figure 3.6, we show a tensor A decomposed into

three tensors, B, % and C, which when contracted together reproduce A. Much like a

contraction, splitting a tensor involves permute and reshape operations to put the tensor

in matrix form. For the SVD or ED, the dimension of the indices , and ,! reflect the rank

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3.1 Basic tensor operations 13

Figure 3.6: An example of the process splitting a tensor

of the matrix. The open indices of A are recovered in the B and C tensors by reshape

operations.

3.1.6 Truncation

Each leg of a tensor describes a vector space addressed by an index. We have seen that

the cost of storing and manipulating tensor networks depends greatly on the dimension

of the tensor legs. In order to keep such costs manageable, we may sometimes reduce the

dimension of the shared bonds of a tensor network, by projecting out tensor coe"cients

corresponding to particular values of a given index. This operation is known as truncation.

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14 Foundations of Tensor Networks

3.2 Tensor Network Descriptions of Quantum States

Mathematically, pure quantum states can be described by a vector in Hilbert space. An

N -body quantum system is described by a Hilbert space that is a tensor product of N

local Hilbert spaces, each spanned by an orthonormal basis of dimension d, and indexed

by i. This means we can express any (pure) N -body quantum state |$N# by the following

expansion

|$N# =#

i1i2i3...iN

Ci1i2i3...iN |i1i2i3.....iN#, (3.2)

where i1, i2, ...., iN span the degrees of freedom within each d dimensional local system.

Note that the number of coe"cients, C, scales exponentially in N , and this is the di"culty

in the well-known quantum many-body problem. Exactly representing and computing

physical characteristics of the N -body quantum state is computationally hard.

For now, one can see that the coe"cients C may be stored in a tensor, C, with N legs

corresponding to the local degrees of freedom. Figure 3.7i) shows a graphical representa-

tion of a tensor corresponding to the state |$N#. We can also represent basic operations

on such a state (figures 3.7ii)-(v)).

To this point, we have merely shown that states can be represented by tensors and that

we can perform basic operations on these states by way of tensor contraction. We also

know that these representations and operations demand computational resources scaling

exponentially in N . The naive approach of representing generic quantum states with

tensor networks does not o!er us anything with which to tackle the many-body problem.

What we need to know is if there are ways we can use tensor networks to describe a subset

of quantum states, which can be e"ciently simulated on a classical computer.

3.2.1 Entanglement in Tensor Networks

We say that a pure N-body state, |$prod#, is a product state if it can be expressed in a

tensor product form, i.e.

|$prod# = |)1# ( |)2# ( |)3# ( .....( |)N# (3.3)

for some |)k#. Otherwise, we say that the state is entangled, i.e.

|$entangled# )= |)1# ( |)2# ( |)3# ( .....( |)N# (3.4)

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3.2 Tensor Network Descriptions of Quantum States 15

Figure 3.7: Some Common TN Quantum State Operations

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16 Foundations of Tensor Networks

One can see that a product state may be represented as a tensor network with N tensors,

each representing a subsystem |)k#. Since the state is given by a product of such subsys-

tems, the tensors are trivially connected by an index of dimension & = 1. The role then

of non-trivial interconnections in a tensor network is to represent entanglement between

subsystems.

Product states have been historically important in approximating quantum many-body

systems. The well known mean-field (MF) approach to quantum many-body ground

states finds the product state with the lowest energy. The number of coe"cients in a

product state description scales linearly in the number of particles, so the mean-field

theory solution is extremely compact. However, we will see that quantum entanglement

is a crucial ingredient when describing quantum phase transitions, and as such mean-field

theory solutions can obtain results that agree poorly with observed behaviour in many

situations.

The Schmidt Decomposition

The Schmidt decomposition is a bipartite representation of a quantum state. Given a

pure state, |$#, of an N -site system, the Schmidt decomposition is given by,

|$# =)#

i

%i

!

!)Ai

" !

!)Bi

"

(3.5)

where the system has been divided into two subsystems, A and B. Here, the state is

described by a sum over tensor products of the orthonormal Schmidt bases!

!)Ai

"

and!

!)Bi

"

, each multiplied by a Schmidt coe!cient %i. The Schmidt bases!

!)Ai

"

and!

!)Bi

"

span some auxiliary spaces, which represent the subsystems A and B. Each basis is

orthonormal and in this respect the representation is maximally compact. The dimension

of the auxiliary space, & is known as the Schmidt rank. If there is no entanglement

between the subsystems A and B (which may themselves be internally entangled), then

|$# =!

!)A" !

!)B"

and & = 1. On the other hand, if & > 1, it means the two subsystems

are entangled with one another.

The Schmidt decomposition arises in many contexts in quantum information theory. In

one-dimensional systems, it is often used to determine the amount of entanglement be-

tween the ‘left’ and ‘right’ halves of a chain. Generally, in any number of spatial dimen-

sions, the Schmidt rank gauges the amount of entanglement between a block of sites and

the rest of the system.

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3.2 Tensor Network Descriptions of Quantum States 17

Measures of Entanglement

The Schmidt rank is a discrete measure of entanglement. It reflects only the number of

non-zero Schmidt coe"cients, and not their relative weights. It is obvious that bipartite

state with roughly comparable Schmidt coe"cients is further from a product state than a

state with one Schmidt coe"cient of significantly greater magnitude than the others. To

reflect this, there exist alternative entanglement measures and we briefly introduce these

here.

Entanglement Entropy

The entanglement entropy (or Von Neumann Entropy) of a block of sites A is given by

SA = 'tr ('A log 'A) (3.6)

Where 'A * tr/%A

(|$# $$|) is the reduced density matrix of the sites A. Here, a value

of SA = 0 determines that the block A is not entangled with the rest of the system,

also known as the environment. A non-zero value means that A is entangled with the

environment and within this definition, as SA increases the block A is said to be more

entangled with the environment.

Bloch Vector Magnitude

For a spin-1/2 system, we can decompose the reduced density matrix for a single site, i

in the following way,

' =I + /ri · /*

2(3.7)

where /* is the vector sum of the spin-1/2 Pauli operators. The vector /ri is known as the

Bloch vector and its magnitude, the Bloch vector magnitude or purity of a single site,

captures the entanglement between the site i and the environment. A purity of 1 signifies

that the site is not entangled with the environment. The amount of entanglement between

the site and the environment is said to increase with decreasing purity.

Geometric Entanglement

Whilst continuous, the entanglement entropy and purity are still bipartite in that they

define a block containing a site or sites and calculate the entanglement between this

block and the rest of the system. A non-bipartite entanglement measure is the geometric

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18 Foundations of Tensor Networks

entanglement. Here, we attempt to find the product state, |$GE# from the set of all

product states, Sprod, that maximally overlaps with our state, |%0#. We can define such

an overlap as the fidelity,

&max(%0) * |$$GE|%0#| = max!%Sprod

|$$|%0#|. (3.8)

In the thermodynamic limit, the quantity $$|%0# can decay rapidly to zero. As a simple

example, consider that both |%0# and |$# are normalised, translationally invariant, N-

body product states, i.e.|%0# = |$#&N , |$# = |0#&N

$%0 | %0# = $$ | $# = 1(3.9)

Then, the overlap is given as

|$$ | %0#| = |$0 | $#|N = -N , (3.10)

where the per-site overlap 0 + - + 1 describes how close the two product states are

locally. In the thermodynamic limit, N & ,, the fidelity exponentially decays to zero

for all - < 1. This problem, whereby we cannot distinguish between states that are very

similar (- - 1) or very di!erent (- - 0) in a local sense, can easily arise for general,

entangled states |%0#.

For this reason, the authors of [WDM+05] prescribed the following intensive quantity.

E(%0) * 'log &2

max(%0)

N, (3.11)

A geometric entanglement, E(%0) = 0 occurs if |%0# is a product state and the amount

of entanglement in the system is said to increase with increasing E(%0).

3.2.2 Area Laws and Quantum Lattice Systems

We have seen from the Schmidt decomposition that entanglement can determine the cost

of representing quantum states. We also know that in a tensor network, the shared bonds

facilitate representation of entangled states. Therefore, in being able to e"ciently rep-

resent quantum states with tensor networks it is important to understand how much

entanglement is present in the ground states we will study. We consider Hamiltoni-

ans that are local in nature. In this document, we only consider those with nearest-

neighbour or next-nearest-neighbour terms. It has been shown that in many circum-

stances, the entanglement entropy of the ground states of such Hamiltonians obeys an

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3.2 Tensor Network Descriptions of Quantum States 19

Figure 3.8: The area law of quantum entanglement for ground states of local Hamiltoni-ans. (i) For a block of L sites in a non-critical 1D system, the entanglement entropy isindependent of L. Critical 1D systems attain a logarithmic correction to this scaling. (ii)For a L " L block of sites in a non-critical 2D system, the entanglement entropy scaleswith L. Critical systems may or may not attain a logarithmic correction.

area law [VLRK03, PEDC05, CEPD06]. More precisely, if we take a contiguous block of

sites in a D-dimensional system, then the entanglement entropy of this block is said to

scale with the size of the (D'1)-dimensional boundary of the block. We show such blocks

for 1D and 2D lattices in fig. 3.8. In 1D, this means that the entanglement entropy of a

length-L block of sites is independent of L. In 2D, the entanglement entropy of an L" L

block scales with L.

The exception to this rule is critical systems, where we sometimes incur a multiplicative

logarithmic correction to the area law. In 1D, the entanglement entropy of a block of

L-sites in a ground state at criticality scales as log(L). Remarkably, in 2D it is understood

that it is only some exotic systems [Wol06, GK06, SMF09] that violate the area law and

incur a logarithmic correction. For tensor network representations of ground states, we

will see that this has implications for the dimension of the tensor bonds.

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20 Foundations of Tensor Networks

Figure 3.9: The representation of a quantum state with coe"cients Ci1i2...iN in an MPSform.

3.2.3 Tensor Networks for Quantum Ground States

We now briefly introduce some tensor network structures for representing ground states

in one and two dimensions. In 1D, the Matrix Product State (MPS) assigns to each

lattice site a tensor. The tensors at each end of the chain have an open physical index of

dimension d and a shared bond index. The remaining tensors have a physical index and

two bond indices. Consider a quantum state expanded in a local basis, i,

|$# =#

i1

#

i2

...#

iN

Ci1i2...iN |i1i2...iN# (3.12)

The MPS representation is to represent the coe"cients, C, as follows,

Ci1i2...iN =)"#

$

)##

"

...

)$#

'

'1$i1%

1$$'

2$"i2%

2""...'

N'iN

(3.13)

The tensors ' are the site tensors, and the indices -,", ...., ' are the bond indices. In this

instance, we have diagonal weight matrices % which may contain the Schmidt coe"cients,

but we could also choose a form without such weights as in the original MPS formal-

ism [FNW92]. The coe"cients C are obtained by contracting all of the shared indices.

Graphically, we may represent the expansion of the state in an MPS form in figure 3.9.

Projected entangled pair states (PEPS)[VC04] are a natural extension of the MPS to two

and higher dimensions. The basic principle of the MPS is retained - we represent our

state with a contractible tensor network with tensors at each site. Contracting the PEPS

network once again returns the coe"cients of the wave vector expanded in the local basis.

We show a PEPS in figure 3.10.

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3.3 E"cient Representations of Quantum States 21

Figure 3.10: The representation of a quantum state with coe"cients Ci1i2...iN in a PEPSform. Here we label a D-dimensional bond index and a d-dimensional physical index.

A similar ansatz, called tensor product states (TPS)[NOH+00, NHO+01, GMN03], had

been employed to compute statistical properties of 3D classical systems at finite temper-

ature. In this document we will use the term PEPS to describe quantum states on 2D

lattices, however the reader should note that generally, the terms PEPS and TPS are

interchangeable.

3.3 E!cient Representations of Quantum States

An e"cient representation of a quantum state satisfies two main properties,

1. It may be stored with a number of coe"cients that grows polynomially in the number

of sites.

2. We can compute, to at least a good approximation, basic properties of the state

and perform basic operations on the state with a computational cost that scales

polynomially in the number of sites. For example, we want to be able to e"ciently

compute local observables, local entanglement measures and simulate the action of

local gates in quantum circuits.

We will see in the following chapters that we satisfy these by bounding the dimension

of the shared indices in the MPS and PEPS to some upper limit. For an MPS we term

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22 Foundations of Tensor Networks

Figure 3.11: The refinement parameter controls the size of the corner of the entire Hilbertspace that is accessible in the tensor network representation. A higher value of refinementparameter means more states are accessible, but the representation is also less compact.

this refinement parameter & and for a PEPS, D. This means we constrain our state to a

low-entanglement region of the entire Hilbert space.

So, whilst states of quantum many-body systems lie in a Hilbert space that grows ex-

ponentially in the system size, ground states of certain Hamiltonians lie in a region of

this Hilbert space where there is relatively little entanglement. Meanwhile, with tensor

networks, we can e"ciently represent corners of the entire Hilbert space. The size of the

corner depends on the value of the refinement parameter - as we increase the refinement

parameter, we gain access to a larger region of Hilbert space, each one capable of describ-

ing more entangled states and a better approximation to the real ground state than the

last (see fig. 3.11). However, in comparison to the size of the entire Hilbert space, these

corners are still exponentially small.

The e!ectiveness of TNs for representing quantum ground states depends on there being

a good overlap between the region of the entire Hilbert space containing the ground states

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3.3 E"cient Representations of Quantum States 23

and those accessed by TNs for manageably small values of the refinement parameter. It

was shown in [TdOIL08, VC06] that in order to represent ground states of 1D systems

at criticality, we only require a & that grows polynomially in the system size. This may

appear daunting, but it is significantly more favourable than an exponential growth of

the Hilbert space. On the other hand, we can represent a non-critical ground state with

finite &. For infinite 2D systems, it is elementary to show an example of a critical ground

state that can be written in PEPS form[VWPGC06]. For us, these form a basis for our

approach for representing quantum systems with tensor networks. With a fixed & or D

we can well approximate the ground state. This representation is more accurate when the

amount of entanglement in the system is relatively small, and less accurate when a large

amount of entanglement is present, as, for example, at criticality.

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24 Foundations of Tensor Networks

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Chapter 4

Dimension and Computational Complexity

In Chapter 2 we established a correspondence in computational complexity for deter-

mining the local statistical properties of a (D + 1)-dimensional classical system at finite

temperature and local observable properties of the ground state of a D-dimensional quan-

tum system. In this chapter we further develop these ideas in the context of tensor

network representations described in Chapter 3.

4.1 D = 0

We firstly consider the case D = 0. This relates to calculating the local properties of a 1D

classical system at finite temperature or the local observable properties of the ground state

of a single quantum particle. We will assume that our classical system is translationally

invariant, and that our quantum Hamiltonian is time-invariant. Consider a classical spin

system where we want to measure the expectation value of the ith spin. That is, we wish

to compute,

$Si# =

$

!Sie""H(!)

Z=

$

!Sie""E!

Z(4.1)

Here, Si . ±1 is the spin at site i, each * is a configuration of the system and Z is the

partition function as defined in equation 2.1.

Now consider a single quantum spin, governed by a Hamiltonian, HQ. We wish to calculate

the expectation value of the spin in the z-direction of the ground state. That is,

25

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26 Dimension and Computational Complexity

$*z# =$$gr|*z |$gr#$$gr | $gr#

= lim##$

$$i| e"H†#*ze"H# |$i#$$i| e"H†#e"H# |$i#

= lim##$

$$i| e"H#*ze"H# |$i#$$i| e"H#e"H# |$i#

(4.2)

Here we have used the technique of imaginary time evolution explained in Chapter 2 to

obtain the ground state, |$gr#, from some random initial state |$i#.

In each case, we have a numerator containing some unnormalised physical information

and a denominator that acts as the normalisation constant. In the classical case, the

normalisation constant is the partition function, Z, whilst for the quantum particle, it is

the norm of the unnormalised ground state.

A standard way to solve the classical problem is to put the numerator and denominator

of equation 4.1 in transfer matrix form. This means finding the matrices Tk,k+1, such that

#

!

Sie""H! = vT

1

%

i"1&

k=1

Tk,k+1

'

(i

%

N"1&

k=i

Tk,k+1

'

vN (4.3)

here, v1 = [1 1 ... 1] and vN = [1 1 ... 1]T . The operator (i ensures that the correct

multiplicative constant, Si, is applied to each term in the summation in the numerator.

Likewise, the normalising partition function can be expressed as,

Z =#

!

e""E! = vT1

%

N"1&

k=1

Tk,k+1

'

vN (4.4)

Graphically, we represent the calculation of $Si# in fig. 4.1. From this, we conclude

1. Equation 4.3 and equation 4.4 each require order N matrix-vector multiplications.

That is, the computational complexity scales linearly in the system size.

2. The two computations di!er only by the insertion of a local operator (i. The

remainder of the computation is common to each expression.

The consequence of the first point is that computing a local statistical property of a 1D

classical system at finite temperature is an elementary task for a classical computer to

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4.2 D = 1 27

Figure 4.1: The computation of the partition function of a 1D classical system. Thematrices T are known as transfer matrices

perform. A consequence of the second point is that computing such a local property is

no more complex than evaluating the partition function. As a result, we sometimes talk

synonymously about the computational complexity of computing local properties of the

classical system and the computational complexity of evaluating the partition function.

If the system is infinite and translationally invariant, then our transfer matrix Tk,k+1 is

identical for all k. We realise that in order to compute local properties, we must first

compute the left and right eigenvectors of T corresponding to the maximum magnitude

eigenvalue. We term each eigenvector the dominant eigenvector. This can be determined

by diagonalising T , or simply performing many matrix-vector multiplications and moni-

toring for convergence.

We make the direct correspondence between this 1D classical problem and computing

local observable properties of the ground state of the quantum particle (equation 4.2) by

dividing the imaginary time evolution into steps of length ,+ . As shown in the tensor

network in figure 4.2, our quantum problem contains imaginary-time evolution operators,

e"HQ&# in place of the transfer matrices. Otherwise, the two problems are computationally

equivalent and are both represented by the contraction of a 1D tensor network.

4.2 D = 1

We now consider the case of computing the local properties of a 2D classical system

or the ground state of a 1D quantum chain. These involve the contraction of a 2D

tensor network. The 4" 4 tensor network in figure 4.3i could represent, for example, the

partition function of a 2D classical system. We need to be able to contract such networks

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28 Dimension and Computational Complexity

Figure 4.2: The imaginary time evolution of a point particle (0-dimensional) governed bythe Hamiltonian, H. The time dimension has been discretized into steps of ,t

to evaluate the local properties of the system. One possible way of contracting such a

network is to isolate a 1D boundary of the system and evolve it under the action of an

adjacent row or column of operators. For instance, we can take the top row of tensors

R11, R12, R13, R14 and calculate the action of the row R21, R22, R23, R24 on it. One then

determines a new boundary state that represents two rows of the lattice, as described by

the tensors B1, B2, B3 and B4 in figure 4.3ii. The problem with proceeding in such a way

is that it is exponentially hard to exactly simulate the evolution of the boundary state - at

each step, the horizontal links in the boundary state need to expand to exactly represent

all of the correlations. From this, we can appreciate that contracting 2D tensor networks

is not as simple for a classical computer as contracting 1D networks. We need to think

of ways to approximate 2D contractions in a computationally e"cient manner.

For an infinite 2D tensor network with open boundary conditions the problem of con-

tracting the network seems even more daunting. In fig. 4.4 we translate the approach

for a translationally invariant, infinite 1D tensor network to a translationally invariant,

infinite 2D tensor network. That is, we describe the boundary state by a single tensor, #

and treat rows of the tensor network as transfer matrices acting on #. After evolving for

many iterations, # converges to the eigenvector corresponding to the maximum eigenvalue

of transfer matrix.

4.2.1 The Infinite MPS approach and iTEBD algorithm

Such an approach for infinite systems presents an obvious computational problem. The

boundary state has infinitely many degrees of freedom making storing and manipulating

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4.2 D = 1 29

Figure 4.3: i)A 4 " 4 2D tensor network that might represent, for example, a partitionfunction of a 2D classical system at finite temperature. ii) A straightforward approachto contracting the network by starting with a boundary and evolving it leads to an accu-mulation of indices. As such, the computational cost of contracting an L" L network insuch a way scales exponentially in L.

Figure 4.4: An infinite, translationally invariant 2D network defined by the four-leggedtensor a. The tensor # describes the boundary state.

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30 Dimension and Computational Complexity

it in the form of a single tensor, #, computationally intractable. So the question is how

to approximately represent the boundary state and its evolution in a computationally

e"cient way, exploiting the translational invariance of the lattice. In doing this, we take

inspiration from the 1D quantum mechanical analogue to this problem.

Finding the ground-state of a 1D quantum chain is in general computationally hard

due to the presence of quantum entanglement. The density matrix renormalization group

(DMRG) algorithm [Whi92] described a means of finding an approximation to the ground

state in a low-entanglement region of the Hilbert space. For infinite translationally in-

variant 1D systems, it was shown that the ground states found by DMRG could be

determined by a variational minimization of the energy over the family of MPS states

[OR95, VPC04]. Later, methods were developed to find such an approximation within

an MPS representation[Vid04] using imaginary-time evolution to converge to the ground

state. The infinite time-evolving block decimation (iTEBD) algorithm approximated the

imaginary-time evolution by a network of near-unitary two-body gates. These were ob-

tained from a Suzuki-Trotter decomposition of the imaginary time evolution operator for

an interval ,# . An example of the resulting network is shown in figure 4.5. Starting with

an MPS of maximum bond dimension &, rows of time evolution gates are successively

applied. After application, the bonds in the MPS are truncated so that the description

of the state remains compact.

In examining the iTEBD algorithm more thoroughly (see [Vid04]), one can see that the

each iteration consists of SVD operations on tensors, followed by truncation of tensor

legs. In the case of the iTEBD algorithm, each step scales as &3d3, where d is the physical

dimension.

Having obtained an MPS representation of the ground state, then computing the local

observable properties of the state is done by contracting the network shown in figure 4.6.

Such a computation falls in the D = 0 class of problems and scales polynomially in the

bond dimension &. This means that we can both find an approximate representation

ground state of a 1D quantum Hamiltonian and compute its properties in a computation-

ally e"cient manner.

Now, return to the problem of contracting the infinite, translationally invariant 2D tensor

network, this time with an infinite, translationally invariant MPS approximating the

boundary state in some diagonal orientation (see figure 4.7). We can express the evolution

of the boundary state in precisely the same schematic form as the iTEBD algorithm. The

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4.2 D = 1 31

Figure 4.5: The imaginary-time evolution of a translationally invariant MPS as definedby the iTEBD approach. The row of G gates collectively approximates the evolution forsome time, ,# .

Figure 4.6: The computation of a local observable quantity, M, of an MPS. Note that theoperation is of the D = 0 class.

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32 Dimension and Computational Complexity

Figure 4.7: The contraction of a 2D tensor network, using the MPS to describe theboundary state

di!erence is the gate characteristics - in one case the contraction of the gate network

approximates an imaginary-time evolution and in the other it constitutes, for example,

the evaluation of the partition function.

An important distinction between the iTEBD algorithm for 1D quantum systems and for

the contraction of generic 2D tensor networks is the near-unitary nature of the imaginary-

time evolution gates. As explained in [Vid04], this means we can rather naively truncate

the MPS, as the near-unitary gates keep the MPS close to the so-called canonical form.

For the 2D network we must perform some additional operations before truncation (see

[OV08] or Appendix A). However, since these operations are either local decompositions,

or network contractions of the D = 0 class, our algorithm remains e"cient.

4.2.2 The Corner Transfer Matrix

An alternative means of contracting a 2D tensor network is the corner transfer matrix

(CTM)[Bax82]. The basic structure of the CTM consists of a unit cell, surrounded by

corner matrices and edge tensors (fig. 4.8b), each of which characterise a certain part of

the environment around the unit-cell (fig. 4.8a). The tensors are connected by bonds of

dimension & (fig. 4.8c) and thus our description is a reduced-rank representation of the

environment.

The corner transfer matrix renormalization group (CTMRG)[NO96] is an algorithm for

determining the tensors C1, C2, C3 and C4 and E1, E2, E3 and E4. We cover the details

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4.3 D = 2 33

of an implementation of a similar algorithm extensively in Appendix B. Here, we merely

wish to convey the basic principle behind it. The CTMRG proceeds by absorbing part of

the unit-cell into the environment. Here, we absorb the unit-cell A and two edge tensors

into the corner C1. Thinking about how this simple step aggregates the correlations in the

system illustrates the idea behind the CTMRG. In figure 4.9a), we represent the combined

action of A and the edge tensors by the L-shaped block L. If we continued this process, we

would obtain an increasingly larger corner matrix. After k steps, we would have a corner

matrix with k+1 horizontal legs and k+1 vertical legs, each of dimension D and a total of

D2k+1 coe"cients. Proceeding in such a fashion is obviously computationally ine"cient.

However, what if the state we are considering has some limited amount of many-body

correlations present, such that after k-steps the unit-cell is only correlated with the last

several legs added to the corner matrix? In fig. 4.9a, this means that the red index

is not correlated with the blue indices. In this case, maintaining all D2k+1 coe"cients is

excessive. The CTMRG algorithm accounts for this by renormalizing the bonds after each

step, such that the aggregated information of all the legs is approximately transmitted

in a bond of maximum dimension &. In this way, we are trying to find a fixed point of

the system as depicted in figure (fig. 4.9b), where the renormalization operators V and

W are determined in some systematic way. In Appendix B, we describe algorithms for

finding these renormalization operators using simple, local contractions or decompositions

of tensors.

4.3 D = 2

So far, it has been seen that we solve problems of the D = 1 class by approximately

contracting 2D tensor networks. Such an approach breaks the problem into a series of

D = 0 matrix-vector eigenproblems and simple local operations. We do this because it

alleviates the exponential computational cost scaling of the D = 1 problem, and because

mathematical software is particularly adept at performing these operations. The philos-

ophy for D = 2 extends upon this. For example, in order to find the ground state of a

2D quantum system, or evaluate the partition function of a 3D classical system at finite

temperature, we firstly break the problem down into a series of D = 1 problems and local

decompositions. Having done this, we can approximately solve the D = 1 problems - by

methods such as the infinite boundary MPS or the CTMRG. In this reductionist manner,

we can approximately solve D = 2 problems. In the next chapter, we outline the iPEPS

algorithm - a scheme for computing the properties of the ground state of 2D quantum

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34 Dimension and Computational Complexity

Figure 4.8: a) The CTM structure, with the regions surrounding a single-site unit-celleach allocated a tensor. b) The resulting CTM structure, showing four corner matricesand four edge tensors. c) A single edge tensor, with bond indices of dimension &.

systems that is based on these very principles.

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4.3 D = 2 35

Figure 4.9: a) A visualization of the basic CTM operation. After each step, the numberof indices on the corner transfer matrix increases, however if some of the indices areuncorrelated, we only need keep a subset of all of the correlations. b) The CTMRGfixed-point problem. Our task is to find the rank-reducing matrices V and W.

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36 Dimension and Computational Complexity

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Chapter 5

Projected Entangled Pair States

In this chapter, we describe the PEPS ansatz in more detail and then develop an algorithm

for finding the ground state of infinite 2D quantum systems described by local Hamilto-

nians. Our intention is to avoid discussion of the low level algorithmic details, instead

deferring these to the appendix. Here, we will discuss the algorithm in terms of the basic

notions introduced in Chapter 4. By now we have a clear understanding that 1D tensor

network structures (D = 0) are easily dealt with on a classical computer, and that 2D

tensor network contractions (D = 1) can be approximated by a series of one-dimensional

contractions and some simple local operations. In this chapter we show that 3D tensor

network contractions (D = 2) can be decomposed in a similar way. We will focus here on

the computation of ground states of 2D quantum systems.

5.1 Projected Entangled Pair States

We briefly introduced the PEPS/TPS ansatz [VC04, NOH+00] in Chapter 3. Historically,

the great success of DMRG and the MPS in e"ciently solving 1D quantum problems

meant that this approach appeared an obvious candidate for solving problems in two

spatial dimensions. It was proposed that one could choose a basis ordering that ‘snakes’

through the lattice and then solve the resulting energy minimization on the 1D ansatz e.g.

[WC07, WS98]. We show an example of such an ordering in fig. 5.1i). When such a 2D

state is represented by a 1D ansatz, one can see in fig. 5.1ii) that the nearest-neighbour

interactions become long-ranged in the 1D picture. Recall the reason underpinning the

success of the iTEBD and DMRG algorithms - that the entanglement entropy grew at

worst logarithmically with the block size in 1D systems - was based on the Hamiltonian

37

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38 Projected Entangled Pair States

having local interactions. On this basis, we might suspect that it is only for relatively

small lattices that the ground state of a 2D system will be well described by an MPS with

a reasonable, fixed bond dimension &.

Of course, we already know from the area law that 1D and 2D systems with nearest-

neighbour interactions have fundamentally di!erent entropy scaling behaviour and we

can use this to formalise our argument against using a 1D ansatz to represent 2D ground

states. Recapping our discussion in Chapter 3, for a 1D system, the entanglement entropy

is independent of the block size o!-criticality, and scales logarithmically with block size

at criticality. For a 2D system, the entanglement entropy scales with the perimeter of

the block for non-critical and some critical ground states. For other critical ground states

the entanglement entropy acquires the multiplicative logarithmic correction. It was also

shown by Vidal[Vid03] that for a tensor network with fixed bond dimension, the maximum

entanglement entropy for a contiguous block of sites is proportional to the number of

tensor bonds crossing the boundary. In fig. 5.1iii) we demonstrate that for an MPS

representation of a 2D system, as we increase the perimeter of a block of sites (black box

to red box) the number of bonds crossing the boundary stays fixed. That is, the size of

the boundary increases, but the maximum entanglement entropy of the larger and smaller

blocks (for fixed &) is identical. This shows that 1D schemes such as the MPS and DMRG

violate the area law and are unsuitable for capturing the physics of strongly correlated

2D systems.

As a result of this realization, a general tensor network ansatz for quantum states in

two and higher dimensions, in which all neighbouring sites were connected by correlation

carrying bonds, became an area of great interest. The resulting PEPS ansatz [VC04] was

shown to have some remarkable properties [VWPGC06]. Most notably, these are:

1. Quantum states expressed as a PEPS are capable of obeying the area law.

2. Every PEPS represents the ground state of some local Hamiltonian.

Combined, these suggest that the PEPS is a powerful ansatz for representing quantum

ground states in two or more dimensions. The authors of [VC04] also presented a varia-

tional algorithm for finding a PEPS representation of the ground state of finite systems.

However, finding the ground state of infinite 2D systems required additional insight. In

the next sections, we take a look at this problem, motivated by the iTEBD algorithm for

the MPS.

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5.2 Problem Overview 39

Figure 5.1: Representing 2D systems with a 1D ansatz. (i) A possible ordering for thebasis of a 4" 4 2D system with nearest-neighbour interactions. (ii) When linearised ontoa 1D ansatz, some of the nearest-neighbour interactions in the 2D system become long-ranged. (iii) For a 2D system described by a 1D ansatz, as the perimeter of a block ofsites increases, the number of bonds crossing the perimeter may stay constant. Followingthe reasoning in [Vid03], the maximum entanglement entropy of the two blocks remainsthe same. As a result, 1D schemes cannot describe 2D states that obey the area law.

5.2 Problem Overview

In this thesis, we are interested in finding the ground states of infinite, translationally

invariant 2D systems. That is, our Hamiltonian will be invariant under some shift of

lattice sites. For the purpose of this discussion, we will assume that the Hamiltonian

contains identical nearest-neighbour terms. That is,

H =#

<i,j>

h[i,j], (5.1)

where i and j are adjacent lattice sites. It should be noted that the translational invariance

of the Hamiltonian may be spontaneously broken in the ground state. However, we will

always assume that some degree of translational invariance remains in the ground state.

As such, our PEPS representation of the ground state will be made up of a repeating

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40 Projected Entangled Pair States

pattern of tensors.

5.2.1 Imaginary-time Evolution Re-visited

We again consider imaginary-time evolution as a means of finding an approximation to

the ground state of a quantum Hamiltonian. In this discussion, we consider the square

lattice, but it will become obvious that the technique is easily adapted to other lattice

geometries. We will assume that our ground state is invariant under shifts of two sites. In

e!ect, this means that our PEPS is formed by an alternating pattern of A and B tensors

(see fig. 5.2i). We may say that the lattice L is composed of two interacting sub-lattices

LA and LB. In this representation, there are only four unique links and we reference these

by the direction they protrude from the tensor A - up, right, down and left (see fig. 5.2i).

Thus, we can rewrite our Hamiltonian in terms of four non-commuting sums of terms, i.e.

H =#

r%LA

h[r,r+y] + h[r,r+x] + h[r,r"y] + h[r,r"x]

=#

r%LA

hru + hr

r + hrd + hr

l

(5.2)

where x and y are lattice unit vectors in the x and y directions and u, r, d and l refer to

up, right, down and left interactions.

Taking the imaginary-time evolution operator corresponding to a time-step ,# , and then

the first-order Suzuki-Trotter expansion, we obtain

e"H&# = e"$

r!LA

hru&#"

$

r!LA

hrr&#"

$

r!LA

hrd&#"

$

r!LA

hrl &#

=&

r%LA

e"hru&#&

r%LA

e"hrr&#&

r%LA

e"hrd&#&

r%LA

e"hrl &# + O(,+ 2)

(5.3)

This closely resembles the iTEBD update in [Vid04], except that we have four distinct

links instead of two. A key di!erence from the iTEBD algorithm is that for a PEPS, there

is no known canonical form. In the iTEBD algorithm, the canonical form of the MPS

was exploited to drastically simplify the update procedure. After applying the imaginary

time-evolution gates, a good approximation to the resulting MPS could be reached by

a local SVD and truncation of the MPS tensors. Since there is no canonical form for

the PEPS, there is no guarantee that a PEPS determined from local split and truncation

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5.3 The iPEPS Algorithm 41

Figure 5.2: (i) The infinite PEPS for a square lattice that is invariant under shifts of twolattice sites. (ii) The four links, up (u), right (r), down (d) and left (l) are distinguishedbased on which direction they protrude from tensor A.

operations will be an e"cient use of the classical resources. In section 5.3.1 we will consider

such a ‘split and truncate’ algorithm, but for now, we consider a more systematic way of

updating the tensors.

5.3 The iPEPS Algorithm

An overview of the iPEPS algorithm is shown in figure 5.3. The combined action of the

gates labeled g in figure 5.3i represents the factor,(

r%LA

e"hru&# in equation 5.3.

The first (a & b) and third (c & d) steps are justified by the same reasoning. Since

the imaginary-time evolution operation is near-identity, we assume that the e!ect of the

gate on the link on which it acts dominates the combined e!ect of the other gates on this

link. This means that in order to update a given site, we can focus on a single link (see

fig. 5.3b) and not worry about the secondary change in correlations introduced by other

gates. Step two involves the determination of new tensors A! and B! that are of maximal

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42 Projected Entangled Pair States

Figure 5.3: The four basic stages of the PEPS algorithm a) The imaginary-time evolutiongate is applied to a sheet of corresponding links. b) Since the action of the gate is near-identity in nature, we focus on a single link and assume that the changes to this linkfrom other gates are negligible. c) We determine the new tensors A! and B! (with D-dimensional interjoining bond) that best represent the action of the gate on the link. d)We enforce the change globally

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5.3 The iPEPS Algorithm 43

bond-dimension D and best represent the action of the gate on A and B in some sense.

Step three involves the replacement of A by A! and B by B! globally. This represents the

update with respect to the ‘up’ link direction. Repeating this for ‘right’, ‘down’ and ‘left’

links means we have evolved the system for some imaginary time, ,+ .

In step 2, we need to determine A! and B! such that they maximise the closeness of (or

minimise the distance between) |$g# and |$A"B"#. In the iTEBD algorithm the split and

truncation operation e!ectively approximated such a requirement due to the canonical

form of the MPS, but here we will need to do so in a more explicit manner. We choose

to minimise the square error between the target state, |$A"B"#, and the gate operation on

the previous state, |$g# * g |$# . i.e.

minA",B"

||$A"B"# ' |$g#|2 * minA",B"

($$g | $g# ' $$A"B" | $g# ' $$g | $A"B"#+ $$A"B" | $A"B"#)

(5.4)

Each of the four terms on the right-hand side may be written as an infinite 2D tensor

network. The first term is a constant and so may be ignored in the minimization problem.

The remaining three terms depend on A! and B! and define our minimization problem.

Since the states |$g# and |$A"B"# are identical apart from the tensors sharing the link being

updated, there is significant computational overlap in the calculation of these terms. We

call the 2D tensor network surrounding the link tensors the environment (see fig. 5.4a).

The environment captures the influence of correlations on the best choice of A! and B!.

In practice, we find an approximation to the environment by contracting the 2D tensor

network into an e!ective six-tensor form (see fig. 5.4b).

So, the main computational components of the iPEPS algorithm are,

1. Contraction of the periodic 2D tensor network into an environment surrounding a

single link.

2. Determination of the new PEPS tensors in such a way that minimises the square

error distance function in equation 5.4

For 1, we can employ either the infinite MPS or CTMRG procedures outlined in Chapter

4 and described in greater detail in Appendix A and B.

For 2, we detail two approaches in Appendix C. The first is a variational update scheme

that updates the matrices A! and B! iteratively, at each step finding the A! or B! that

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44 Projected Entangled Pair States

Figure 5.4: i) The tensor contraction defining the environment around a given link. ii)The six-tensor form approximating the environment.

minimises the metric. This method is computationally simple and e"cient, but is prone to

becoming trapped in local minima. Furthermore, it uses the inverse of the environment

to calculate the new tensors and as such small errors in the environment can become

greatly magnified. As a result, the new tensors A! and B! from this approach have been

seen to introduce spurious correlations into the system. An alternative approach is based

on the well-known conjugate gradient (CG) algorithm. Here, we use the gradient of the

distance metric with respect to the PEPS tensor coe"cients to guide us towards the

minima. Importantly, both tensors A! and B! are updated together, helping us to avoid

local minima. Furthermore, the gradient is linear in the environment, and so errors in

small environment spectral components are not amplified.

5.3.1 The Simplified Update

Although there exists no canonical for for the PEPS, some authors [JWX08], inspired

by the simplicity of the iTEBD algorithm, described the analogous local update for the

PEPS. Here, on each site of the PEPS resides a tensor ' with a physical index and

bond indices and on each bond resides a diagonal weights tensor, %. The imaginary-time

evolution proceeds by applying the gate to the PEPS and making the update based only

on local information. That is, we assume that the information normally encoded in the

environment is stored in the % tensors, and for the link being updated the new ' and %

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5.3 The iPEPS Algorithm 45

tensors are determined in the same way as in the iTEBD algorithm - by contraction of the

gate and two tensors, followed by SVD and truncation, recovering the interjoining bond.

Whilst there is no formal reasoning to suggest such an approach is a near-optimal use of the

PEPS coe"cients, empirical evidence suggests it does a remarkable job of finding ground

states in certain circumstances. In particular, since the long-range correlations encoded

in the environment are not taken into account, this update favours states with short range

correlated behaviour. On the other hand, in systems with diverging correlation length,

such as those near a continuous phase transition, this scheme will struggle to capture the

physical properties. We will see the di!ering fortunes of the simplified update in Chapters

7 and 9.

The advantage of using such a method is the reduction in computational cost. Typically,

the most expensive part of a PEPS imaginary-time evolution step is the contraction of the

environment. This approach avoids this computational work during the time evolution,

as here we only compute the environment when we need to obtain observable properties

of the system. In a minimalist sense observables need only computed once in the final

stage. Thus, by using the simplified update, we can access higher values of D, at the

expense of limiting ourselves to states of a more local nature (for a given D).

5.3.2 Computational Complexity of the iPEPS algorithm

On first analysis, it would appear that the leading order computational complexities of

the infinite-MPS and CTMRG lattice contraction schemes scale identically as &3D6 +

&2D6d per step. Each lattice contraction requires many steps for the coe"cients of the

infinite-MPS or the CTM to converge. We label these number of steps S* and SCTMRG

respectively. In practice, it is seen that the infinite-MPS contraction scheme is slower than

the CTMRG approach for equivalent & and D, and this can be attributed to the presence

of an iterative procedure within each step of the infinite-MPS algorithm (see Appendix

A for details). We label the number of steps required in this procedure SLR. Though the

per-step computational cost of this routine is sub-leading order, SLR typically takes on

values between 10 and 40 for the values of & and D we have considered in our studies,

making its overall cost significant. Following the convergence of the infinite-MPS pair,

two finite eigenvectors are converged in another iterative routine of Sv iterations, however

this should be of secondary significance if a sensible limit is put on the value of Sv.

The update of a single link has two stages. The initial out-of-loop preparatory contractions

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46 Projected Entangled Pair States

incur a one-o! cost, scaling as &3D4d2+&2D6d2+&2D4d4+D2d6. This process is identical

for both the variational and CG schemes. For the variational scheme, each of the SV U

steps of the update loop scales as D5d5 + D6d3. Within the CG update loop, we need to

perform SCG % D2d2 steps each of which involves an iterative line minimization. Each

line minimization scales as SLMD4d6, where SLM is the number of steps required to find

the minimum.

The simplified update can be performed by using an iterative sparse SVD routine, where

each step scales as D2d3 +Dd4. We assume that SS % D steps are required. In optimizing

the tensors for such an update, we must perform some decompositions on the PEPS

tensors, each of which scales as D5d2.

These complexities are summarized in Table 5.1.

As a rule of thumb, we estimate that & % D2 and D / d. Under these assumptions,

the critical path of the iPEPS algorithm is in the computation of the environment, an

operation that scales as D12. For this reason, when using the full environment at every

iteration, we can generally only access D = 2, 3 and 4 on a standard quad-core desktop

computer. For the simplified update, we can access D as high as 8, as the environment

only needs to be computed once.

5.4 Computing Physical Properties of PEPS States

Having computed a PEPS representation of the ground state, we wish to extract infor-

mation about the properties of the state. If by changing some Hamiltonian parameter

our ground state undergoes a phase transition, we wish to distinguish between the phases

of the system and approximate where such a transition occurs. Traditionally, the the-

ory of phase transitions has centred around Landau’s symmetry breaking theory[Lan37].

Landau proposed that a phase transition occurs when a symmetry is broken, and that

as such the di!erent phases could be distinguished by a local order parameter. Whilst

modern physics has uncovered classes of phase transitions that depart from this picture

[KT73, Lau83] it is apparent that for a large number of many-body Hamiltonians, local

observables can tell us much about the ground state and the order of the phases in which

the system can exist. Additionally, within the subset of symmetry-breaking QPTs, con-

tinuous phase transitions exhibit a diverging length scale at the transition point. This

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5.4 Computing Physical Properties of PEPS States 47

Scheme Leading order cost of Leading order cost oftensor update computing environment

iMPS

&2D6d2 + &3D4d2 S*(&3D6 + &2D6dVariational update +&2D4d4 +SLR (&3D4)) +

+SV U(D5d5 + D6d3) Sv(&3D4 + &2D6d)

Conjugate gradient &2D6d2 + &3D4d2

update +&2D4d4

+SCGSLMD5d5

CTMRG

&2D6d2 + &3D4d2

Variational update +&2D4d4

+SV U(D5d5 + D6d3) SCTMRG(&3D6 + &2D6d)

Conjugate gradient &2D6d2 + &3D4d2

update +&2D4d4

+SCGSLMD5d5

One-time computationSimplified SS(D2d3 + Dd4) using iMPS orupdate +D5d2 CTMRG algorithm

(see above)

Table 5.1: Leading order computational complexity of various iPEPS algorithms for thesquare lattice. Here, SV U denotes the number of iterations for the variational update algo-rithm (see Appendix C). SCG denotes the number of iterations for the conjugate gradientupdate algorithm and SLM denotes the number of iterations for the line minimizationcontained in each conjugate gradient iteration (see Appendix C). S* denotes the numberof iterations required for the infinite boundary MPS to converge (see Appendix A) andSLR denotes the number of iterations required to compute the left and right scalar productmatrices per infinite MPS update (see Appendix A). Sv denotes the number of iterationsrequired to converge the finite left and right eigenvectors per infinite MPS update (seeAppendix A). SCTMRG denotes the number of iterations required to converge the cornertransfer matrix (see Appendix B). SS denotes the number of iterations required in thesparse iterative SVD routine in the simplified update.

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48 Projected Entangled Pair States

means that spatial correlation functions decay polynomially, rather than exponentially

as in a first-order QPT. Thus, computing spatial correlation functions sheds light on the

order a given symmetry-breaking phase transition. In this section, we describe how these

well-known quantities - and additional measures rooted in quantum information theory -

can be e"ciently approximated for PEPS ground states.

5.4.1 Computation of Local Observables

Consider the computation of the expectation value of a local observable M for an infinite,

translationally invariant state, |$#, represented by a PEPS. By local observable, we mean

that in our local basis it acts as the identity everywhere except for a single site, i.

We show the tensor network representation for the computation of $$|M |$# in figure

5.5. In figure 5.5i, we show the three parts of the tensor network ‘sandwich’ comprising

the calculation of $$|M |$#. In figure 5.5ii, we show that the quantity can be easily

expressed as a 2D tensor network (with bond dimension D2). Recall that our 2D tensor

network contraction algorithms work for translationally invariant networks. The presence

of the observable M means that the translational invariance is disturbed, but importantly

it is only disturbed in a local region. The environment around the site is comprised of

identical tensors a and b and as for the iPEPS update we can approximately contract

the surrounding region. In figure 5.5iii, we represent the approximate computation of

$$|M |$#, with the contraction of the purple 6-tensor environment and the remaining

PEPS and observable tensors.

For computing the expectation value of an observable acting on an L" L block of sites,

the computational complexity scales exponentially in L.

5.4.2 Spatial Correlation Functions

A similar rationale can be applied to the computation of spatial correlation functions. We

can contract 2D tensor networks without translational invariance, so long as the distur-

bance to the translational invariance is confined to some easily contractible finite region

of the network. In figure 5.6, we show the computation of two-point spatial correlators,

$#1#2# along a horizontal or diagonal lattice direction. Here, we contract the infinite,

translationally invariant sections of the lattice via an infinite MPS technique. Then, the

disturbance introduced by the operators #1 and #2 is confined to a 1-dimensional tract

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5.4 Computing Physical Properties of PEPS States 49

Figure 5.5: i) The computation of a local observable as a tensor network contraction. Herethe red tensor at site i is the local observable operator. ii) Contraction along the physicalindices leaves a 2D tensor network. iii) The observable is approximated by approximatingthe environment by the means described in Chapter 4 and Appenidix A and B.

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50 Projected Entangled Pair States

of the network, which is easily e"ciently contracted. So whilst we may not be able to

compute, for example, arbitrary three-point correlation functions, we can compute certain

two-point correlation functions quite easily.

For an MPS, it is apparent that the correlation function will decay exponentially in sepa-

ration distance. The reason for this is that the number of exponentially decaying terms in

the computation of the correlation function is capped by &. For an infinite PEPS, there is

an infinite number of paths between two lattice points. Thus, our correlation function is

the sum of an infinite number of exponentially decaying terms. It is therefore possible to

represent states with polynomially decaying correlation functions with a PEPS, such as

quantum critical ground states. However, since we use an MPS with finite & to contract

the PEPS lattice, we will never in practice be able to reproduce a polynomially decaying

correlation function with our approach. That said, we can still draw conclusions from

studying the comparative rate of exponential decay of two ground states. Furthermore,

the rate of polynomial decay of a critical system can be estimated by computing the cor-

relator for various values of D and & and then observing the asymptotic behaviour of the

correlator.

5.4.3 Fidelity Measures

It was suggested in [ZB07, ZPac06] that certain fidelity measures are useful in the study

quantum phase transitions in one and two dimensions. That is, the fidelity captures infor-

mation about the macroscopic similarity of two ground states corresponding to di!erent

values of some Hamiltonian parameter. Recall that traditionally, many quantum phase

transitions are detected by a local order parameter, an observable that has value zero in

one phase and non-zero in another. The choice of order parameter, a priori, is not always

obvious. Fidelity measures on the other hand can suggest if two ground states are in

the same phase without reference to any kind of order parameter. Sampling the fidelity

between ground states of the phase diagram and creating a so-called fidelity diagram, one

can judge whether the QPT is first or higher-order in nature.

We define the fidelity in the standard sense,

F (%,%!) = |$$0 (%!) | $0 (%)#| (5.5)

where |$0 (%!)# and |$0 (%)# are ground states of Hamiltonians parameterized by the co-

e"cients %! and % respectively.

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5.4 Computing Physical Properties of PEPS States 51

Figure 5.6: The calculation of spatial correlation functions with PEPS. (above) Corre-lation functions along a horizontal (or vertical) direction can be computed by evolvinginfinite-MPS boundary states from above and below. Then, the remaining horizontal tractof tensors can be e"ciently contracted. (below) Likewise, correlation functions along adiagonal lattice direction can be computed by employing infinite-MPS boundary statesin a diagonal orientation.

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52 Projected Entangled Pair States

Figure 5.7: The expression of the fidelity as a 2D tensor network. Here, the states |$0 (%)#and |$0 (%!)# are translationally invariant and contain tensors A,B and C,D respectively.

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5.4 Computing Physical Properties of PEPS States 53

If two states di!er only by a global phase, then F (%,%!) = 1. On the other hand, it is

quite possible that for di!erent states, the fidelity F (%,%!) decays exponentially in the

system size. For infinite systems, many di!erent states have a fidelity of zero, regardless

of how close they are in a local sense. For large many-body systems, it was proposed in

[ZB07, ZPac06] that the rate at which the fidelity between two di!erent ground states

tends to zero can e!ectively express how close two states are in phase space. Such an

intensive quantity is defined by considering that the fidelity scales as

F (%,%!) = [d (%,%!)]L , (5.6)

where L is the number of sites. The term d is known as the fidelity per lattice site. Taking

the natural logarithm of both sides, we obtain,

d (%,%!) =ln (F (%,%!))

L(5.7)

It was shown in [ZOV08] that the computation of d for infinite (L&,) lattice systems is

realized by capturing eigenvalues arising during the contraction of 2D networks like that

shown in figure 5.7ii.

5.4.4 Computation of Entanglement Measures

We can also compute some of the elementary entanglement measures described in Chapter

3 with a PEPS. Take for instance the entanglement entropy of an adjacent pair of sites.

To compute this, we need to firstly compute the reduced density matrix for the site-

pair. This involves a trivial modification to the steps in the computation of an observable

(see figure 5.5), leaving the physical indices of the site-pair open instead of contracting

them with the observable. The approximate reduced density matrix is now represented

by the contraction of the network in figure 5.8. From this reduced density matrix, it is

simple to compute the entanglement entropy. However, the computational complexity of

computing the reduced density matrix for a contiguous block of sites scales exponentially

in the number of sites, and so it is generally only possible to compute the entanglement

entropy of relatively small blocks of sites.

The Bloch vector was related to the reduced density matrix in equation 3.7. There, it was

stated that there was a precise mapping between the reduced density matrix for a single

site and the Bloch vector, and that the Bloch vector could be determined from a set of

local observables:

r = $*x# i + $*y# j + $*z# k (5.8)

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54 Projected Entangled Pair States

Figure 5.8: The tensor contraction that gives the four-legged tensor containing the coef-ficients of the two-site reduced density matrix.

The magnitude of r, or purity, is thus computable from the expectation value of the

spin-1/2 Pauli operators.

The geometric entanglement of a PEPS state, as defined in equation 3.11, is the maximal

fidelity per lattice site between the PEPS state and a product state. In our case, we have a

PEPS representation of the ground state that is translationally periodic by some integer-

site shift in a vertical or horizontal direction. Since translational invariance undergirds

our ability to contract infinite 2D tensor networks and compute the fidelity per lattice

site, the domain of our fidelity maximisation will be restricted to the set of product states

that are themselves translationally invariant. For algorithmic simplicity, we choose |0# to

be a D = 1 PEPS with the same periodicity as |$#. A variational algorithm for computing

the geometric entanglement is described in more detail in Chapter 11.

5.5 Concluding Remarks

In this chapter, we have described the iPEPS algorithm - a technique approximately

computing the ground state of local 2D Hamiltonians. For a low-level treatment of certain

important stages of the algorithm, we refer the reader to Appendices A, B and C. We

have also described some important ways in which we can manipulate the PEPS to extract

physical information about the state. In the following chapters, we apply the iPEPS

algorithm to lattice models, firstly benchmarking our algorithm against analytical results

and models well studied by existing numerical schemes, and then applying our algorithm

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5.5 Concluding Remarks 55

to problems beyond the reach of other algorithms.

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56 Projected Entangled Pair States

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Chapter 6

The 2D Classical Ising Model

6.1 Introduction

The Ising model is the most well-known example of a spin system for which rich macro-

scopic behaviour results from a simple microscopic description. The Hamiltonian describes

a nearest neighbour spin interaction where each spin is confined to two values, nominally

called up and down. The interaction may be ferromagnetic or antiferromagnetic in nature,

depending on whether the system energetically favours neighbouring spins pairing in the

same or opposite directions. On the square lattice, this distinction is largely superficial,

as the essential statistical physics of the two systems is identical up to a sign. As a result,

in this chapter we confine our discussion to the ferromagnetic case. At zero temperature

the classical solution is trivial - the system spontaneously breaks the spin-up/spin-down

(Z2) symmetry and lies in a state in which all of the spins align in either the up or down

direction. As temperature is increased, the Boltzmann weight of states with flipped spins

increases and the partition function contains contributions from a greater number of clas-

sical states. At infinite temperature, the Boltzmann weights of all possible classical spin

states are equal and the Z2 symmetry evident in the Hamiltonian is restored. The key

question of Ising’s original thesis [Isi25] was whether the system exhibited a classical phase

transition. That is, was there a finite temperature point in the phase diagram where, in

the thermodynamic limit, the partition function of the system switched from exhibiting

magnetic order (spontaneously broken Z2 symmetry) to disorder (symmetry restored).

He determined that in one dimension there is no such transition and incorrectly assumed

that this would hold for two and higher dimensions as well. In two dimensions, Onsager

famously determined that a phase transition does in fact occur and obtained analytical

57

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58 The 2D Classical Ising Model

expressions for properties such as the magnetization of the state at a given temperature

[Ons44]. The order-disorder transition at inverse temperature " =log(1+

'2)

2 - 0.4407

was seen to be a second order phase transition. In particular, this meant that spatial

correlation functions of the classical statistical ensemble decayed polynomially with site

separation.

In Chapter 4 we explained that determining the statistical properties of 2D classical sys-

tems at finite temperature amounted to contracting an infinite 2D tensor network. There,

we also described some approximate schemes for contracting infinite, translationally in-

variant 2D tensor networks. This operation was also seen to be key to computing the

environment in the iPEPS algorithm. In this chapter (see also [OV08]) we show in detail

how we can encode the partition function of the classical Ising model in a 2D tensor

network, and describe how the basis structure can be altered to compute local statistical

properties. By computing such properties and comparing with Onsager’s exact solution,

we can benchmark our tensor network contraction algorithm. In doing so, we will not

only be able to validate an important part of the iPEPS algorithm, but also demonstrate

some basic notions of tensor network representations of strongly correlated states.

6.2 The Model

We introduce the classical (ferromagnetic) Ising model Hamiltonian

H2DC = '#

<i,j>

SiSj, (6.1)

where i and j represent adjacent lattice sites. The Hamiltonian is Z2 symmetric, as it is

invariant under a transformation that flips all of the spins, i.e. Sk & 'Sk, 0k. At inverse

temperature " = 1/T, the partition function, is defined in the usual sense

Z =#

!

e""H(!), (6.2)

where each * = {S1, S2, S3, ...., SN} is a unique spin configuration of the system.

As mentioned, our intention is to recast the evaluation of the partition function as the

contraction of a 2D tensor network. This is done by firstly finding the matrices,

T =

%

e" e""

e"" e"

'

(6.3)

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6.3 Results 59

that when multiplied (contracted) in a certain way compute the partition function. The

arrangement of T matrices is shown in fig. 6.3i). In the spirit of Onsager’s solution, an

entire row (or column) of the T matrices forms a transfer matrix. Next, we split each T

by SVD (fig. 6.3ii) and reform the tensors at the lattice sites (fig. 6.3iii). This creates an

infinite square tensor network which when contracted computes the partition function of

the 2D classical Ising model.

To thoroughly justify the connection to the iPEPS algorithm, one can completely repro-

duce the PEPS structure, as described in [VWPGC06]. The statistical properties and

spatial correlation function for the classical system can be calculated using the very same

algorithms that we developed for quantum states represented by a PEPS. This simple and

seemingly benign procedure is actually of tremendous importance. It shows that a state

with polynomially decaying correlation functions (i.e. a critical ’quantum’ state) can be

exactly encoded in an infinite PEPS with bond dimension D = 2.

6.3 Results

The calculation of local expectation values can be performed in either the straightforward

2D tensor network representation or the iPEPS representation of the system. As an

example, we may wish to compute the average spin at a given site, i.

$Si# =

$

!Si(*)e""H(!)

Z(6.4)

This quantity is similar to the partition function, except that microstates with a down

spin at site i acquire a negative sign in the summation. In the straightforward approach,

such a computation amounts to replacing the A tensor in fig. 6.3iii) at site i with a

modified tensor A0. In the PEPS approach, we simply insert the *Z Pauli operator along

the physical indices at site i, just as we would to compute a quantum observable.

Magnetization

A plot of the magnetization against the inverse temperature " is shown in figure 6.2. The

exact solution is also plotted. Here, the only varying parameter is the value of & used

for the boundary state. For each &, the solid line represents a result computed with the

CTMRG method, and the dotted line a result computed with the infinite MPS approach.

There are two conclusions to draw from this plot.

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60 The 2D Classical Ising Model

Figure 6.1: The process for expressing the partition function as a 2D square tensor net-work. i) The partition function in terms of interaction matrices, T . ii) The splitting of theT matrices by singular value decomposition iii) The recombination of the split matricesinto the tensor A, creating a translationally invariant, square 2D network.

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6.3 Results 61

Firstly, as & increases, the PEPS magnetization generally becomes closer to the analytic

result. This is not universally true - in our plot it can be seen that the infinite-MPS result

for & = 20 is actually inferior to that for & = 16. This is most likely explained by power-

of-two values of & retaining some symmetry in the boundary state, and is usually only

observed for small values of &. Secondly, for the same &, one can see that CTMRG gives

better magnetization results. This may suggest that the CTMRG more e"ciently stores

correlations of an infinite network. One possible reason for such a result is that at each step

of the infinite-MPS boundary state evolution, we keep information that holds correlations

between infinite halves of the network. Furthermore, say the MPS is horizontal - then

horizontal and vertical correlations are treated quite di!erently. On the other hand, our

CTMRG algorithm (Appendix B) proceeds by adding sites to the environment one at

a time and renormalizing to retain a finite set of correlations. The local correlations,

those closest in a radial sense to the unit-cell, are strongest and favoured to be retained.

This may be why the CTMRG performs better in the computation of local observable

quantities.

A second parameter of interest in contracting infinite systems is the number of iterations

until convergence of the boundary state. In figure 6.2 we used 10,000 iterations to converge

the boundary state. In figure 6.3 we show a plot of the magnetization for constant

& = 48 and boundary iterations from 100 up to 10,000. One can see a marked change

in the magnetization error profile as the number of iterations increases. Even at 10,000

iterations, it can be seen that the computed magnetization is still some distance from the

exact magnetization near the critical point. This reflects the great di"culty in simulating

critical systems with tensor networks. As the system becomes more strongly correlated,

in order to accurately describe the physics there is a simultaneous requirement for both

increased tensor bond dimension and an increased number of algorithm iterations.

From the above results, it can be seen that the classical Ising model provides a very useful

toolbox for testing the contraction of an infinite 2D tensor network. When contracting an

infinite 2D network, we are faced with a trade-o! between accuracy and computational

time - larger & and more boundary evolution steps means more accurate results at the

cost of computational cycles.

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62 The 2D Classical Ising Model

0.44 0.4402 0.4404 0.4406 0.4408 0.441 0.44120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

β

Mag

netiz

atio

n

exactχ = 10χ = 16χ = 20χ = 32χ = 48

0.44 0.4402 0.4404 0.4406 0.4408 0.441 0.44120

0.1

0.2

0.3

0.4

0.5

β

Mag

netiz

atio

n Er

ror

χ = 10χ = 16χ = 20χ = 32χ = 48

Figure 6.2: A plot of the magnetization and magnetization error (below) of the 2D classicalIsing model for various &, along with the exact solution. The dotted plots indicate thatthe infinite MPS boundary state technique was used. The solid results are derived fromCTMRG. Note that i) large & generally results in a smaller magnetization error. ii)CTMRG outperforms the iMPS approach.

Two-point correlation function

We compute the two-point correlation function

$Sx,ySx+i,y# =

$

!Sx,ySx+i,y (*)

Z(6.5)

as a function of the horizontal site separation |i|.

At the critical temperature, it is known that this correlation function should decay poly-

nomially, with exponent 1 = 1/4. The plot in figure 6.4 shows the plot of the two-point

correlator for various values of &, along with the exact correlator. Once again, it can

be seen that the correlation function more closely approximates the exact solution as &

increases. For & = 60, the correlator is almost indistinguishable from the exact result

for separations of over 1000 sites. Importantly however, we cannot fully reproduce the

polynomially decaying correlation function with finite &. This confirms a key limitation

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6.3 Results 63

0.44 0.4402 0.4404 0.4406 0.4408 0.441 0.44120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

β

Mag

netiz

atio

n

exactN = 100N = 500N = 1000N = 5000N = 10000

Figure 6.3: A plot of the magnetization of the 2D classical Ising model for various numbersof boundary state iterations. Note that as the number of iterations increases, we moreclosely track the exact solution.

of the iPEPS algorithm described in Chapter 5. Here, we have an exact representation

for a critical correlation function, but by computing it using an infinite-MPS with fixed

&, we can only capture exponentially decaying correlations.

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64 The 2D Classical Ising Model

100 101 102 103 10410−6

10−5

10−4

10−3

10−2

10−1

100

Separation, x = |i − j|

C(x

)

exactχ =10χ = 20χ = 40χ = 60

Figure 6.4: A plot showing the two-point correlation function of the classical Ising modelfor various & along with the exact solution. Note that as & increases, we obtain resultsthat better approximate the polynomially decaying exact solution. Also note that thoughall of the results here are exponentially decaying in the infinite limit, as we increase & wesee a larger window of approximately polynomial decay.

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Chapter 7

The 2D Quantum Ising Model

7.1 Introduction

In this chapter, we study the quantum Ising model on the square lattice, a simple example

of a non-trivial model of quantum magnetism. The quantum Ising Hamiltonian takes the

form,

H = '#

<i,j>

*iz*

jz ' %x

#

i

*ix (7.1)

where *z and *x are Pauli operators. The system here is tunable by an external transverse

magnetic field, %x. We refer to the eigenstates of *z as |+#z and |'#z. The Hamiltonian is

Z2 symmetric as the energy is invariant under a flip of every spin in the system, i.e. |+#z &|'#z , |'#z & |+#z. Equivalently, the Hamiltonian is invariant under the substitution

*z & '*z. In the Landau theory of phase transitions [Lan37], the phases are distinguished

by whether the symmetry of the Hamiltonian is conserved or broken in the ground state.

Since the Hamiltonian is invariant under the substitution *z & '*z, a symmetric ground

state, |%sym#, should be marked by having zero magnetization in the z-direction, i.e.

$%sym|*z |%sym# = $%sym|' *z |%sym#

= '$%sym|*z |%sym#

= 0

By contrast, a state with broken symmetry will have non-zero magnetization in the z-

direction. The longitudinal magnetization mz(%x) = $%+x|*z|%+x# is therefore an order

parameter of the system. This magnetization can be positive or negative in value, de-

pending on whether the average spin is up or down. When the magnetization is positive,

65

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66 The 2D Quantum Ising Model

we may say that the ground state lies in a spin up region of phase space. Conversely,

when the magnetization is negative, we say the ground state lies in a spin-down region of

phase space.

At %x = 0, the ground is in a product state configuration, where either every spin is the

|+#z state or every spin is in the |'#z state. The system exhibits a phenomenon known

as spontaneous symmetry breaking, where it randomly chooses either the spin up or spin

down configuration. In the limit of infinite %x, the state is in the |+#x eigenstate of *x,

and the Z2 symmetry is restored. The task is to determine at what value of %x the ground

state changes between these types of behaviour, and to describe the phase diagram of the

system.

Before proceeding, it should be noted that the transverse magnetic field %x plays an

analogous role to temperature in the classical Ising model, in that it encourages disorder in

the *z-basis of the state. Indeed, the quantum Ising Hamiltonian in D spatial dimensions

is derivable from the classical Ising Hamiltonian in (D+1) spatial dimensions, by treating

one of the spatial dimensions in the classical problem as representing imaginary-time (see

e.g. [FS78]). This makes concrete the quantum-classical correspondence of Chapter 2,

the result being that certain (universal) properties of the 2D quantum Ising model map

exactly to certain (universal) properties of the 3D classical Ising model.

Although there exists no exact solution for the quantum Ising model in two spatial di-

mensions, it has been well-studied for finite systems by various numerical techniques such

as quantum Monte Carlo (QMC) [BD02], series expansion (SE) [HHO90] and exact diag-

onalization (ED). Since the QMC treatment of the quantum Ising model does not su!er

from the sign problem, finite size scaling of QMC simulations produces numerical data

thought to be of an extremely high accuracy. From these, it has been estimated that the

phase transition occurs at %x - 3.044. Moreover, it is strongly suggested that the phase

transition is of second order (as in the 1D quantum Ising model) with a critical exponent

of " - 0.327 [BD02]. Our intention here is to benchmark the iPEPS algorithm against

these results, to establish that the iPEPS algorithm can be used to e!ectively study the

phase diagram of a 2D quantum system.

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7.2 Results 67

7.2 Results

We perform the imaginary-time evolution for this model with the algorithm described in

Chapter 5. As a first attempt, we use the infinite-MPS method to contract the boundary

state. Having obtained the ground state PEPS representation for values of %x between 0

and 5, we plot the energy-per-link,

el =1

4

#

,k

$%+x|h[,r,,r+,k]l |%+x#, /r . LA, /k = {x,'x, y,'y} (7.2)

where LA is the sub-lattice defined as in section 5.2.1, and

h[,i,,j]l = *

,iz*,jz +

%x

4*,ix +

%x

4*,jx, (7.3)

and the transverse magnetization,

mx(%x) = $%+x|*x|%+x#, (7.4)

in fig. 7.1. Here, we have compared the plotted PEPS points against series expansion

results, obtaining a very good agreement between the two. One can see that the first

derivative of the energy appears continuous, suggesting that the QPT is a continuous

phase transition. The discontinuity in the first derivative of mx suggests there could be a

phase transition at this point.

In fig. 7.2, we plot the longitudinal magnetization,

mz(%x) = $%+x|*z|%+x#, (7.5)

against %x. We say that the phase with non-zero longitudinal magnetization is ordered

and the phase with zero longitudinal magnetization disordered. The inset shows clearly

that the D = 3 solution predicts a magnetization closer to the QMC result than the

D = 2 solution. This is in-line with D = 3 producing a lower energy approximation to

the ground state.

Critical exponents describe how certain physical quantities behave near a phase transition.

They are universal quantities in the sense that many critical systems with non-trivial

di!erences in their microscopic description have identical critical exponents. We say that

such critical systems are in the same universality class and that the long-range behaviour

of systems in the same universality class is identical. The critical exponents depend on

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68 The 2D Quantum Ising Model

Figure 7.1: Transverse magnetization mx and energy per site e of the quantum Isingmodel as a function of the transverse magnetic field h. The continuous line shows seriesexpansion results (to 26th and 16th order in perturbation theory) for h smaller and largerthan hc - 3.044 [HHO90]. Increasing D leads to a lower energy per site e. For instance,at h = 3.1, e(D = 2) - '1.6417 and e(D = 3) - '1.6423.

properties such as the spatial dimension of the system and lattice geometry, and the spin

dimension, Hamiltonian symmetries and the range of interactions. For the Ising model,

the order parameter, mz scales near the critical point as,

mz(%x) = mx0 |%x ' %x,crit|" (7.6)

Here, the parameter " is the critical exponent. With PEPS, we obtain an estimate of

" = 0.328, which is within 2%, of the QMC estimate.

The results so far have been determined using the infinite-MPS contraction scheme to

determine the environment. We now wish to investigate the e!ect of using the corner

transfer matrix to compute the environment for the PEPS update, or alternatively using

the simplified update scheme in which the link updates proceed without determination

of any environment tensors. For the classical Ising model, it was seen that the use of

CTMRG resulted in smaller error in the magnetization (for the same &). This suggested

that the CTM better represents the correlations between the environment and the unit-

cell. On the other hand, it is believed that the computationally inexpensive simplified

update should perform well only in regions where the correlations are very local in nature.

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7.2 Results 69

Figure 7.2: Magnetization mz(%) of the quantum Ising model as a function of the trans-verse magnetic field %. Dashed lines are a guide to the eye. We have used the diagonalscheme for (D,&) = (2, 20), (3, 25) and (4, 35) (the vertical/horizontal scheme leads tocomparable results with slightly smaller &.) The inset shows a log plot of mz versus|%' %c|, including our estimate of %c and ". The continuous line shows the linear fit.

Near the critical point, the simplified scheme is expected to struggle to capture the correct

physics due to the diverging correlation length of the system.

We want to see how the iMPS, CTMRG and simplified update variants of the algorithm

perform for computing the order parameter and critical exponent of the quantum Ising

model. The results are shown in figure 7.3 and quite resoundingly a"rm the idea that the

CTM stores correlations more e"ciently for the computation of local properties. Here,

the CTM with D = 2 predicts a critical point of %c = 3.08 and a D = 3 CTM predicts it

as %c = 3.04. Meanwhile, it can be seen that the simplified update scheme performs quite

well far from the critical point, but di!ers significantly near where QMC and the other

methods suggest a phase transition.

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70 The 2D Quantum Ising Model

2.6 2.7 2.8 2.9 3 3.1 3.2 3.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

λx

<mz>

D = 2 iMPSD = 3 iMPSD = 2 CTMRGD = 3 CTMRGD = 3 simplifiedD = 5 simplifiedMFT

Figure 7.3: A comparison of the order parameter for iMPS, CTMRG and the simplifiedupdate.

QMC D=2 D=3 D=2 D=3 D=3Ref. [BD02] iMPS iMPS CTMRG CTMRG VDMA Ref. [NOH+00, NHO+01, GMN03]

%c 3.044 3.10 3.06 3.08 3.04 3.2" 0.327 0.346 0.332 0.333 0.328 –

Table 7.1: Critical point and exponent " as a function of D.

Figure 7.4: Two-point correlator Sxx(l) of the quantum Ising model near the criticalpoint, % = 3.05. For nearest neighbors, the correlator quickly converges as a function ofD, whereas for long distances we expect to see convergence for larger values of D.

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7.2 Results 71

Figure 7.5: Fidelity diagram of the quantum Ising model, computed from a catalogue ofD = 2 ground states

7.2.1 Two-Point Correlation Functions

Figure 7.4 plots the two-point correlation function Sxx = $$|*[r]x *[r+i]

x |$#, where i is some

displacement in a horizontal direction. The states used correspond to a magnetic field near

criticality, %x = 3.05. On this plot, it is clear that both correlations decay exponentially

(a polynomial decay would appear linear on a log-log plot). Furthermore, the D = 2

correlation decays at a rate faster that the D = 3 plot. However, the D = 3 plot shows

a greater tendency towards polynomial decay, more evidence that as D increases, the

qualities of the ground state more closely match the physical properties.

7.2.2 Fidelity plot

The fidelity plot for the 2D quantum Ising model is shown in fig. 7.5. Our results here are

determined from PEPS ground states with bond dimension D = 2. Such a plot was also

presented in [ZOV08]. There, it was suggested that the characteristic ”pinch-point” of the

surface near the the point (%1x,%

2x) = (%c,%c) is indicative of a continuous phase transition.

We will see a quite di!erent picture for a first-order phase transition in Chapter 9.

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72 The 2D Quantum Ising Model

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Chapter 8

The Hard-Core Bose-Hubbard Model

8.1 Introduction

The physics of interacting bosons at low temperature has since long attracted consider-

able interest due to the occurrence of Bose-Einstein condensation [DGPS99]. The Bose-

Hubbard model, a simplified microscopic description of an interacting boson gas in a

lattice potential, is commonly used to study related phenomena, such as the superfluid-

to-insulator transitions in liquid helium [FWGF89] or the onset of superconductivity in

granular superconductors [JHOG89] and arrays of Josephson junctions [BD84]. In more

recent years, the Bose-Hubbard model is also employed to describe experiments with cold

atoms trapped in optical lattices [JBC+98, GBM+01, GME+02].

In this chapter we initiate the exploration of interacting bosons in an infinite 2D lattice

with tensor network algorithms. We use the iPEPS algorithm explained in Chapter 5 and

[JOV+08] to characterize the ground state of the hard-core Bose-Hubbard (HCBH) model,

namely the Bose Hubbard model in the hard-core limit, where either zero or one bosons

are allowed on each lattice site. Although no analytical solution is known for the 2D

HCBH model, there is already a wealth of numerical results based on mean-field theory,

spin-wave corrections and stochastic series expansion [BBM+02]. These techniques have

been quite successful in determining some of the properties of the ground state of the 2D

HCBH model, such as its energy, particle density or condensate fraction. Our goal in this

chapter is twofold. Firstly, by comparing our results against those of Ref. [BBM+02], we

aim to benchmark the performance of the iPEPS algorithm in the HCBH model. Secondly,

once the validity of the iPEPS algorithm for this model has been established, we use it

73

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74 The Hard-Core Bose-Hubbard Model

to obtain results that are harder to compute with (or simply well beyond the reach of)

the other approaches. These include the analysis of entanglement, two-point correlators,

fidelities between di!erent ground states[ZPac06, ZB07, ZOV08], and the simulation of

time evolution.

We note that the present results naturally complement those of Ref. [MVC07] for finite

systems, where the PEPS algorithm [VC04] was used to study the HCBH model in a

lattice made of at most 11" 11 sites.

8.2 Model

The Bose-Hubbard model [FWGF89] with on-site and nearest neighbour repulsion is

described by the Hamiltonian

HBH = ' J#

(i,j)

)

a†iaj + a†

jai

*

'#

i

µni

+#

i

V1ni (ni ' 1) + V2

#

(i,j)

ninj,

where a†i , ai are the usual bosonic creation and annihilation operators, ni = 'i * a†

iai

is the number (density) operator at site i, J is the hopping strength, µ is the chemical

potential, and V1, V2 / 0. The four terms in the above equation describe, respectively, the

hopping of bosonic particles between adjacent sites (J), a single-site chemical potential

(µ), an on-site repulsive interaction (V1) and an adjacent site repulsive interaction (V2).

Here we shall restrict our attention to on-site repulsion only (V2 = 0) and to the so-

called hard-core limit in which this on-site repulsion dominates (V1 & ,). Under these

conditions the local Hilbert space at every site describes the presence or absence of a single

boson and has dimension 2. With the hard-core constraint in place, the Hamiltonian

becomes

HHC = 'J#

(i,j)

)

a†iaj + a†

jai

*

'#

i

µni , (8.1)

where a†i , ai are now hard-core bosonic operators obeying the commutation relation,

+

ai, a†j

,

= (1' 2ni) ,ij .

A few well-known facts of the Hard Core Bose Hubbard (HCBH) model are:

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8.2 Model 75

(i) U(1) symmetry.— The HCBH model inherits particle number conservation from the

Bose-Hubbard model,

[HHC, N ] = 0, N *#

l

nl , (8.2)

and it thus has a U(1) symmetry, corresponding to transforming each site l by ei-nl ,

0 . [0, 22).

(ii) Duality transformation.— In addition, the transformation al & a†l applied on all

sites l of the lattice maps HHC(µ) into HHC('µ) (up to an irrelevant additive constant).

Accordingly, the model is self-dual at µ = 0, and results for, say, µ > 0 can be easily

obtained from those for µ < 0.

(iii) Equivalence with a spin model.— The HCBH model is equivalent to a quantum spin12 model, namely the ferromagnetic quantum XX model,

HXX = 'J

2

#

(i,j)

*xi *

xj +*y

i *yj +

µ

2

#

i

*zi , (8.3)

which is obtained from HHC with the replacements

al =*x

l + i*yl

2, a†

l =*x

l ' i*yl

2,

where *x, *y and *z are the spin 12 Pauli matrices. In particular, all the results of this

paper also apply, after a proper translation, to the ferromagnetic quantum XX model on

an infinite square lattice.

(iv) Ground-state phase diagram.— The hopping term in HHC favors delocalization of

individual bosons in the ground state, whereas the chemical potential term determines

the ground state bosonic density ',

' *1

N

#

i

$a†iai# .

For µ negative, a su"ciently large value of |µ| forces the lattice to be completely empty,

' = 0. Similarly, a large value of (positive) µ forces the lattice to be completely full,

' = 1, as expected from the duality of the model. In both cases there is a gap in

the energy spectrum and the system represents a Mott insulator. When, instead, the

kinetic term dominates, the density has some intermediate value 0 < ' < 1, the cost

of adding/removing bosons to the system vanishes, and the system is in a superfluid

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76 The Hard-Core Bose-Hubbard Model

phase [FWGF89]. The latter is characterized by a finite fraction of bosons in the lowest

momentum mode ak=0 * (1/N)$

i ai, that is by a non-vanishing condensate fraction '0,

'0 * $a†k=0ak=0# =

1

N2

#

i,j

$a†jai# .

In the thermodynamic limit, N &,, a non-vanishing condensate fraction is only possible

in the presence of o!-diagonal long range order (ODLRO) [Yan62], or $a†jai# )= 0 in the

limit of large distances |i' j|, given that

'0 = lim|i"j|#$

$a†jai#. (8.4)

(v) Quantum phase transition.— Between the Mott insulator and superfluid phases, there

is a continuous quantum phase transition [FWGF89], tuned by µJ .

8.3 Results

In this section we present the numerical results obtained with the iPEPS algorithm.

Without loss of generality, we fix the hopping strength J = 1 and compute an approxi-

mation to the ground state |%GS# of HHC for di!erent values of the chemical potential µ.

Then we use the resulting PEPS/TPS to extract the expectation value of local observ-

ables, analyze ground state entanglement, compute two-point correlators and fidelities, or

as the starting point for an evolution in real time.

In most cases we only report results for µ + 0 (equivalently, density 0 + ' + 0.5) since

due to the duality of the model, results for positive µ (equivalently, 0.5 + ' + 1) can be

obtained from those for negative µ.

8.3.1 Local observables and phase diagram

Particle density '.— Fig. 8.1 shows the density ' as a function of the chemical potential

µ in the interval '4 + µ + 0. Notice that ' = 0 for µ + '4, since each single site is

vacant. Our results are in remarkable agreement with those obtained in Ref. [BBM+02]

with stochastic series expansions (SSE) for a finite lattice made of 32 " 32 and with a

mean field calculations plus spin wave corrections (SW). We note that the curves '(µ) for

D = 2 and D = 3 are very similar.

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8.3 Results 77

Energy per site (.— Fig. 8.1 also shows the energy per site ( as a function of the density

'. This is obtained by computing ((µ) and then replacing the dependence on µ with '

by inverting the curve '(µ) discussed above. Again, our results for ((') are in remarkable

agreement with those obtained in Ref. [BBM+02] with stochastic series expansions (SSE)

for a finite lattice made of 32" 32. They are also very similar to the results coming from

mean field calculations with spin wave corrections (SW) of Ref [BBM+02], and for small

densities reproduce the scaling (valid only in the regime of a very dilute gas) predicted

in Ref. [Sch71, HFM78] by using field theory methods based on a summation of ladder

diagrams. Once more, the curves ((') obtained with bond dimension D = 2 and D = 3

are very similar, although D = 3 produces slightly lower energies.

Condensate fraction '0.— In order to compute the condensate fraction '0, we exploit

that the iPEPS algorithm induces a spontaneous symmetry breaking of particle number

conservation. Indeed, one of the e!ects of having a finite bond dimension D is that the

PEPS/TPS that minimizes the energy does not have a well-defined particle number. As

a result, instead of having $ai# = 0, we obtain a non-vanishing value $ai# )= 0 such that

'0 = lim|i"j|#$

$a†jai# = |$ai#|2. (8.5)

In other words, the ODLRO associated with the presence of superfluidity, or a finite

condensate fraction, can be computed by analysing the expectation value of al,

$al# =1'0e

i*, (8.6)

where the phase ) is constant over the whole system but is otherwise arbitrary. The

condensate fraction '0 shows that the model is in an insulating phase for |µ| / 4 (' = 0, 1)

and in a superfluid phase for '4 < µ < 4 (0 < ' < 1), with a continuous quantum

phase transition occurring at |µ| = '4, as expected. However, this time the curves '0(')

obtained with D = 2 and D = 3 are noticeably di!erent, with D = 3 results again in

remarkable agreement with the SSE and SW results of Ref. [BBM+02].

8.3.2 Entanglement

The iPEPS algorithm is based on assuming that a PEPS/TPS o!ers a good description

of the state |%# of the system. Results for small D will only be reliable if |%# has at most

a moderate amount of entanglement. Thus, in order to understand in which regime the

iPEPS algorithm should be expected to provide reliable results, it is worth studying how

entangled the ground state |%GS# is as a function of µ.

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78 The Hard-Core Bose-Hubbard Model

-4 -3 -2 -1 0µ

0

0.2

0.4

ρ

SSEMFSWLadderAndersenD = 2D = 3

0 0.2 0.4ρ

-1.2-1

-0.8-0.6-0.4-0.2

0

ε

0 0.2 0.4ρ

0

0.05

0.1

0.15

0.2

0.25

ρ 0

Figure 8.1: Particle density '(µ), energy per lattice site ((') and condensate fraction'0(') for a PEPS/TPS with D = 2, 3. We have also plotted results from Ref.[BBM+02]corresponding to several other techniques. Our results follow closely those obtained withstochastic series expansion (SSE) and mean field with spin wave corrections (SW).

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8.3 Results 79

−4 −2 00.88

0.9

0.92

0.94

0.96

0.98

1

µ

r

−4 −2 00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

µ

S L

L = 1L = 2L = 4

D = 2D = 3

Figure 8.2: Purity r and entanglement entropy SL as a function of the chemical potentialµ. The results indicate that the ground state is more entangled deep inside the superfluidphase (µ = 0) than at the phase transition point (µ = '4). Notice that the more entangledthe ground state is, the larger the di!erences between results obtained with D = 2 andD = 3 (see also Fig. 8.1).

To do this, we compute the purity for a single site, as defined in Chapter 3. Recall the

convention - if the site is unentangled with the rest of the system, then its purity, r, is

1. As the entanglement between a site and the rest of the system increases, the purity

decreases. Fig. 8.2 shows the purity r as a function of the chemical potential. In the

insulating phase (µ + '4), the ground state of the system consists of a vacancy on each

site. In other words, it is a product state, r = 1. Instead, For µ > '4 the ground state is

entangled. Several comments are in order:

(i) The purity r(µ) for D = 3 is smaller than that for D = 2 by up to 3%. This is

compatible with the fact that the a PEPS/TPS with larger bond dimension D can carry

more entanglement.

(ii) Results for D = 2, 3 seem to indicate that the ground state is more entangled (r is

smaller) deep into the superfluid phase (e.g. µ = 0) than at the continuous quantum

phase transition µ = '4. This is in sharp contrast with the results obtained e.g. for

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80 The Hard-Core Bose-Hubbard Model

the 2D quantum Ising model [JOV+08], where the quantum phase transition displays the

most entangled ground state. However, notice that in the Ising model the system is only

critical at the phase transition whereas in the present case criticality extends throughout

the superfluid phase. Each value of µ in the superfluid phase corresponds to a fixed

point of the RG flow. That is, in moving away from the phase transition we are not

following an RG flow. Therefore, the notion that entanglement should decrease along an

RG flow[LLRV05], as observed in the 2D Ising model, is not applicable for the HCBH

model.

(iii) Accordingly, we expect that the iPEPS results for small D become less accurate as

we go deeper into the superfluid phase (that is, as we approach ' = 0.5). This is precisely

what we observe: the curves '0(') for D = 2 and D = 3 in Fig. 8.1 di!er most at ' = 0.5.

We also compute the entanglement entropy for the reduced density matrix 3L (L = 1, 2, 4)

corresponding to one site, two contiguous sites and a block of 2"2 sites respectively. The

entanglement entropy vanishes for an unentangled state and is non-zero for an entangled

state. The curves S(3L) confirm that the ground state of the HCBH model is more

entangled deep in the superfluid phase than at the quantum phase transition point.

8.3.3 Correlations

We compute the two point correlation function C(s),

C(s) * $a†iai+sx# ' $a†

i#$a[i+sx]# , (8.7)

for pairs of sites separated by s lattice spacings along the horizontal direction x. For

points in the gapless superfluid phase, one expects to see correlation functions that decay

polynomially with the s. One can see in fig. 8.3, C(s) for PEPS representations of

a superfluid ground state, µ = 0. Just as for the Ising model, we obtain correlation

functions that decay exponentially in s, but as D increases, we see a tendency towards a

polynomial decay and agreement for small s.

The results show that while for short distances s = 0, 1, 2 the correlator C(s) is already well

converged with respect to D, for larger distances s the correlator still depends significantly

on D. This seems to indicate that while the iPEPS algorithm provides remarkably good

results for local observables already for a!ordably small values of D, a larger D might be

required in order to also obtain accurate estimates for distant correlators.

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8.3 Results 81

0 2 4 6 8 10 12

10−4

10−3

10−2

10−1

100

s

C(s

)

D = 2

D = 3

D = 4

Figure 8.3: Two-point correlation function C(s) versus distance s (measured in latticesites), along a horizontal direction of the lattice. For very short distances the correlatorfor D = 2, 3, 4 are very similar whereas for larger distances they di!er significantly.

8.3.4 Fidelity

Given two ground states |%GS(µ1)# and |%GS(µ2)#, corresponding to di!erent chemical

potential µ, the fidelity per site f [ZB07], defined through

ln f(µ1, µ2) = limN#$

1

Nln |$%GS(µ1)|%GS(µ2)#| ,

can be used as a means to distinguish between qualitatively di!erent ground states

[ZPac06, ZB07]. In the above expression, N is the number of lattice sites and the thermo-

dynamic limit N &, is taken. Importantly, the fidelity per site f(µ1, µ2) remains finite

in this limit, even though the overall fidelity |$%GS(µ1)|%GS(µ2)#| vanishes. In a sense,

f(µ1, µ2) captures how quickly the overall fidelity vanishes.

As explained in Chapter 5, the fidelity per site f(µ1, µ2) can be easily computed within the

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82 The Hard-Core Bose-Hubbard Model

µ1

µ2

−5 −4 −3 −2 −1 0 1 2 3 4 55−5

−4

−3

−2

−1

0

1

2

3

4

5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 8.4: Fidelity per lattice site f(µ1, µ2) for the ground states of the HCBH model.Notice the plateau f(µ1, µ2) = 1 (white) for µ1, µ2 + '4 (also for µ1, µ2 / 4) correspond-ing to the Mott insulating phase, and the pinch point at µ1, µ2 = '4 (also at µ1, µ2 = 4)consistent with a continuous quantum phase transition.

framework of the iPEPS algorithm [ZOV08]. In the present case, before computing the

overlap each ground state is rotated according to ei*!z/2, where ) is the random condensate

phase of Eq. 8.6. In this way all the ground states have the same phase ) = 0. The

fidelity per site f(µ1, µ2) is presented in Fig. 8.4. The plateau-like behavior of f(µ1, µ2) for

points within the separable Mott-Insulator phase (µ1, µ2 + '4 or µ1, µ2 / 4) is markedly

di!erent from that between ground states in the superfluid region ('4 + µ1, µ2 + 4),

where the properties of the system vary continuously. Moreover, similarly to what has

been observed for the 2D quantum Ising model [ZOV08] or in the 2D quantum XYX model

[LLZ09], the presence of a continuous quantum phase transition between insulating and

superfluid phases in the 2D HCBH model is signaled by pinch points of f(µ1, µ2) at

µ1 = µ2 = ±4. That is, the qualitative change in ground state properties across the

critical point is evidenced by a rapid, continuous change in the fidelity per lattice site as

one considers two ground states on opposite sides of the critical point and moves away

from it.

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8.3 Results 83

0 5 10 15 20 25 30 35 40−0.55

−0.5

<Ho>

0 5 10 15 20 25 30 35 40−0.56

−0.55

−0.54

<H>

0 5 10 15 20 25 30 35 400

0.5

1

ρ

0 5 10 15 20 25 30 35 400.2

0.21

0.22

ρ 0

Time

Figure 8.5: Evolution of the energies $H0# and $H#, the density ', and condensate fraction'0 after a translation invariant perturbation V is suddenly added to the Hamiltonian.

8.3.5 Time evolution

Using a slight modification on the algorithm for imaginary-time evolution, we can simulate

the Hamiltonian evolution of PEPS states. A first example of such simulations with the

iPEPS algorithm was provided in Ref. [ODV09], where an adiabatic evolution across the

phase transition of the 2D quantum compass orbital model was simulated in order to show

that the transition is of first order.

The main di"culty in simulating a (real) time evolution is that, even when the initial state

|%(0)# is not very entangled and therefore can be properly represented with a PEPS/TPS

with small bond dimension D, entanglement in the evolved state |%(t)# will typically grow

with time t and a small D will quickly become insu"cient. Incrementing D results in

a huge increment in computational costs, which means that only those rare evolutions

where no much entanglement is created can be simulated in practice.

For demonstrative purposes, here we have simulated the response of the ground state

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84 The Hard-Core Bose-Hubbard Model

|%GS# of the HCBH model at half filling (' = 0.5 or µ = 0) when the Hamiltonian HHC is

suddenly replaced with a new Hamiltonian H given by

H * HHC + .V, V * 'i#

k

)

ak ' a†k

*

, (8.8)

where . = 0.2 and, importantly, the perturbation V respects translation invariance. As

the starting point of the simulation, we consider a PEPS/TPS representation of the ground

state with bond dimension D = 2, obtained as before through imaginary time evolution.

Fig. 8.5 shows the evolution in time of the expectation value per site of the energies $HHC#and $H#, as well as the density ' and condensate fraction '0. Notice that the expectation

value of H should remain constant through the evolution. The fluctuations observed in

$H#, of the order of 0.2% of its total value, are likely to be due to the small bond dimension

D = 2 and indicate the scale of the error in the evolution. The simulation shows that, as

a result of having introduced a perturbation V that does not preserve particle number,

the particle density ' oscillates in time. The condensate fraction, as measured by |$al#|2,is seen to oscillate twice as fast.

8.4 Conclusion

In this chapter we have initiated the study of interacting bosons on an infinite 2D lattice

using the iPEPS algorithm. We have computed the ground state of the HCBH model on

the square lattice as a function of the chemical potential. Then we have studied a num-

ber of properties, including properties that can be easily accessed with other techniques

[BBM+02], as is the case of the expected value of local observables, as well as properties

whose computation is harder, or even not possible, with previous techniques.

Specifically, using a small bond dimension D = 2, 3 we have been able to accurately

reproduce the result of previous computations using SSE and SW of Ref. [BBM+02]

for the expected value of the particle density ', energy per particle ( and condensate

fraction '0, throughout the whole phase diagram of the model, which includes both a Mott

insulating phase and a superfluid phase, as well as a continuous phase transition between

them. Interestingly, in the superfluid phase the PEPS/TPS representation spontaneously

breaks particle number conservation, and the condensate fraction can be computed from

the expected value of the annihilation operator, '0 = |$al#|2.

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8.4 Conclusion 85

We have also conducted an analysis of entanglement, which revealed that the most entan-

gled ground state corresponds to half filling, ' = 0.5. This is deep into the superfluid phase

and not near the phase transition, as in the case of the 2D quantum Ising model[JOV+08].

Furthermore, inspection of a two-point correlator at half filling showed much faster con-

vergence in the bond dimension D for short distances than for large distances. Also,

pinch points in plot of the fidelity f(µ1, µ2) were consistent with continuous quantum

phase transitions at µ = ±4.

Finally, we have also simulated the evolution of the system, initially in the ground state

of the HCBH model at half filling, when a translation invariant perturbation is suddenly

added to the Hamiltonian.

Now that the validity of the iPEPS algorithm for the HCBH model (equivalently, the

quantum XX spin model) has been established, there are many directions in which the

present work can be extended. For instance, one can easily include nearest neighbour

repulsion, V2 )= 0, (corresponding to the quantum XXZ spin model) and/or investigate a

softer-core version of the Bose Hubbard model by allowing up to two or three particles

per site.

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86 The Hard-Core Bose-Hubbard Model

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Chapter 9

The Quantum Potts Model

9.1 Introduction

The classical Potts model was introduced as a generalisation of the classical Ising model

to an arbitrary number of local spin components [Pot52]. Instead of being confined to

merely two directions, the spin at site i is free to exist in one of q directions #ni= 2.ni

q ,

where ni = 0, 1, 2, ..., q ' 1. The general form of the classical Potts Hamiltonian is

H =#

<ij>

Jij, (9.1)

where Jij describes some nearest-neighbour coupling between spins.

Potts considered two forms for Jij. In the first, he chose:

Jij = ' cos-

#ni' #nj

.

(9.2)

This so-called planar Potts model or clock model is invariant under the action of the Zq

symmetry group, corresponding to a cyclic rotation of each spin by 2.kq , where the global

constant k = 0, 1, ..., q'1. It is obvious that for q = 2, the Ising Hamiltonian is recovered.

Potts determined the critical temperature for the model on the square lattice for q = 3

and q = 4 but was unable to progress any further. Domb suggested a simplification to

the model, where the coupling takes a second form:

Jij = ', (ni, nj) (9.3)

For this model, Potts was able to determine critical temperatures for all values of q and

it became known as the standard Potts model. The four-component instance of this

87

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88 The Quantum Potts Model

model had previously been studied by Ashkin and Teller[AT43], and for this reason the

generalization is sometimes called the Ashkin-Teller-Potts model.

The Potts model is of interest for several reasons. Firstly, it extends the same general

behaviour as the Ising model for all q. In two or more dimensions, the system exists

in a ferromagnetic state at low temperatures, where the symmetry of the Hamiltonian is

broken. As temperature increases, the system undergoes a phase transition to a disordered

paramagnetic state with the symmetry restored. Secondly, many properties of the classical

system are known exactly in two dimensions (D = 1) [Bax73], meaning that the rich

character of the phase transition can be explored and compared for di!erent q, providing

broad insight into the theory of phase transitions. Thirdly, the system has been seen to

have wider application in areas such as lattice statistics and percolation theory [KF69,

Wu78, KW78, Wu82], studies of dilute spin glasses [Wu82] and in the investigation into

self-dual lattice gauge theories [KPSS80] and the structure of QCD [SY82, BB07].

A particularly interesting aspect of the Potts model is the way in which the nature of

the phase transition changes with q. In the classical Potts model on the square lattice,

the phase transition is continuous for q + 4 and first-order for q > 4. Whilst there exists

no analytical solution in three spatial dimensions (D = 2) the system has been explored

with Monte Carlo [BBD08, ABV91] and series expansion [PYJM06] techniques and the

phase transition is thought to be continuous for q = 2, weakly first-order for q = 3 and

first-order for q / 4. Treating q as a continuous parameter leads to the concept of a

‘critical q’, qc, where the change from a first-order to continuous phase transition occurs

[Wu82]. For example, for the 2D classical Potts model, it is known that qc = 4. The

exact value of qc is not known for the 3D classical Potts model, but has been estimated

as qc - 2.6 ± 0.1 [KS81].

In this chapter, we apply the iPEPS algorithm to the quantum Potts model in two spatial

dimensions (D = 2). The universal behaviour of this model is the same as the 3D classical

model, and hence we expect the same characterisation of the phase transition with q. Our

interest is as much in profiling the physical characteristics of the quantum Potts model as

it is to benchmark the iPEPS algorithm on systems with continuous and first-order phase

transitions.

In Chapter 7, we studied the quantum Ising model (q = 2 quantum Potts model). There,

we observed that the presence of quasi-long-range correlations near the phase transition

had profound algorithmic implications. In particular, we observed the following (see fig.

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9.1 Introduction 89

7.3):

1. The most dramatic improvement in the estimation of the order parameter with

increasing D occurred near the phase transition.

2. Use of the full environment in the PEPS update improved the estimation of the order

parameter when compared with the simplified update, again most dramatically near

the phase transition.

3. Mean-field theory produced results that agreed very poorly with PEPS, apart from

far from the phase transition.

These observations only confirmed what we already knew about continuous quantum

phase transitions. As the system approaches criticality, the amount of quasi-long-range

correlations in the ground state increases. This meant that a purely local approximation

(MFT) was unable to describe the physics of the ground state. Furthermore, an imaginary-

time evolution of an iPEPS guided by mostly local information (the simplified update)

performed significantly worse than an evolution guided by the full information of the

state. In this chapter, we aim to assess the relative performance of the full-update, the

simplified update and mean-field theory for the quantum Potts model.

9.1.1 First-order phase transitions

The identifying characteristics of a first-order phase transition include:

1. A sharp phase transition corresponding to an energy eigenvalue crossing. A transi-

tion between phases well separated in phase space, and with substantial uniformity

within each phase.

2. A ground state energy that is discontinuous in its first derivative with respect to

the Hamiltonian parameter at the phase transition.

3. A non-diverging correlation length, 4.

These have considerable implications on how we approach the system with PEPS, and

what we might expect to observe in the simulation results. The first point poses well

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90 The Quantum Potts Model

known problems for numerical investigation of first-order phase transitions. The two

phases of the system at zero temperature are not connected at any order of perturbation

theory, nor are they connected in the normal sense of a Metropolis walk in quantum Monte

Carlo simulation. For such techniques, the location of the eigenvalue crossing is often

either poorly estimated or not detected at all. As such, our treatment of such a system

with tensor networks demands special consideration. Traversing the phase diagram by say

increasing % and reusing evolved PEPS ground states as subsequent starting points means

that we will likely stay in a given phase and be unable to detect the phase transition.

Even using a random PEPS as an initial point will naturally favour one phase over the

other close to the phase transition. For this reason, we approach the phase transition from

above and below - firstly using an initial state we know to be in the disordered phase and

computing the ground states for decreasing magnetic field, then using an initial state we

know to be in the ordered phase and computing the ground states for increasing magnetic

field. Plotting the ground state energy obtained from each of the two approaches should

give rise to the discontinuity in the first derivative of the energy stated in the second point.

The third point suggests that quasi-long-range correlations will not be as significant in

describing the ground state. So even though the local physical dimension, q is greater than

in the spin-1/2 Ising model (and hence the computational cost for a given bond-dimension,

D, is higher), the correlation length of the ground state is finite and the system may be

comparatively better characterised for small D. Furthermore, we may suggest that this

model will be relatively well described by the simplified update outlined in section 5.3.1,

where locally correlated states are favoured in the imaginary-time evolution. Finally, we

expect the weakly first-order q = 3 phase transition to be less emphatic in its first-order

characteristics than the q = 4 model and to show some tendencies towards a second-order

phase transition.

In the following sections, we firstly define the quantum Potts model and then aim to

determine the phase diagram for q = 3 and q = 4. For both values of q we will compute

the energy of the system at varying q and its first derivative to i) locate the phase transition

and ii) observe in each case that it is first-order in nature. Thereafter, we will compute

relevant local observables, including the order parameter. Finally, we will examine the

fidelity diagram, correlation function and entanglement properties of the system.

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9.2 The quantum Potts Model 91

9.2 The quantum Potts Model

The quantum Potts model typically refers to a quantum mechanical version of the stan-

dard Potts model. As a generalisation of the quantum Ising model, the quantum Potts

Hamiltonian can be written in the form,

Hpotts = '1

q

#

<i,j>

q"1#

k

*ix,k*

jx,q"k ' %z

#

i

*iz (9.4)

where %z is again an external magnetic field.

In line with the discussion in [SP81] we define the following operators,

) =

/

0

0

0

0

0

0

0

1

1 0 0 . . . 0

0 w 0 . . . 0

0 0 w2 . . . 0...

......

. . ....

0 0 0 . . . wq"1

2

3

3

3

3

3

3

3

4

, M =

/

0

0

0

0

0

0

0

1

0 1 0 . . . 0

0 0 1 . . . 0

0 0 0 . . . 0...

......

. . ....

1 0 0 . . . 0

2

3

3

3

3

3

3

3

4

(9.5)

where 5 = e2%&q , and 6 =

1'1.

For a representation diagonal in the coupling, we make the replacements,

*x,k = )k, *z =q"1#

k

Mk (9.6)

Alternatively, we can transform to a basis where the external field operator is diagonal,

in which case,

*x,k = Mk, *z = R =

/

0

0

0

0

1

q ' 1 0 . . . 0

0 '1 . . . 0...

.... . .

...

0 0 . . . '1

2

3

3

3

3

4

(9.7)

The Hamiltonian is Zq invariant under a cyclic rotation of the basis. In the representation

given by eqn. 9.6, this is proven by the commutator:

5

Hpotts,Mk6

= 0,. k = 0, 1, ...q ' 1 (9.8)

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92 The Quantum Potts Model

D=3 D=4 D=5 D=6 D=3 lowest energysimplified simplified simplified simplified CTMRG D=3 CTMRG and D = 6 simplified

%z,pt 0.8732 0.8732 0.8718 0.8716 0.8749 0.8722

Table 9.1: The location of the phase transition for various versions of the algorithm anddi!erent D. The ’lowest energy’ solution is taken from the lowest energy ground states oneither side of the phase transition.

9.3 q = 3 Results

In fig. 9.1 we plot the energy for the q = 3 Potts model. We show results for a mean-field

theory calculation, as well as an iPEPS with D = 3, D = 6 using the simplified update

and D = 3 with the full environment. For each D, we record the energy crossing point in

table 9.1.

Comparing the first derivative of the energy curves as we approach from above and below

we can quantify to what extent the phase transition is first-order. In fig. 9.2 we plot

the derivative as calculated by a finite di!erence method. The black dotted vertical line

shows the position of the phase transition as determined from the energy crossing and it

is clear for all values of D the first derivative is discontinuous at the phase transition. For

the D = 6 results, the magnitude of the discontinuity at the phase transition is - 0.207.

We define an order parameter for a q-level Potts system as,

( =

7

8

8

9

1

q ' 1

q"1#

k

:

*ix,k

" :

*jx,q"k

"

, (9.9)

where i and j are adjacent sites. This quantity is non-zero when the Zq symmetry of the

Hamiltonian is broken. For the q = 3 Potts model, we have plotted the order parameter

and magnetization in the direction of the magnetic field in figs. 9.3 and 9.4 respectively.

At the phase transition we see a discontinuous jump in both quantities. Moreover, we

see a good convergence with D in the simplified PEPS results, and good agreement with

those generated with the full environment. Examining the insets, it can be seen that

it is only in the neighbourhood of the phase transition that the full environment yields

noticeably di!erent local observable properties. This contrasts greatly with the results for

the quantum Ising model where there was little convergence amongst the results obtained

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9.3 q = 3 Results 93

0 0.2 0.4 0.6 0.8 1 1.2 1.4−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

λZ

Ener

gy p

er la

ttice

site

simplified D = 3MF

0.871 0.8715 0.872 0.8725 0.873 0.8735 0.874 0.8745 0.875 0.8755 0.876−8

−6

−4

−2

0

2

4

6

8

10

12x 10−4

λZ

ΔEn

ergy

per

latti

ce s

ite

simplified D = 3simplified D = 6CTMRG D = 3

simplified D = 3λZ = 0.8732

CTMRG D = 3λZ = 0.8749

simplified D = 6λZ = 0.8716

lowest energy(CTMRG D = 3 & simplified D = 6)

λZ = 0.8722

Figure 9.1: (above) The energy of the Potts model, showing mean-field results and resultsfor a PEPS using the simplified update with D = 3. At this scale, the energies for D= 4, 5 and 6 ground states computed with the simplified update, or for D = 3 groundstates computed with the full environment (CTMRG), are indistinguishable. (below) Amagnified picture, with simplified update and CTMRG results, and energies relative tothe D = 3 CTMRG results. These results clearly depict a first-order transition as weapproach the transition from above (dotted) and below (dashed). The points where thetwo curves cross marks the phase transition. We have labeled such points for a few valuesof D and also the point corresponding to the crossing of the lowest energy solutions formagnetic fields above (simplified PEPS, D = 6) and below (CTMRG, D = 3) the phasetransition.

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94 The Quantum Potts Model

0.865 0.87 0.875 0.88 0.885

−1.85

−1.8

−1.75

−1.7

−1.65

−1.6

−1.55

−1.5

−1.45

λZ

dE/dλ Z

simplified D = 3simplified D = 4simplified D = 5simplified D = 6CTMRG D = 3

Phasetransition

Figure 9.2: Plot showing the first derivative of the energy per lattice site with respect tothe external field, %Z , as determined by a finite di!erence method. The vertical dottedline marks the phase transition. The derivative is clearly discontinuous at this point. Thedashed plots show the trajectory of the derivatives after the crossing.

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9.3 q = 3 Results 95

0.2 0.4 0.6 0.8 1 1.2

0

0.2

0.4

0.6

0.8

1

λZ

Θ

simplified D = 3simplified D = 4simplified D = 5simplified D = 6CTMRG D = 3MF

0.8 0.82 0.84 0.86 0.88 0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 9.3: Plot showing the order parameter of the q = 3 Potts model as a function ofexternal field, %Z .

with the simplified update, and it was only with the full environment that our results

agreed well with other numerical results. Furthermore, the Potts model mean-field theory

estimation of the order parameter - which involves no entanglement - is comparatively

much closer to the best PEPS estimate. This suggests there is far less entanglement in

the ground state of the q = 3 quantum Potts model than the quantum Ising model.

Fidelity Diagram

The fidelity diagram for the q = 3 Potts model is shown in fig. 9.5. In comparison with

the corresponding diagram for the Ising model (fig. 7.5), the fidelity diagram of the q = 3

Potts model exhibits a sharper drop o! at the phase transition. This clearly represents

a first-order transition between two phases with quite di!erent macroscopic properties.

By contrast, in the quantum Ising model, we observed a smooth roll-o! around the phase

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96 The Quantum Potts Model

0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.5

0

0.5

1

1.5

2

λZ

σZ

simplified D = 3simplified D = 4simplified D = 5simplified D = 6CTMRG D = 3MF

0.8 0.82 0.84 0.86 0.881.31.41.51.61.71.81.9

Figure 9.4: Plot showing the magnetization of the q = 3 Potts model in the direction ofthe magnetic field, as a function of %Z .

transition and a characteristic pinching at the phase transition.

Correlation Functions

For the q = 3 Potts model, we consider the following spatial correlation function given in

[FFGP07],

C)

/lx*

=1

(q ' 1)

;

q"1#

k

*'ix,k*

'i +

'l x

x,k

<

(9.10)

Where'

l x represents some displacement from the site i in the horizontal direction. In

figure 9.6 we plot the correlation function for the D = 3 ground states computed with the

full environment, along a row of the lattice for separations of up to 20 sites.

The correlation length along a row can be defined as in [Bax82],

4 =1

ln)

"1"2

* , (9.11)

Page 125: Studies of Infinite Two-Dimensional Quantum Lattice …Studies of Infinite Two-Dimensional Quantum Lattice Systems with Projected Entangled Pair States By Jacob Jordan B.Eng. (Hons

9.3 q = 3 Results 97

!Z

1

!Z2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0.7

0.75

0.8

0.85

0.9

0.95

1

0

0.5

1

1.5

0

0.5

1

1.5

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

!Z

1!Z

2

0.75

0.8

0.85

0.9

0.95

1

Figure 9.5: Fidelity diagram for the q = 3 Potts model, computed from D = 3 groundstates evolved with the simplified update.

where &1 and &2 are the eigenvalues of the column transfer matrix of largest and second-

largest magnitude respectively. It should be noted that if our system possessed a con-

tinuous phase transition, the transfer matrix would be degenerate in its first and second

eigenvalues at the critical point. This means that the correlation length is infinite at the

phase transition in, for example, the quantum Ising model. In the inset of figure 9.6, we

plot the correlation length against the magnetic field for D = 3 ground states computed

with the full environment and & = 10 (blue), 20 (red) and 30 (green). One can see that

the correlation length appears finite and well converged in & at the phase transition, where

4 - 0.56. However, since we have not computed ground states for higher D and the full

environment, we cannot say our results are converged in D. As such, our results for the

correlation function and correlation length can only be considered a first approximation.

Entanglement Entropy

The entanglement entropy for a pair of neighbouring sites is shown as a function of the

external field, %Z in fig. 9.7. Here, the solid lines trace the entanglement entropy of

the lowest energy eigenstate of the Hamiltonian. The dotted lines indicate the trajectory

of the entropies for the two sectors after the spectral crossing. The phase transition is

marked by the vertical dotted black line. Here it can be seen that there is a sharp jump

in the entanglement entropy between the two lowest eigenstates at the phase transition.

Furthermore, the transition point does not correspond to a crossing of the entanglement

Page 126: Studies of Infinite Two-Dimensional Quantum Lattice …Studies of Infinite Two-Dimensional Quantum Lattice Systems with Projected Entangled Pair States By Jacob Jordan B.Eng. (Hons

98 The Quantum Potts Model

100 10110−6

10−5

10−4

10−3

10−2

10−1

100

No. lattice sites, | i − j |

C(|

i − j

|)

0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.87

0.4

0.5

0.6

0.7

λZ

ξλZ = 0.5

λZ = 0.7

λZ = 0.8

λZ = 0.85

λZ = 0.87

λZ = 0.872

Figure 9.6: Two point correlation function of the q = 3 Potts model for various values ofthe external field, %Z . These have been computed from D = 3 ground states computedwith the full environment and & = 30. (inset) The correlation length of the q = 3 Pottsmodel, as defined in equation 9.11 against %Z for D = 3 ground states computed with thefull environment and & = 10, 20 and 30.

entropies of the two eigenstates, nor does it represent a point at which these entropies

peak. In the inset, we see a slightly increased entropy for the CTMRG solution. As we

move away from the phase transition, the CTMRG ground state entropy converges with

the simplified update entropy.

9.4 q = 4 Results

We now perform the same simulations for the q = 4 Potts model. We again use the

simplified update to generate ground states corresponding to PEPS states with D = 4, 5

and 6. Additionally, we perform a simulation guided by the full environment (CTMRG)

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9.4 q = 4 Results 99

0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

λZ

S 2

simplified D = 3simplified D = 4simplified D = 5simplified D = 6CTMRG D = 3

0.82 0.84 0.86 0.88 0.9 0.92 0.940.1

0.2

0.3

0.4

Figure 9.7: Entanglement entropy of the q = 3 Potts model. The dashed lines plot thetrajectory of the entropy after the phase transition.

for D = 4. The energy plot is shown in fig. 9.8. It is quite evident here that there

exists a first-order transition, with a discontinuity in the first derivative of the energy at

%Z,pt - 0.61. In the inset of fig. 9.8, one can see once again that the first derivative of the

dotted and solid energy lines di!er at the crossing. Moreover, in contrast to the q = 3

transition in fig. 9.1, we see that the results for di!erent D are almost indistinguishable.

The system appears well converged for a simplified update with D = 4.

The first derivative of the energy is plotted in fig. 9.9. At the phase transition, marked by

the vertical black dotted line, the di!erence in the magnitude of the derivative immediately

above and below the transition is - 0.694. This exceeds the same result for the q = 3

Potts model, supporting the accepted notion that phase transition in the q = 3 Potts

model is weakly first order in comparison.

The order parameter and magnetization in the direction of the external field are shown

in figs. 9.10 and 9.11 respectively. Once again we observe that the mean-field solution is

very close to the PEPS results.

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100 The Quantum Potts Model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3

−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

λZ

Ener

gy p

er la

ttice

site

simplified D = 4simplified D = 5simplified D = 6CTMRG D = 4MF

0.58 0.59 0.6 0.61 0.62 0.63−2

−1.95

−1.9

−1.85

Figure 9.8: (above) The energy of the q = 4 Potts model, showing mean-field results andresults for a simplified PEPS with D = 4, 5, 6. One can see that even magnified, theresults for di!erent D are almost indistinguishable. The results here also seem to agreewell with the mean-field theory results.

Fidelity Diagram

The fidelity diagram for the q = 4 Potts model is shown in fig. 9.12. In comparison to

the equivalent figure for the q = 3 model, one can see that the drop o! around the phase

transition is much more severe, and that surface for %1Z ,%2

Z < %Z,pt is much flatter. This

is in line with a phase transition that is strongly first-order.

Correlation Functions

In figure 9.13, we plot the correlation function for the q = 4 Potts model. Inspection

of the correlation function shows the q = 4 correlator decaying slightly faster than for

q = 3. In particular, it can be seen that at the phase transition, the q = 4 correlation

function appears to decay faster than the q = 3 correlation function. The inset shows a

good convergence of the correlation length (as defined in eqn. 9.11) with & and a final

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9.4 q = 4 Results 101

0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68−3

−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

λZ

dE/dλ Z

sPEPS D = 4sPEPS D = 5sPEPS D = 6PEPS CTMRG D = 4

Phase transition

Figure 9.9: The first derivative of the energy-peer-site of the q = 4 Potts model withrespect to the magnetic field, %Z , as calculated by a finite di!erence method

estimate of 4 - 0.51, a value smaller than that found for q = 3. However, once again we

cannot say that our results are converged in D and as such the accuracy of this result is

quite uncertain.

Entanglement Analysis

The entanglement entropy for a pair of neighbouring sites, S2, is plotted against the

external field, %Z , in fig. 9.14. The magnitude of S2 at the phase transition is comparable

to the q = 3 result in fig. 9.7. However, compared to the q = 3 plot, the entropy varies

less with increasing D (see inset). This once again suggests that the D = 4, result for the

q = 4 quantum Potts model is very close to the ground state.

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102 The Quantum Potts Model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λz

<σz>

simplified D = 4simplified D = 5simplified D = 6CTMRG D = 4MF

0.5 0.55 0.6

0.6

0.7

0.8

0.9

Figure 9.10: A plot of the order parameter of the q = 4 Potts model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

0

0.5

1

1.5

2

2.5

3

λz

<σz>

simplified D = 4simplified D = 5simplified D = 6CTMRG D = 4MF

0.55 0.6 0.65

1.5

2

2.5

3

Figure 9.11: A plot of the magnetization in the direction of the magnetic field for theq = 4 Potts model

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9.4 q = 4 Results 103

0

0.5

1

0

0.5

1

0.7

0.8

0.9

1

1.1

!Z

1!Z

2

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

!Z

1

!Z2

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Figure 9.12: The fidelity diagram for the q = 4 Potts model, using the simplified updateD = 4 ground states.

100 10110−6

10−5

10−4

10−3

10−2

10−1

100

No. lattice sites

C(|

i − j

|)

λZ = 0.51

λZ = 0.55

λZ = 0.59

λZ = 0.6

λZ = 0.61

0.52 0.54 0.56 0.58 0.60.2

0.3

0.4

0.5

0.60.6

λZ

ξ

Figure 9.13: Two point correlation function of the q = 4 Potts model for various values ofthe external field, %Z for ground states computed with the full-environment and D = 4.(inset) The correlation length of the q = 4 Potts model, as defined in equation 9.6 against%Z for & = 8, 16 and 24.

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104 The Quantum Potts Model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

λz

S 2

simplified D = 4simplified D = 5simplified D = 6CTMRG D = 4

0.55 0.6 0.650

0.2

0.4

0.6

Figure 9.14: Entanglement entropy for the q = 4 Potts model. The dashed lines show thetrajectory of the entropies after the transition.

9.5 Conclusion

The results of this chapter in conjunction with those from Chapter 7 provide some inter-

esting insights into the PEPS algorithm and its application to first-order and continuous

quantum phase transitions. These results reinforce the central idea in tensor network

theory - that the computational di"culty of simulating a quantum system is deeply con-

nected to the degree and nature of entanglement in the system. In this chapter, we

have considered the quantum Potts model with increasing local dimension, q, and some-

what paradoxically determined that the computational resources required to describe the

ground state of such systems reduces with increasing q.

The explanation for this lies in the fact that the ground states of these systems possess

vastly di!erent structures of entanglement. In the quantum Ising model, the phase tran-

sition is continuous and at the phase transition the system is critical and the ground state

possesses a large amount of entanglement. The spatial correlations decay as a power law.

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9.5 Conclusion 105

i) *)

dEd+Z

*

ii) Estimated 4 iii) Di!erence in iv) Percentage error

order parameter, in location,at PT iPEPS vs MFT of PT iPEPS vs. MFT

Ising (q = 2) 0 ,† 0.63 31.2%Potts q = 3 0.207 0.56 0.26 14.7%Potts q = 4 0.694 0.51 0.18 1.8%

Table 9.2: A summary of results for the quantum Ising model and q = 3 and q = 4quantum Potts models. One can see that i) the discontinuity in the first-derivative of theenergy increases with q, whilst ii) the correlation length decreases. In iii) and iv), it can beseen that as q increases, the accuracy of mean-field theory results improve, suggesting thatthe amount of entanglement in the ground state is decreasing. † An infinite correlationlength can not be reproduced by a PEPS computation of the kind suggested in thisthesis, however it is accepted that the quantum Ising model has a second-order phasetransition, and hence a diverging correlation length. The results for the quantum Pottsmodel correlation length are not converged in D, and should only be seen as indicative ofthe possible behaviour of 4 with q.

As a result, the system is rather poorly characterised by mean-field theory. Due to the

long-range nature of correlations in the ground state, an e!ective PEPS simulation of this

system requires the full environment in the tensor update. By contrast, as q increases,

the phase transition of the system changes to being of a first-order character, and the

amount of entanglement in the system decreases greatly. The systems become progres-

sively better described by mean-field theory as q increases, and for a PEPS solution, the

simplified update becomes increasingly e!ective, significantly reducing the computational

burden of accurately describing the ground state.

In table 9.2 we present a final summary of the results obtained for the quantum Potts

model with PEPS. For each q, we record i) the magnitude of the discontinuity in the

first derivative of the energy at the phase transition, ii) the correlation length, iii) the

di!erence in the order parameter as estimated by iPEPS and MFT at the iPEPS phase

transition and iv) the percentage error in the location of the phase transition as estimated

by iPEPS and MFT. In these key metrics, one observes that as q increases, the system

takes on an increasing first-order nature, and a corresponding decrease in the scale of

correlations and increase in the e!ectiveness of MFT results.

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106 The Quantum Potts Model

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Chapter 10

The J1-J2 Model

10.1 Introduction

The study of geometrically frustrated spin systems remains a great challenge in compu-

tational quantum many-body physics. Quantum Monte Carlo - a very e!ective method

for non-frustrated systems - su!ers from the well-known sign problem when applied to

frustrated systems [HS00]. Furthermore, exact techniques such as exact diagonalization

are still limited by poor computational cost scaling with system size. In light of these

di"culties, determining the e!ectiveness of tensor network approaches to such problems

is of considerable consequence.

Geometrically frustrated systems occur when the Hamiltonian contains terms that com-

pete with each other in such a way that the energetic minimization of one or more terms

is in disagreement with the energetic minimization of other terms, as a result of the na-

ture of the interactions and the geometric structure of the system. Consider the simple

classical example shown in fig. 10.1. Here, we have three sites arranged on a triangle.

Each pair of sites interact in accordance with either a ferromagnetic (fig. 10.1i) or anti-

ferromagnetic Ising interaction (fig. 10.1ii). One can see that in the ferromagnetic case,

we can simultaneously minimise the energy contributions from each of the links with-

out conflict. In the anti-ferromagnetic case, we observe that minimising with respect to

the link A-B means that we find it di"cult to choose the appropriate spin orientation

for site C. Whilst this simple treatment is illustrative of the problem, it does not fully

reproduce its complexity. Considering the statistical properties of such a system in the

thermodynamic limit at finite temperature [Wan50], one begins to sense the immensity

107

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108 The J1-J2 Model

Figure 10.1: A simple example of a frustrated system. The three sites in i) are actedupon by a ferromagnetic interaction. The minimum energy state is straightforward. Thethree sites in ii) are acted upon by an anti-ferromagnetic interaction. Whilst it is easy tominimise the energy for the link A-B, it is not clear then what state spin C should take.

of the problem. For frustrated quantum spin systems at zero temperature, quantum fluc-

tuations give rise to enormously complex descriptions of the ground state [ML04]. In

modern condensed matter physics, frustrated models such as the XY model and Heisen-

berg antiferromagnet on the triangular lattice [And73, JPPU80], the Heisenberg Kagome

antiferromagnet [Sac92] and the Shastry-Sutherland model [SS81] continue to generate

great interest. For example, in studying the phase diagram of oxide superconductors,

Anderson connected magnetic frustration to the mechanism of high-Tc superconductiv-

ity [And87]. More generally, frustrated systems arouse the possibility of systems with

rich phase diagrams containing exotic and elusive types of ground states, and this alone

motivates their study.

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10.2 The J1-J2 Hamiltonian 109

10.2 The J1-J2 Hamiltonian

In this chapter, we study the J1-J2 model, also known as the frustrated Heisenberg an-

tiferromagnet on the square lattice, one of the most fundamental models of a frustrated

quantum spin system.

The Hamiltonian for the J1-J2 model is,

H = J1

#

<i,j>

Si · Sj + J2

#

<<i,j>>

Si · Sj (10.1)

where /Si = (Sxi , Sy

i , Szi ). The J1 term describes an isotropic antiferromagnetic nearest

neighbour interaction and the J2 term describes a next-to-nearest neighbour interaction.

To elucidate the competing forces, consider a classical treatment of the model at zero

temperature. For J2J1

< 0.5, the J1 term dominates and the system lies in a Neel order

ground state with (2, 2) periodicity. For J2J1

> 0.5, the J2 term dominates and the system

breaks into two diagonal sub-lattices each with its own Neel order. Globally the system is

said to exist in a collinear phase, with (2, 0) or (0, 2) periodicity. At J2J1

= 0.5 a classical

critical point exists.

We wish to study the e!ect of quantum fluctuations on this picture. Furthermore, we wish

to study the suitability of PEPS for simulating such a system. Previous investigations

have helped develop something of a general picture for the phase diagram. In the limits

of low-J2J1

and high-J2J1

, we expect two respective phases with Neel and collinear order.

These phases are depicted in fig. 10.2. In each case, the global SU(2) symmetry of the

Hamiltonian is broken down to a U(1) symmetry, and the phases may be distinguished

by the momentum space representation of the spin-spin correlation function [MVC09],

S(/q) =1

N2

#

kl

ei,q(,rk",rl)=

/Sk · /Sl

>

(10.2)

where /rk and /rl are the spatial lattice vectors of two sites k and l. For the Neel ordered

state, S(/q) peaks at /q = (2,2), whilst for the collinear state, S(/q) peaks at either /q = (0,2)

or /q = (2, 0). S(/q) is known as a structure factor, as computing it for these values of /q

can tell us whether our state has Neel or collinear order.

In between these phases, the increased e!ects of frustration make it di"cult to classify

the ground state, although finite-size scaling of exact diagonalization results has suggested

that at J2J1- 0.35-0.4, the system undergoes a continuous phase transition from the Neel

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110 The J1-J2 Model

Figure 10.2: (left) An example of a Neel order state. The momentum space representationof the spin-spin correlation function peaks at /q = (2,2). (right) An example of a collinearstate. Here, since the diagonal interactions dominate, the system splits into two Neelordered sub-lattices. Here, the red spins form one sub-lattice, marked by the pink dottedlines, and the black spins form another sub-lattice, marked by the grey dotted lines. S(/q)peaks at /q = (0,2) or /q = (2, 0). In this example we show the /q = (0,2) state.

ordered state to a paramagnetic state, which remains until a first-order transition to

the co-linear ordered ground state at around J2J1- 0.6-0.65. A general picture of the

phase diagram is shown in figure 10.3. There have been several candidate ground states

suggested for this paramagnetic region, including:

1. A spin-liquid with no broken translational or rotational (SU(2)) symmetries.[FKK+90]

2. A dimerized ground state, where the translational symmetry is broken, but the

SU(2) symmetry preserved.[GSH89, SN90, SWHO99, ZU96, KOSW99]

3. A twisted, or spiral ordered ground state.[DM89]

4. A chrial (parity breaking) ordered ground state.[SZ92]

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10.2 The J1-J2 Hamiltonian 111

Figure 10.3: The generally accepted phase diagram for theJ1-J2 model.

We aim in this chapter to present a first picture of what an iPEPS treatment of the

system suggests for the J1-J2 model in the thermodynamic limit. It is generally thought

that spin-liquid ground states will be di"cult to converge to with a PEPS simulation, due

to the fact that the PEPS algorithm quite systematically breaks translational symmetries

by imposing a multi-site unit-cell structure. Additionally, there has been much emphasis

on novel crystalline structures in the intermediate region, and since this is a preliminary

study of the model, we will restrict ourselves to answering a few select questions. Firstly,

we wish to determine if the ground state phase diagram computed with iPEPS displays

an intermediate paramagnetic region. If so, we wish to determine approximately where

the boundaries of such a region are located. Lastly, we wish to investigate whether

the intermediate region is characterised by any of the suggested dimerized ground state

patterns. Results for the finite J1-J2 model with PEPS, for lattices with up to 14 " 14

spins and open boundary conditions have already been published in [MVC09]. The authors

there detected some evidence of a dimer ground state in the intermediate region, but could

not reach a definite conclusion. In the following sections, we will define two of the most

commonly suggested dimer patterns. Then, we will describe some important algorithmic

considerations, before presenting our results for the phase diagram.

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112 The J1-J2 Model

10.3 Columnar and Plaquette Ordered Ground States

Many authors have suggested the existence of ground states that resemble a regular ar-

rangement of two-body singlets. The most common types of order suggested are the

columnar dimer order and the plaquette resonating valence bond (RVB) order. We illus-

trate these two basic foundational states in fig. 10.4.

In some schemes, such as the reduced subspace exact diagonalization in [MLPM06], the

ground states are computed and their properties are checked for agreement with the

properties of these candidate ground states. Series expansion schemes [SWHO99] have

gone even further, and taken as a starting point a Hamiltonian that gives rise to one of

the candidate ground states. That is, the J1-J2 Hamiltonian is considered in the form

(10.3),

H+ = H$ + %H" (10.3)

where the ground state of H$ is either the columnar dimer or plaquette RVB state, and

as % is increased from 0 to 1, we recover the J1-J2 Hamiltonian. In this iPEPS study, we

initialise our imaginary-time evolution with both random initial states and initial states

biased toward some dimer order. We also define schemes for which the structure of the

tensor network quite deliberately favours a dimerized state in a columnar arrangement.

10.4 Algorithmic Considerations

The presence of a next-to-nearest neighbour interaction in our Hamiltonian poses a chal-

lenge for our iPEPS formalism. Firstly, due to the diagonal interactions, it is clear that

the minimal unit cell will be a 2-by-42 block A-B-C-D. More importantly, in other models

we have studied, there has been a PEPS bond allocated to every term in the Hamilto-

nian. Enforcing this for the J1-J2 model requires that each PEPS tensor has 8 shared

bonds. Each tensor then contains D8d components, and contracting the environment for

such a network is more complex and more expensive computationally. A standard iPEPS

algorithm, using the full environment update is simply not a!ordable. For this reason,

we needed to consider modifications to this approach. Specifically, these are:

1. A square (4-bond) iPEPS and the full environment update. The Hamiltonian is

expressed in four-body terms operating on a 2x2 plaquette. For each update, the

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10.4 Algorithmic Considerations 113

Figure 10.4: The suggested VBC states. i) A columnar dimer arrangement. ii) A plaquetteRVB arrangement iii) The plaquette RVB for a single plaquette in terms of nearest-neighbour singlets

four PEPS tensors are updated based on the environment surrounding the plaquette.

This scheme is of broader interest, as it can be used for Hamiltonians with general

4-site plaquette interactions.

2. Using a PEPS with 8 shared bonds on each tensor, but using the simplified update

scheme to perform imaginary time evolution.

3. A simplified update scheme for a square PEPS (i.e. 4 shared bonds) with an exten-

sion allowing the update of diagonal links. This approach was used in a study of

2D fermionic models with next-nearest neighbour interactions. [CJV10]

4. Joining adjacent pairs of sites in a way that would be favourable if the ground state

possessed columnar order, and once again using 4-body plaquette terms.

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114 The J1-J2 Model

Figure 10.5: The four alternative PEPS variants we have chosen. i) A square PEPSwith updates on 2x2 plaquettes. There are four distinct plaquettes, each one marked bya di!erent colour. The Hamiltonian is written in terms of four-site plaquette operatorshp1, hp2, hp3 and hp4. ii) An iPEPS with a bond for each Hamiltonian interaction. Tominimise the computational load, the simplified update is used and so each link bears adiagonal % matrix. iii) A square PEPS updated by a simplified update capable of handlinginteractions between next-nearest neighbours. iv) A scheme designed to favour a columnarordered ground state. Here, pairs of sites are grouped into tensors in a columnar fashion.Bonds along a column are of dimension D1, whereas bonds connecting columns are ofdimension D2. The Hamiltonian is written in terms of four-site operators. There are twoflavours of update, those on columns (red) and those between columns (yellow).

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10.4 Algorithmic Considerations 115

These are shown and in figure 10.5. For scheme 1 (10.5i), the cost of computing the

environment (via the CTM method) at each timestep scales as &3D6 + &2D6d - the same

as for a PEPS with A-B-B-A periodicity. This means that environments of D = 2, 3 and 4

PEPS should be able to be computed with the method in reasonable time. However, the

four body conjugate-gradient update scales with a one-time cost of &3D7d2 + &2D10d4 +

D8d8, followed by a per-iteration cost of &3D6+&2D8+D10d4. This means that simulating

with a D = 4 PEPS becomes excessive. Furthermore, since correlations between next-

nearest neighbour sites flow along horizontal and vertical links of the PEPS, the correlation

carrying bonds become quickly saturated. As a result, it was seen that results for D = 2

converged poorly and that for this scheme the only reasonable value was D = 3.

For scheme 2 (10.5ii), the simplified update of the PEPS scales as D9d2 + D3d4, but to

compute observables, one needs to develop an alternative CTMRG scheme to take into

account renormalization of diagonal bond indices. This can be done at a cost per CTMRG

step of &3D18 + &2D15d, which is excessive, and so we use a reduced tensor rank scheme

which scales as &3D12 + &2D13d. Furthermore, the cost of calculating a four-site reduced

density operator from the A-B-C-D unit-cell scales as &3D20 + &2D24d + &2D16D8. Even

though this only needs to be performed once, it is still a very costly computation and as

such the scheme is only computationally viable for a bond dimension D = 2.

For scheme 3 (10.5iii), the cost of each diagonal update scales as D6d4+D5d6. This allows

one to converge PEPS for comparatively large values of D. The one-time computation of

the environment scales as &3D6 + &2D6d. The computation of four-site reduced density

operators scales as &3D6d4 + &2D4d8. For this scheme, one can converge and compute

observables for up to D = 6. The disadvantage of this scheme is that it cannot reproduce

long-range correlations as each update is e!ectively local. We saw in Chapters 7 and

9 that the degree to which this e!ects the accuracy of results depends on the scale of

entanglement in the system and, by virtue of this, on the order of any quantum phase

transitions.

Scheme 4 (10.5iv) is biased towards representing the correlations present in states with a

columnar order and so will work best if this is the type of order in the ground state. It uses

an environment to update the PEPS, with each step of the CTMRG algorithm scaling

once again as &3D6 + &2D6d2. The four-site reduced density operator can be computed

with leading order cost &3D6d4+&2D8d5+&2D4d8. Once again, the Hamiltonian is written

in four-site terms with the most costly update being for the sites bounded by the yellow

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116 The J1-J2 Model

Scheme Cost of contracting Cost of performingenvironment or computing update

reduced density matrix1 Environment: S1(&3D6 + &2D6d) &3D7d2 + &2D10d4 + D8d8

Four-site RDM†: &3D6d4 + &2D4d8 +L1(&3D6 + &2D8 + D10d4)2 Environment†: S2(&3D12 + &2D13d) D9d2 + D3d4

Four-site RDM†: &3D20 + &2D24d + &2D16D8

3 Environment†: S3(&3D6 + &2D6d) D6d4 + D5d6

Four-site RDM†: &3D6d4 + &2D4d8

4 Environment: S4(&3D6 + &2D6d2) D8d12 + &2D10d8 + &3D8

Four-site RDM†: &3D6d4 + &2D8d5 + &2D4d8 +L4(&2D8 + &3D6 + D10d8)

Table 10.1: Leading order computational cost of four PEPS schemes for the J1-J2 model.Operations that need only be computed once per simulation are marked with †. Theparameters S and L represent the number of steps for contracting the environment andperforming an iterative update respectively. Note that there is not necessarily equivalencebetween the parameter D across the schemes. For instance, as scheme 2 dedicates a bondto every Hamiltonian interaction, it is likely that it will require a relatively smaller valueof D to represent the correlations in the system. However, it also possesses a much highercomputational cost scaling in D.

square in fig. 10.5iv). During this conjugate gradient routine, we incur a one-time cost of

D8d12 + &2D10d8 + &3D8, plus a per-gradient computation cost of &2D8 + &3D6 + D10d8.

The computational costs of the four schemes are summarised in Table 10.1.

In figure 10.6 we plot a brief comparison of the results obtained from the techniques de-

scribed above. One can see that in the intermediate phase, the simplified update produces

the lowest energy results. One can also compare these results with the columnar (*) and

plaquette RVB (+) series expansion results of [SWHO99] and see that the D = 6 PEPS

produces an energy at the point J2/J1 = 0.5 that is 0.5% lower than the columnar expan-

sion result and 2% lower than the plaquette RVB expansion result. For these schemes, we

also tried biasing the imaginary-time evolution towards the columnar dimer and plaquette

RVB ordering by starting in the states given in fig. 10.4. However, initialising the states

in such a way led to ground states with slightly higher energies.

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10.5 Results 117

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.7

−0.65

−0.6

−0.55

−0.5

−0.45

J2/J1

Ener

gy p

er la

ttice

site

Scheme 1, D = 3Scheme 2, D = 2Scheme 3, D = 6Scheme 4, D1 = 4, D2 = 2

0.45 0.5 0.55−0.51

−0.5

−0.49

−0.48

−0.47

Figure 10.6: A plot comparing the energies given by the four PEPS algorithm variants.The + and * symbols mark the energies for series expansions around plaquette RVB andcolumnar dimer ground states respectively [SWHO99].

10.5 Results

10.5.1 Energy and structure factors

Using scheme 3 as our method for characterizing the J1-J2 model, we derive the phase

diagram for various values of D. In figure 10.7, we show the plot of the energy-per-site

diagram for various values of D.

We next plot the Neel (fig. 10.8) and collinear (fig. 10.9) structure factors in order to

demonstrate that the results comply with the well-understood behaviour in the low-J2 and

high-J2 limits. We use the Neel and collinear structure factors as defined in [MVC09].

For an infinite translationally invariant system, the structure factors reduce to:

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118 The J1-J2 Model

0 0.2 0.4 0.6 0.8 1−0.7

−0.65

−0.6

−0.55

−0.5

−0.45

J2/J1

Ener

gy p

er la

ttice

site

, e0

Plot showing J1J2 energies vs J2

D = 3D = 4D = 5D = 6

3 4 5 6

−0.495

−0.49

−0.485

D

e 0

Figure 10.7: A plot of the energy per lattice site vs J2J1

for various values of D. (inset)

Convergence of the energy per lattice site with the bond dimension, D, at J2J1

= 0.5.

S =1

16

;

2#

x=1

2#

y=1

2#

x"=1

2#

y"=1

!xyx"y"Sxy·Sx"y"

<

(10.4)

where x, y, x’ and y’ define the relative row and column positions within some A-B-

C-D plaquette. For the Neel structure factor, !xyx"y" = ej(x+y"x""y").. For the collinear

structure factor, !xyx"y" = ej(x"x"). or !xyx"y" = ej(y"y")., depending on whether the state

breaks into a (0, 2) or (2, 0) order.

In our plot it can be seen that as J2 increases from 0, the Neel order reduces steadily and

in the intermediate region the state displays little or no evidence of Neel order. Likewise,

as we decrease J2 from J2 = 1, there is a steady reduction in the amount of collinear

ordering in the ground state. For D = 5, it appears as though Neel order disappears

by J2 = 0.55. However, for D = 6, remnants of Neel order reappear. The D = 6 plot

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10.5 Results 119

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

J2/J1

S(π,

π)

D = 3D = 4D = 5D = 6

Figure 10.8: A plot of the Neel order parameter vs J2J1

for various values of D

shows a curious hump at around J2 = 0.4 that is not evident for other values of D. This

behaviour is interesting as it is occurs in a region where a continuous phase transition is

widely predicted. Some studies of the phase diagram predict as many as three continuous

phase transitions between J2 = 0.34 and J2 = 0.5, [SOW01]. Recall that for the quantum

Ising model, the simplified update estimated the critical magnetic field and the local

observable properties in the neighbourhood of the phase transition poorly (see fig. 7.3).

It is possible that this hump is evidence of a continuous transition, that - for a simplified

PEPS scheme - would only be properly described with very large D. Figure 10.9 shows the

collinear structure factor. For increasing D, it appears as though the collinear structure

factor disappears at progressively higher values of J2.

A most interesting observation from our simulations is the dependence of the final ap-

proximation to the ground state on the initial state. As mentioned previously, initialising

the simulation in one of the dimer states did not lead to an approximation to the ground

state with a lower energy than a randomly initialised simulation. This suggests that the

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120 The J1-J2 Model

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

J2/J1

S(π,

0)

D = 3D = 4D = 5D = 6

Figure 10.9: A plot of the co-linear order parameter vs J2J1

for various values of D

energy minimization becomes trapped in local minima. The magnetic order parameters of

the ground states from each initialization agree that the system exists in a Neel ordered

phase for low-J2J1

and a collinear ordered phase for high- j2J1

. In between, the magnetic

structure factors decay and there is the possibility of a paramagnetic phase appearing for

increasing D. For random initial states, the actual values of the magnetic observables

can fluctuate over multiple runs of the energy minimization and a smoother picture is

found by initializing a D = Q simulation with the PEPS found for D = Q' 1. However,

this initialization also results in an energy that is generally slightly higher. Additionally,

the update itself can be sensitive to a lattice rotational symmetry, or explicitly break the

symmetry (see [CJV10]) but this seems to have no observable e!ect on the determined

ground states.

This all points towards the PEPS selecting from a rich set of states with similar energies.

For the values of D that we can access, it appears as though the ground state picture

is di"cult to determine and that the imaginary-time evolution often becomes trapped in

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10.5 Results 121

local minima. A more in-depth study may incorporate symmetries in order to restrict

the ground states to subspaces of the Hilbert space invariant under global U(1) or SU(2)

operations. However, as a first approximation, it can be said that our method produces

a ground state energy in the intermediate region that is lower than that obtained by

series expansion around suggested VBC configurations. So the question we might like to

answer is: to what degree does our solution show properties consistent with a columnar

or plaquette ordered VBC?

10.5.2 Columnar and Plaquette Order Parameters

Here, we aim to use the order parameters suggested in [MVC09] to determine if the ground

states we find in the intermediate region are of columnar dimer or plaquette RVB form.

Columnar order

To detect columnar order in the intermediate regime, we compute the spin-spin correlator/Si · /Sj for all neighbouring sites i and j. Columnar order would be evidenced by particular

parallel pairs of links having a much stronger correlation than other neighbouring links.

We show in fig. 10.10 the nearest-neighbour correlation values for the links of an A-B-C-D

plaquette for the D = 6 ground states. For low J2J1

, one can see that the Neel ordered

ground state exhibits roughly equal correlation values on all vertical and horizontal links.

For J2J1

= 1, one can see that the correlations along horizontal links are roughly equal

and contribute negatively to the energy. Meanwhile, the correlations on vertical links

contribute positively to the energy. This is indicative of (2, 0) collinear order. In between,

there appears to be an abrupt change between the two magnetic orders consistent with a

first-order phase transition, and little evidence of a columnar dimer order appearing.

Plaquette order

To detect a plaquette RVB state, we compute the expectation value of the cyclic permu-

tation operator,

QABCD =1

2

-

PABCD + P"1ABCD

.

(10.5)

where PABCD is a cyclic permutation operator on a 2x2 block of sites. For a pure plaquette

state, the plaquette order parameter should be 1 on the four sites of the plaquette, and

1/8 elsewhere. Fig 10.10 shows the plaquette order parameters for various values of J2.

One can see that across the entire phase diagram, there is no evidence of a translational

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122 The J1-J2 Model

symmetry breaking of the kind that would suggest a plaquette RVB type order. Instead,

for a given J2J1

the plaquette order parameter is similar on each plaquette in the system.

There is a steady increase in the plaquette order parameter from the Neel ordered state

at J2J1

= 0 before suddenly dropping at J1J2- 0.6 in a manner consistent with a first order

phase transition.

10.5.3 Entanglement Entropy

The plot of the four-site entanglement entropy, averaged over all four plaquettes, is shown

in fig. 10.11. It is di"cult to extract a general picture of the J1-J2 entanglement from this

plot, other than to say that the entanglement entropy increases as J2 increases, and peaks

at the point where the energy plot suggests a first-order phase transition. At J2 = 0.35

there again appears to be a slight rise in the entanglement entropy of the D = 6 results,

which again could be a very early indication of a continuous phase transition in this region.

10.6 Concluding Remarks

In this chapter, we studied the J1-J2 model with the iPEPS algorithm. We presented en-

ergies for several versions of the algorithm, and found that for simulations of an acceptable

duration, a version of the simplified PEPS algorithm for a square PEPS with diagonal

interactions gave best results. One of our main objectives here was to test the hypothesis

proposed elsewhere that in the regime where the system is most frustrated, the ground

state exhibits a tendency towards a valence bond crystal ordering. We examined our

ground states for tendency toward either a columnar dimer or plaquette RVB ordering,

but could not see the onset of either, even when the system was coerced to favour such

an order. Instead, our results favoured a first order phase transition from the Neel order

ground state to the collinear ground state at J2 - 0.6' 0.625.

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10.6 Concluding Remarks 123

Figure 10.10: A plot of the nearest-neighbour spin-spin expectation values and plaquetteorder parameters. The labels in near the links give the expectation value of the spin-spincorrelator, /Si · /Sj, for each of the eight distinct links. The labels enclosed in the boxgive the expectation value of the plaquette order parameter, QABCD, for each of the fourdistinct plaquettes. As can be seen, there is no tendency toward any VBC order, but anabrupt change from Neel order to collinear order.

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124 The J1-J2 Model

0 0.2 0.4 0.6 0.8 13

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

J2/J1

S 4

D = 3D = 4D = 5D = 6

Figure 10.11: A plot of the entanglement entropy for the J1-J2 model.

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Chapter 11

Geometric Entanglement

11.1 Introduction

How well can we represent the ground state of a strongly correlated quantum many-body

system without using any quantum correlation at all? This insightful question lies at the

root of a well established method in condensed matter physics: the mean field (MF) theory

approach. Using product states as a first step to study quantum many-body systems can

provide some qualitative and quantitative answers about the behavior of the system at

hand.

For an N -body system governed by a Hamiltonian H, the mean-field approximation to

the ground state is the product state that minimises the energy, i.e.

eH * $$H|H|$H# = min!$$|H|$# (11.1)

Here, |$# = |0[1]# ( |0[2]# ( · · ·( |0[N ]# is a product state of the N bodies and |$H# is the

mean-field solution.

While the energy obtained in this way is still higher than the true ground state energy

(unless the ground state is a product state), it is the best possible estimation of the

ground state energy that can be achieved just by using a product state. Nevertheless,

this approach may not be satisfactory if the aim is to reproduce other properties of the

system, such as local order parameters.

An alternative question to ask, motivated by the notion of the fidelity of two quantum

states, is: what is the product state that maximises the overlap with the ground state,

125

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126 Geometric Entanglement

|%0#? That is, we want to find the product state |$G#, such that

&max(%0) * |$$G|%0#| = max!

|$$|%0#|. (11.2)

Here, we call |$G# the closest product state to |%0#.

If we can determine this product state, we can ask several more interesting questions, such

as: In what way does such a state di!er from the mean-field solution, if at all? If they

are di!erent, how do their estimation of the energy and order parameter, among other

quantities, compare? Answering such questions provides objective feedback on the ability

of mean-field theory to describe strongly correlated systems. Also, recall from section

3.2.1 that the overlap between a given state and its closest product state gives rise to a

uniquely macroscopic measure of entanglement - the so-called geometric entanglement of

the state:

E(%0) * 'log &2

max(%0)

N, (11.3)

Here, a geometric entanglement, E = 0, occurs when the ground state, |%0#, is a product

state and as E increases, the state is said to become more entangled [WDM+05, SOFZ10,

OW09]. Thus, determining |$G# sheds further light on the structure of entanglement in

the ground states of quantum many-body systems.

In this chapter, we present an algorithm for computing |$G# for ground states of infi-

nite 2D lattice models. Our approach is based around obtaining PEPS approximations,

|%0(D)#, to the ground state for increasing bond dimension D and observing for con-

vergent behaviour in the properties of |$G#. In particular, we present results for three

models: i) the quantum Ising model on the hexagonal lattice ii) the quantum Ising model

on the square lattice and iii) the q = 3 Potts model on the square lattice. Our approach

is thus broken into two parts:

1. Finding PEPS ground states of infinite, translationally invariant Hamiltonians.

2. Maximising the overlap between |%0(D)# and a variational product state |$G#.

We have extensively outlined the iPEPS algorithm for finding ground states of local

Hamiltonians in Chapter 5. We will not review it in this chapter, except to say that since

we are interested in the quantum Ising model, we make use of the CTMRG algorithm and

the full environment to converge to the ground state (refer to fig. 7.3 for justification). For

the second task, we are inspired by the fidelity-per-lattice-site outlined in [ZB07, ZPac06,

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11.2 Computing the Closest Product State 127

ZOV08]. These works formalised the idea that the closeness of infinite, translationally

invariant PEPS states could be captured in an intensive quantity. In the following sections,

we describe our variational algorithm for determining |$G#, before presenting results for

the three models we have studied.

Our approach allows us to touch on a particularly interesting idea from quantum informa-

tion theory - the relation between the monogamy of entanglement [CKW00, Ter04, OV06]

and the connectivity in a quantum many-body system. Monogamy of entanglement states

that entanglement in a many-body system is a shared resource. As the number of entan-

gled bodies increases, the degree to which any two bodies are entangled decreases. Thus,

one might expect for lattice systems that as the connectivity of the lattice increases, the

geometric entanglement of the ground state decreases. We will provide some tentative

numerical evidence suggesting that this is the case for a PEPS study of the quantum Ising

system on the hexagonal and square lattices. In doing so, we reflect on results for 1D

quantum systems [SOFZ10, OW09, OW10].

11.2 Computing the Closest Product State

We now outline a method to numerically compute |$G# e"ciently. Assume that we have

an infinite PEPS describing the ground state of a quantum lattice model. Assume that

this ground state is invariant under shifts by one site, or equivalently that our PEPS is

characterised by the same tensor A at each lattice site (see fig. 11.1a). We start our

search for |$G# with a random infinite product state |$0#. We make the assumption

that our product state is also translationally invariant, i.e. |$0# = |00#&$. Such an

assumption is vital from a computational point of view, but is not always theoretically

sound. It is relatively easy to design a ground state for which the closest product state

has periodicity on a longer scale than the ground state itself. However, our algorithms

rely on translational invariance and so we make this restriction. In this form, |$0# can be

described by a D = 1 PEPS made up of tensors 0 (see fig. 11.1b).

To update |$0#, we take the following steps:

(i) We define the distance between our product state, |$0#, and ground state, |%0#, by

the square error, !SE,

!SE = ||%0# ' |$0#|2 = $%0 | %0# ' $$0 | %0# ' $%0 | $0#+ $$0 | $0# (11.4)

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128 Geometric Entanglement

Figure 11.1: The variational algorithm for finding the closest product state on the squarelattice. For schematic clarity, we show the process for a system with periodic boundaryconditions. a) The PEPS is defined by a single tensor A with four bond . b) The productstate is defined by the vector 0. c) The tensor a formed as the contraction of A with theconjugate of 0. d) The updated (unnormalised) product state tensor µ may be expressedas the contraction of a tensor network that contains a at every site except one.

(ii) We choose to make a local modification to our product state at site m, |0[m]0 # & |0[m]

1 #,such that the square error is minimised. That is, we find the state 0[m]

1 such that /(SE

/-[m]#1

= 0.

It can be shown that an unnormalised solution for 0[m]1 is given by the vector µ1 as

computed by the contraction of the 2D tensor network in figure 11.1d.

(iii) We obtain the normalised solution, |0[m]1 # = |µ#/

?

$µ|µ# and form a new translation-

ally invariant product state, |$1# = |01#&$.

At each step k, the fidelity-per-lattice site limN#$

log((!k|#0))N is computed by the same pro-

cedure as outlined in [ZOV08]. Our algorithm iterates until this quantity converges.

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11.3 Results 129

11.3 Results

Our simulations have been performed for the following 2D models in the thermodynamic

limit:

(i) Quantum Ising model in a transverse field,

H = '#

(,r,,r")

*[,r]x *[,r"]

x ' %z

#

,r

*[,r]z , (11.5)

on the hexagonal and square lattices. In the above equation, *[,r]$ is the -th Pauli matrix

at site /r of the 2D lattice. According to quantum Monte Carlo calculations, the system

undergoes a quantum phase transition from a Z2-symmetric phase to a broken phase at

critical points %z,c 2 2.13 for the hexagonal lattice and %z,c 2 3.04 for the square lattice

[CA80].

(ii) Quantum 3-Potts model,

H = '#

(,r,,r")

)

*[,r]x *[,r"]2

x + *[,r]2x *[,r"]

x

*

' %z

#

,r

*[,r]z , (11.6)

where the Potts matrices at site /r are defined as

*[,r]x =

/

0

1

0 1 0

0 0 1

1 0 0

2

3

4, *[,r]

z =

/

0

1

2 0 0

0 '1 0

0 0 '1

2

3

4. (11.7)

We have studied this model in the square lattice, which is known to undergo a weakly

first order transition as a function of the transverse field %z [BBD08].

The di!erent results that have been obtained by simulating the above models can be

classified as follows:

11.3.1 Energy versus order parameter

In Figs. 11.2, 11.3 and 11.4 we show our results for the energy and order parameters

as computed with i) the MF approximation to the ground state, |$H# ii) the iPEPS

approximation to the ground state, |%0(D)#, for several values of D, and iii) the closest

product state to each of the iPEPS states, |$G(D)#. Remarkably, the two product state

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130 Geometric Entanglement

representations are seen to have very di!erent properties. The MF state energy is closer

to the PEPS energy than the closest product state. On the other hand, the closest

product state in each case more accurately represents the order parameter (and its critical

exponent), and in this sense one can say that it more accurately captures the phase

diagram of the system.

There are a few comments in order. Firstly, the two product states are least accurate

when the system is most entangled. For instance, the MF state predicts the order pa-

rameter most poorly near the phase transition, where the quantum Ising model is critical.

Similarly, the closest product state predicts the energy poorly in this region. By com-

parison, the same can be said for the quantum Potts model where, whilst there is no

critical point, there has been witnessed an increase in entanglement entropy approaching

the phase transition (see Chapter 9). However, in an overall sense, the product state rep-

resentations do a much better job of representing the quantum Potts model than they do

the quantum Ising model, and this may be attributed to the lesser degree of entanglement

in the quantum Potts system.

11.3.2 Geometric entanglement and local fidelities

In Fig. 11.5 we show our results for the density of global geometric entanglement E in the

thermodynamic limit for di!erent values of the bond dimension D. In the insets we show

di!erent local fidelities, calculated in the sense of [ZOV08], between the infinite PEPS for

several D and the MF approximations |$H# and |$G(D)#.

As a first comment, it should be noted that the results are not completely converged

with D, and this is especially true at the phase transition. Moreover, for the hexagonal

lattice, our results for D = 3 appear very slightly more entangled than the D = 4

results. This is most likely because the D = 4 PEPS could not be fully converged due

to computation time restraints. Alternatively, it could be that the variational algorithm

for finding |$G(D = 4)# becomes stuck in local minima. With this in mind, we will make

some tentative observations about the results.

Our simulations indicate that the density of global geometric entanglement peaks at the

critical point. This is in contrast to the second-order quantum phase transitions in the

1D quantum Ising model, where the peak in the geometric entanglement is displaced with

respect to the critical point, while its derivative is divergent at criticality [WDM+05].

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11.3 Results 131

1.95 2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4

−2.5

−2.4

−2.3

−2.2

−2.1

−2

−1.9

λZ

Ener

gy p

er s

ite

|ΦH>

|Ψ(2)>,|Ψ(3)>,|Ψ(4)>|ΦG(2)>

|ΦG(3)>

|ΦG(4)>

1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

λZ

Ord

er p

aram

eter

|ΦH>

|Ψ(2)>|Ψ(3)>|Ψ(4)>|ΦG(2)>

|ΦG(3)>

|ΦG(4)>

Figure 11.2: Hexagonal lattice quantum Ising model: (a) expectation value of the Hamil-tonian, and (b) order parameter.

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132 Geometric Entanglement

2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4−2.7

−2.6

−2.5

−2.4

−2.3

−2.2

−2.1

−2

−1.9

λZ

Ener

gy p

er s

ite

|ΦH>

|Ψ(2)>,|Ψ(3)>,|Ψ(4)>|ΦG(2)>

|ΦG(3)>

|ΦG(4)>

2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

λZ

Ord

er p

aram

eter

|ΦH>

|Ψ(2)>|Ψ(3)>|Ψ(4)>|ΦG(2)>

|ΦG(3)>

|ΦG(4)>

Figure 11.3: Square lattice quantum Ising model: (a) expectation value of the Hamilto-nian, and (b) order parameter. For the expectation value of the Hamiltonian, the resultsobtained with |%0(2)# and |%0(3)# are indistinguishable in the scale of the plot.

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11.3 Results 133

0.5 0.6 0.7 0.8 0.9 1 1.1

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

λZ

Ener

gy p

er s

ite

0.5 0.6 0.7 0.8 0.9 1 1.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λZ

Ord

er p

aram

eter

0.86 0.87 0.88 0.89 0.9−1.95

−1.9

−1.85

−1.8

−1.75

0.82 0.84 0.86 0.88

0.55

0.6

0.65

0.7

|ΦH>|Ψ(3)>|ΦG(3)>

|ΦH>|Ψ(3)>|ΦG(3)>

Figure 11.4: Square lattice quantum 3-Potts model: (a) expectation value of the Hamil-tonian, and (b) order parameter.

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134 Geometric Entanglement

Also, notice in the inset, that even though the closest product state has a higher overlap

with the actual PEPS than the MF product state, the MF state still has a quite large

local overlap (> 0.97) even close to the phase transition.

11.3.3 Monogamy of entanglement

Examination of of figs. 11.2, 11.3, and 11.5(a,b), seems to indicate that MF approxima-

tions are more accurate in the case of the quantum Ising model on the square lattice than

in the hexagonal one. This could be interpreted in the context of the monogamy of entan-

glement and as a consequence of the di!erence in coordination number, z. For the square

lattice, each node has four nearest neighbors (z = 4), whereas for the hexagonal lattice

each node has three (z = 3). The decrease in the density of geometric entanglement with

the coordination number z can be seen in Figs. 11.5(a,b), and also in Table 11.1, where

we show the value of E at the quantum critical point of the quantum Ising model in a

1D chain (from [WDM+05]) and the 2D hexagonal and square lattices. We record these

results in table 11.1. These lend some support to the monogamy of entanglement as it

applies to quantum lattice systems. However, a strong conclusion could only be reached

by:

1. Obtaining results that are converged in D.

2. Obtaining results for lattices with higher coordination numbers.

Unfortunately, due to the inhibiting computational cost scaling of the iPEPS algorithm

with D, and the way in which this rapidly worsens with increasing z, such a comprehensive

study seems out of reach at this point in time.

z = 2 (1D) z = 3 (2D hex.) z = 4 (2D sq.)

critical E 0.0631 0.0285 0.0226

Table 11.1: Density of global geometric entanglement at the critical point of the quantumIsing model for a 1D chain (z = 2), a 2D hexagonal lattice (z = 3), and a 2D squarelattice (z = 4).

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11.3 Results 135

0 0.5 1 1.5 2 2.50

0.005

0.01

0.015

0.02

0.025

0.03

λZ

E

1.8 2 2.2 2.4

0.97

0.98

0.99

1

λZ

fidel

ity p

er la

ttice

site

D = 2D = 3D = 4

0.5 1 1.5 2 2.5 3 3.5

0.005

0.01

0.015

0.02

0.025

λZ

E

b)

a)

2.6 2.8 3 3.2 3.4

0.970.975

0.980.985

0.990.995

1

λZ

fidel

ity p

er la

ttice

site

D = 2D = 3D = 4

Figure 11.5: (a) Geometric entanglement for the hexagonal lattice quantum Ising model.In the inset, we show the local fidelity between |%0(D)# and |$G(D)# for D = 2, 3 and 4and between |%0(2)# and |$H#; (b) Geometric entanglement for the square lattice quantumIsing model. In the inset, we show the local fidelity between |%0(D)# and |$G(D)# for D= 2 and 3 and between |%0(2)# and |$H#

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136 Geometric Entanglement

11.4 Concluding Remarks

In this chapter we described an algorithm for computing the closest product state, |$G#,to an iPEPS representation of a ground state. We then studied the properties of such a

state in the context of three quantum lattice models. This allowed us to make several

interesting observations. Firstly, we saw that |$G# has quite di!erent observable properties

to the mean-field theory solution, |$H#. However, it should be noted that |$G# is not an

alternative to mean-field theory as it cannot be computed from a Hamiltonian description

of the system. Obtaining |$G# relies on having an existing PEPS representation of the

ground state and inaccuracies in this PEPS representation may well manifest in |$G#.Secondly, we saw that the e!ectiveness of product states in representing the ground state

properties depends on the amount of entanglement present in the ground state. Ground

states with relatively little entanglement were well described by both mean field theory and

the closest product state. On the other hand, critical systems were comparatively poorly

approximated. This idea is formalised by a macroscopic measure of entanglement known

as the geometric entanglement. Lastly, we saw some results for lattice systems studied with

the techniques outlined in this chapter, suggesting that the peak geometric entanglement

for the quantum Ising model decreases with increasing coordination number. This could

be interpreted as evidence supporting the notion of the monogamy of entanglement in the

context of quantum lattice systems.

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Chapter 12

Conclusion

12.1 Thesis Review

This thesis was comprised of three major sections. In the Introduction and Chapters 2 and

3, we gave some historical context to the problem of simulating many-body lattice systems

and reviewed some foundational ideas. In particular, Chapter 2 described a well-known

correspondence between quantum systems in D dimensions and classical systems in D+1

dimensions. This established that, whilst the scope of this document was largely focused

on developing algorithms for quantum systems in two spatial dimensions, the techniques

discussed could also be applied to classical systems in three spatial dimensions. Chapter

3 presented some basic definitions on tensor networks and the way in which they are

commonly manipulated. Further to this, we introduced many-body entanglement in the

context of tensor networks and motivated their use for representing ground states of

certain Hamiltonians.

The next section of the thesis involved the development of an algorithm for determining the

ground state of infinite quantum systems in two dimensions and techniques for extracting

relevant physical information. Our reasoning followed a hierarchical philosophy, firstly

showing in Chapter 4 that finding the ground state of a point particle is trivial for a

classical computer. The idea then was that problems in higher dimensions could be

approximately solved, inheriting this computational simplicity. One-dimensional quantum

problems could be approximately cast as a series of zero-dimensional problems, and two-

dimensional quantum problems could be approximately cast as a series of one-dimensional

problems. Such an approach helps to overcome the exponential cost associated with

137

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138 Conclusion

exactly computing ground states of quantum systems. In Chapter 5, we described the

basic stages of our iPEPS algorithm in terms of these notions and then demonstrated the

power of the ansatz, by showing e"cient means of computing a range of useful physical

quantities.

The final section of this work validated the algorithm by applying it to a range of real

physical problems. There were three main motivations. Firstly, we wanted to show that

our algorithm for approximately contracting 2D tensor networks was e!ective, and that

the degree of the approximation can be improved by changing a refinement parameter,

&. We did this in Chapter 6 by studying the Classical Ising model on the square lattice.

The results obtained from our infinite-MPS and CTMRG network contraction algorithms

were benchmarked against Onsager’s analytical solution. In doing so, we validated a key

component of the iPEPS algorithm.

Chapters 7 and 8 determined the ground state of the quantum Ising model and hard-core

Bose-Hubbard model respectively. Here we saw that our results closely followed those

obtained from finite-size scaling QMC studies of the models, which in this regime are seen

as near exact results. Additionally, we computed a range of physical quantities that gave

an insight into the structure of entanglement across the phase diagram, including standard

entanglement measures, fidelity diagrams and spatial correlation functions. Finally, the

scale of correlations in the ground state was seen to be of great importance. Specifically, in

the quantum Ising model, the results generated with the full environment gave far better

approximations to the ground state than those generated with the simplified update.

In Chapter 9, we studied the q-state Potts model for q = 3 and q = 4. Our motivation

in this chapter was to determine how e!ectively PEPS could simulate systems with first-

order phase transitions, and in particular to see if PEPS could reproduce the increasingly

first-order nature of the phase transition with increasing q. Comparing our results with

those for the quantum Ising model, it was seen that the simplified update was very

e!ective in describing the properties of the ground state. Additionally, by computing the

first derivative of the ground state energy, fidelity diagrams and a comparison with the

mean-field solutions, we demonstrated the characteristic behaviour of the quantum Potts

model with increasing q.

Chapter 10 provided a first treatment of the J1-J2 model on the infinite lattice. This

frustrated model is out of reach for standard QMC methods due to the sign problem. We

tried several di!erent PEPS schemes, eventually determining that a modified version of

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12.2 Final Comments 139

the simplified update returned ground states with the lowest energy. Recent studies have

focused mostly on the possibility of a valence bond crystal ground state in the intermediateJ2J1

regime, where the competition between Hamiltonian terms is most significant. Typ-

ically, authors have presumed that the ground state possesses either a columnar dimer

or resonating valence bond structure, and performed either perturbation analysis or a

restricted subspace diagonalization to confine their solution to such an ordering. Interest-

ingly, our study failed to detect either of these two valence bond crystal structures in the

maximally frustrated regime, whilst finding lower energy approximations to the ground

state. This should be interpreted as an indication of the enormous competition between

low-energy states with a di!erent characteristic order in the J1-J2 model, and a measure

of the continuing di"culty facing numerical methods in solving frustrated many-body

systems.

Chapter 11 presents a study of the geometric entanglement of two-dimensional systems

with di!erent lattice geometries. Here we introduced the notion of the closest product

state in context of PEPS and described a variational algorithm for computing it. Doing

this for di!erent systems presented an interesting picture of the e!ectiveness of mean-field

theory and the role of quantum entanglement in describing strongly correlated lattice

systems. Finally, we suggested that the peak geometric entanglement could be compared

for lattices with di!ering coordination number, and that such an approach could be used

to numerically justify the idea of the monogamy of entanglement for quantum lattice

systems.

12.2 Final Comments

As a final comment, it is important to put PEPS and tensor networks in an appropriate

place in the context of numerical physics. The work in this thesis built upon the idea

that entanglement is the main source of complexity in quantum many-body systems, and

that for many systems of interest this entanglement is limited by fundamental physical

laws. Tensor networks networks extend the central idea of DMRG - that in choosing how

to approximate quantum many-body states, one should account for the entanglement in

a systematic way. This paints tensor networks in a di!erent light to other numerical

methods. For example, QMC in its simplest form samples the wavefunction without

consideration of the structure of entanglement in the state.

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140 Conclusion

The main strength of tensor networks is the richness with which they describe many-body

states. The algorithm produces a representation of the state that can be manipulated to

return a wealth of information. In this thesis alone we have demonstrated computation of

local observables, correlation functions and various entanglement measures, and compared

states by computing characteristic fidelity measures. Furthermore, tensor network states

can be tailored to better represent states with certain suggested properties. For example,

recent work has focused on the tensor network description of quantum states that possess

global or gauge symmetries. In this light, TNs are a powerful and highly configurable

ansatz for representing quantum states.

The major problem for PEPS and all present tensor network approaches to two and

higher dimensional systems is the extremely high computational cost scaling in the bond

dimensions. To put this in perspective, if one takes the current iPEPS algorithm and

assumes that the computational throughput of modern computers continues to scale with

Moore’s law, then a desktop computer running a D = 10 iPEPS simulation with the full

environment is some 25-30 years away. Whilst there may be some room for optimization

and parallelization of the algorithms, it is likely that the success of these algorithms will

depend on gaining greater understanding of the structure of entanglement in many-body

systems.

PEPS is not alone in its predicament. All numerical approaches to solving the many-body

problem are presently held back by intrinsic di"culties, raising fundamental physical,

philosophical and biological arguments about the degree to limits to which humans -

even armed with computers - are able to describe Nature. No one numerical approach

has emerged as a consensus method for studying many-body systems, but by using them

together it is possible to look at a problem from several points of view. This is the spirit in

which PEPS should currently be appreciated. How successful PEPS ultimately becomes

as a tool for studying many-body physics is not possible to predict, but at the very least

it presently provides a fresh perspective on many problems of interest and more generally

on the role of entanglement in quantum many-body systems.

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Appendix A

Infinite MPS Methods for Computing the

Environment

In Chapter 4, we discussed methods of approximately contracting infinite, translationally

invariant 2D tensor networks. Such a procedure is an important part of the iPEPS

algorithm. In particular, it is required to compute the environment, which in turn can be

used to compute observables and various other properties of quantum states. The first

method for contracting 2D TNs set about describing the boundary state by an infinite

MPS. Then, the MPS was evolved under the action of gates formed from the tensors

(see fig. 4.7). In Chapter 4, we skipped over the low level details of this technique. In

this appendix, we describe a solution to the problem of evolving an infinite MPS under

the action of a repeating set of two-body gates. This corresponds, for example, to the

evolution of the diagonal boundary state for the square tensor network in fig. 4.7. We then

go on to describe how this and similar algorithms can be used to compute the environment

in the iPEPS algorithm.

The first part of this appendix details the approach developed by Orus and Vidal [OV08],

which the author helped to implement and which formed the basis of the original iPEPS

algorithm [JOV+08]. The approach is described in detail as it establishes for the reader

some important ideas in tensor network theory. In particular, it reproduces the theoretical

framework underpinning the iTEBD algorithm for finding the ground state of 1D systems

and justifies why it does not perform well for general gates. This augments the discussion

of Chapter 4, shedding further light on the comparative ease of approximately computing

the ground states of 1D quantum systems due to the availability of a canonical form, and

the necessary modifications needed to compute the environment of 2D quantum systems.

141

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142 Infinite MPS Methods for Computing the Environment

Figure A.1: The operation of the two-body gates a and b on the boundary MPS!

!)[0]"

.Here the accumulated action of the ’a’ (’b’) gates is represented by the transfer matrixTA (TB).

A.1 Problem Overview

The general problem is outlined in fig. A.1. We assume that our initial boundary state,!

!)[0]"

, is translationally invariant under shifts of two sites. It is described by an infinite

MPS composed of four repeating tensors, '[0]C , %[0]

C , '[0]D and %[0]

D . The % matrices in this

representation are diagonal operators. The boundary state is operated on by alternate

sets of gate tensors, where each gate is labeled a or b. For now, we assume that these

gates are completely arbitrary in their coe"cients and make no connection between the

gate and the PEPS tensors.

A single step of the boundary evolution involves the application of a row of a gates,

followed by a row of b gates, obtaining a new boundary MPS of the same structure as in

fig. A.1. We call these rows TA and TB respectively. Applying each of these gates k times,

we obtain the evolved boundary state!

!)[k]"

and its constituent tensors '[k]C , %[k]

C , '[k]D and

%[k]D . Applying the gates an infinite number of times, we obtain the dominant eigenvector

|$# as described below:

|$# =!

!)[$]"

= limk#$

(TBTA)k!

!)[0]"

, (A.1)

where:

$!

! )[0]"

)= 0. Thus, if we obtain a general procedure for finding!

!)[1]"

=

TBTA

!

!)[0]"

, we can iterate this procedure until we see convergence in the boundary state.

In fact, since the structure of TA and TB is identical apart from a translation of one lattice

site, we need only develop a method for computing the half-step evolution |)!!# = TA

!

!)[0]"

.

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A.1 Problem Overview 143

Unfortunately, an exact general solution for this eigenvalue problem is computationally

intractable as the number of coe"cients required to represent the boundary state MPS

can increase exponentially step upon step. We instead define an e!cient method for

approximating the the boundary MPS as one that meets the following criteria:

1. The number of coe"cients required to describe the infinite boundary MPS is bounded

by some fixed upper limit at any stage.

2. If the exact evolved boundary MPS is represented with more coe"cients than this

limit, there exists a method for computing a justifiably good approximation to this.

In the first half of this appendix, we will establish an approach for finding such an ap-

proximation by deriving some properties of the boundary MPS and its evolution under

general, non-unitary gates.

Lemma I: The bond dimension of the boundary MPS may increase exponen-

tially in the number of steps unless truncated.

Proof of this establishes that exactly evolving the boundary state is computationally

ine"cient. Consider the operation of the gate a on the tensors '[0]C , %[0]

C , '[0]D as is fig. A.2.

The bond indices of these tensors are bounded by the parameter &. After contracting

the tensors and the gate into a (maximally) rank-(&D2) tensor M , and splitting M by

SVD, some of the bonds are now of dimension &D2. Iterating this procedure with gates

operating on alternating links, one can see that the maximal bond dimension scales as

&D2N , where N is the number of steps performed.

Lemma II: If the MPS is in the canonical form, the optimal update of the state

after the application of a single gate, g, is given by a local decomposition.

The canonical form of the MPS prescribes that the Schmidt form may be reproduced

about any % operator. That is, our MPS,!

!)[k]"

is of the form:

!

!)[0]"

=#

$

%$$!

!$left$

"

!

!

!$right"

>

(A.2)

where the bases!

!$left$

"

and!

!$right$

"

are orthonormal by construction. As an example,

consider the MPS in fig. A.3i. If this MPS is in the canonical form, then we obtain

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144 Infinite MPS Methods for Computing the Environment

Figure A.2: The exact evolution of the boundary state with an MPS may lead to anincrease in the MPS bond dimension. Here, a gate is contracted with MPS tensors havingphysical indices of dimension D and bond indices of dimension &. Splitting the resultingtensor leads to an interjoining link of dimension up to &D2.

the orthonormal bases for the left and right sections by contracting all the tensors to

the left and right of %[0]C . A special property of an MPS in the canonical form is that

the contraction of the left or right section with its conjugate along the physical indices

returns the identity, a direct result of the orthonormality of the states!

!$left$

"

. We say

that the scalar product matrix M$$" ==

$left$"

!

!

!$left$

>

is the identity. In a general sense,

this matrix corresponds to the left eigenvector of the structure surrounded by the purple

box in fig. A.3ii. We can e"ciently compute it by starting with a random vector and

operating on it with this structure until convergence.

An important property of an orthonormal basis is that we can transform it under the

action of some unitary operator, and the orthonormality is preserved. In fig. A.3iii we

show the state$

$U%$!

!$left$

"

also gives rise to an identity scalar product.

Consider that we have a state |)# that we know to be in the canonical form, that has

bond dimensions bounded by &. We also assume without loss of generality that |)# is

normalised. We want to prove that the canonical form is helpful in computing the state

|)!# (see fig. A.4ii), which is the best approximation to the state g[C,D] |)# (see fig. A.4i)

that can be achieved without any bond dimension exceeding of |)!# exceeding &. Formally,

we want to maximise the fidelity :

f ()!) =$)!| g[C,D] |)#$)! | )#

(A.3)

The numerator in this expression is shown as a tensor network in fig. A.4iii.

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A.1 Problem Overview 145

Figure A.3: Overview of Lemma 2, part 1. i) The MPS can b expressed in a bipartiteform by contacting together all of the tensors to the left and right of some % tensor. ii) Ifthe MPS is in the canonical form, then the bases of each partition are orthonormal andcorresponding scalar product matrix is the identity. iii) The orthonormality of each basisis not disturbed by the application of some unitary operator on the open bond.

The canonical form allows us to make a drastic simplification to the fidelity tensor network

and hence its optimisation. Since the sections to the left and right of the sites are described

by the scalar product matrix, we may replace them with the identity matrix, as shown in

fig. A.5i. Here we have also included the denominator of eqn. A.3.

With this simplification, one can show that the optimal update is given by a simple local

decomposition and truncation. Firstly, we take the tensors and the gate surrounded by

the red box in fig. A.5i. Then, we contract these together, forming a tensor T , and split

them by singular value decomposition (see fig. A.5ii). Next, we truncate the inner bonds

with the projectors P), by only retaining the subspace pertaining to the & largest singular

values of Q. It is possible to show by invoking the properties of the SVD that the new

truncated tensors '!C , %!

C and '!D, as shown in A.5iii, maximise the fidelity in eqn. A.3.

Lemma III: If the set of gates operating on an infinite MPS are near-unitary,

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146 Infinite MPS Methods for Computing the Environment

Figure A.4: Overview of Lemma 2, part 2. i) The state g[C,D] |)# formed by the applicationof a single two-body gate to the state |)#. ii) The proposed new state |)!#. iii) Theunnormalised fidelity $)!| g[C,D] |)# as a tensor network.

then the local update is a good approximation to the optimal update.

We want to describe the update of an infinite-MPS that is in the canonical form under

the action of a repeated set of gates. An example of this is the transfer matrix TA acting

on the state!

!)[0]"

in fig. A.1. However, for now we make the restriction that the gates

are near unitary with respect to their input and output legs. The task is to find a good

approximation to the evolved state |)!# = TA |)# (see fig. A.6i), by maximising the fidelity(*"|TA|*)(*"|*") . Since our |)# is translationally invariant and the gates are identical, we assume

that |)!# is also translationally invariant, and is described by the tensors '!C , %!

C , '!D and

%!C . To do this, we modify the problem in a subtle way. We focus on the update of one

of the repeated sections of |)#, and leave the rest of the MPS acted on by the remaining

gates. This state, which we call |)!!#, is shown in fig. A.7i. We want to find the tensors

'!!C , %!!

C , '!!D and %!!

C that maximise the fidelity (*""|g|*)(*""|*"") . This quantity is shown in fig. A.7ii.

Recall that:

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A.1 Problem Overview 147

Figure A.5: Overview of Lemma 2, part 3. i) The expression of the normalized fidelity ofeqn. A.3 as a tensor network. ii) The local decomposition operation iii) The determinationof the new MPS tensors. Here, the operators P) truncate the bond, retaining the subspacecorresponding the the & largest values of ST .

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148 Infinite MPS Methods for Computing the Environment

Figure A.6: Overview of Lemma 3, part 1. i) The application of a row of two-body gatesto the state |)#. ii) The MPS, |)!#, approximating the updated state TA |)#.

1. The gates g are near-unitary.

2. The MPS representing |)# is in the canonical form.

The first property means that the contraction of the gate with its conjugate transpose in

fig. A.7ii may be approximated by the identity (see fig. A.7iii). As the state is in the

canonical form, this means the scalar product matrices to the left and right of the orange

box in fig. A.7ii are approximately the identity matrix. We then invoke the reasoning of

Lemma 2 to see that '!!C , %!!

C and '!!D may be determined by the same local decomposition.

Next, we update the entire MPS by making the substitutions 'C & '!!C , %C & %!!

C and

'D & '!!D.

The conclusion is that for states in the canonical form, we may approximate the action

of a set of repeated near-unitary gates by carrying out a local decomposition. This is

the backbone of the TEBD algorithm [Vid04, Vid07]. A remarkable aspect of the TEBD

algorithm is that if the MPS starts in the canonical form, then state formed by the iterated

action of transfer matrices TA and TB (see fig. A.1) containing near-unitary gates seems to

stay acceptably close to the canonical form for the algorithm to be e!ective. This is related

to the property g†g - I. Designing an initial MPS in the canonical form that has some

overlap with the eventual dominant eigenvector is usually quite easy (take, for instance,

the product state, (|0#+ |1#)&N , where the computational basis [|0#,|1#)] represents some

local physical degree of freedom).

If the gates were not near-unitary, the e!ectiveness of the local update could not be

guaranteed. It is possible that the finite-& MPS representation of the evolved state would

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A.1 Problem Overview 149

Figure A.7: Overview of Lemma 3, part 2. i) The MPS form of the proposed updatedstate, |)!!#. ii) The unnormalised overlap between |)!!# and TA |)#. iii) If the gate is near-unitary, then the contraction of the gate with its conjugate transpose is approximatelythe identity operator.

only be acceptably close to the actual evolved state for very large &. Furthermore, since

for a general gate g†g )= I, the evolution will cause the MPS to drift away from the

canonical form faster than in TEBD. This means that the local update is not only sub-

optimal for the first iteration of the algorithm, but its performance becomes increasingly

unpredictable as the evolution progresses. It is clear that for the contraction of general

infinite, translationally invariant 2D tensor networks, where the gate is not near-unitary,

we need to think deeper about how to update the infinite MPS boundary state.

Lemma IV: The physical state represented by a tensor network states is in-

variant under the insertion of full-rank operator-inverse pairs on bonds.

We show a justification for this simple notion in fig. A.8. The diagram shows the infinite

MPS representation of some state |)#. The coe"cients of the state expanded in a local

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150 Infinite MPS Methods for Computing the Environment

Figure A.8: Overview of Lemma 4. The state |)# is invariant under an operation on atensor bond that resolves the identity.

physical basis are returned by the contraction of the bonds of the infinite MPS. Since the

insertion of some operator pair T -T"1 on the bond does not change the object returned

by this contraction, the state is invariant under this insertion.

Lemma V: Infinite, translationally invariant MPS states can be transformed

into a canonical form if the scalar product matrix can be computed.

Lemma II stated that the scalar product matrices of partitions of the MPS could tell use

whether the MPS was in the canonical form. It also stated that the orthonormality of the

auxiliary bases were invariant under unitary operations of the cut MPS bond. Lemma

III stated that if the MPS was in the canonical form, then the update of the MPS by

a near-unitary set of gates could be well approximated by some local decomposition of

the repeated sections. Lemma III also stated that the canonical form is approximately

maintained under near-unitary evolution, but would likely drift from the canonical form

under non-unitary evolution, making it di"cult to iterate the evolution. However, we

know from Lemma IV that the state is invariant under certain transformations along the

MPS bonds. The question we want to answer here is - for an arbitrary translationally

invariant infinite MPS, can we find a set of transformations that restore an MPS to the

canonical form?

Figure Figure A.9i shows an MPS representing some state |)!!#. The MPS is composed of

tensors '!!C , %!!

C , '!!D and %!!

D. The MPS is not in the canonical form, and by virtue of this,

the scalar product matrices of the left and right partitions!

!$left$

"

and!

!

!$right"

>

(enclosed by

red boxes in fig. A.9i) are not the identity matrix, but the positive semidefinite matrices

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A.2 Evolution of an Infinite MPS by Non-unitary Operators 151

L and R, shown in fig. A.9ii.

Consider the following decomposition of L and R.

L = ULDLU †L, R = URDRU †

R, (A.4)

and the following assignments,

TL = D"1/2L U †

L, TR = D"1/2R U †

R, (A.5)

Then it is clear that the modified states!

!

!$left$

>

= TL

!

!$left$

"

and!

!

!$right"

>

= TR

!

!

!$right"

>

give rise to identity scalar product matrices.

Using Lemma IV, we introduce TL, TR and their inverses into our MPS (see fig. A.10i),

leaving the overall state invariant. Then, we take the three actions in fig. A.10ii:

1. Contracting T"1L , %!!

C and T"1R into a matrix Q, and splitting Q via SVD into UQ,

SQ and VQ.

2. Contracting '!!C , TL and UQ to form 'C

3. Contracting VQ, TL and '!!D to form 'D

Introducing 'C , 'D and %C = SQ/tr)@

S2Q

*

uniformly through the MPS (see fig. A.10iii,

we have a state |)# = |)!!# (since all the introduced operations resolve the identity), that

is in the canonical form when considered at the links between 'C and 'D. To see this,

consider that the left and right partitions in the red boxes in fig. A.10iii are UQ

!

!

!$left$

>

and VQ

!

!

!$right"

>

respectively. Since applying unitary operators to an orthonormal basis

preserve this orthonormality (see Lemma II), our states to the left and right of %C are

spanned by an orthonormal basis in this representation.

To complete the transformation of the state into the canonical form at any partition, we

need to apply the same procedure to the links between 'D and 'C .

A.2 Evolution of an Infinite MPS by Non-unitary

Operators

In this section, we use the principles set down in Lemmas I-V to describe the evolution

of an infinite MPS by repeating, non-unitary gates g. It is clear by now that there are a

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152 Infinite MPS Methods for Computing the Environment

Figure A.9: Overview of Lemma 5, part 1. i) The state |)!!# is here expressed as an MPS.ii) Since the MPS is not in the canonical form, the scalar product matrices are not theidentity, but the positive definite operators L and R.

couple of essential points to understand.

1. The evolution of an MPS with respect to a single, local gate can be well-approximated

by a local decomposition and truncation (Lemma III), as long as the partitions to

the left and the right of the gate are described by an orthonormal basis.

2. Partitions of the MPS can be orthonormalised, leaving the overall state invariant,

as long as we can compute the scalar product matrix of the partition e"ciently.

(Lemma V)

In fig. A.11 we detail an approach building on these principles. Fig. A.11i depicts the

state |)!!# = TA |)# formed by applying the gates g to the state |)#. Then, we compute

the scalar product of the left and right partitions of the state. In fig. A.11 we show the

tensor contraction to compute the L, the scalar product of the left partition. Having done

this, we orthonormalise the state (fig. A.11iii), such that the sections to the left and right

of the orange box in fig. A.11iv are spanned by an orthonormal basis. Then, we can find

the new tensors '!!C , %!!

C and '!!D by the local decomposition and truncation outlined in

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A.2 Evolution of an Infinite MPS by Non-unitary Operators 153

Figure A.10: Overview of Lemma 5, part 2. i) The partitions!

!

!$left$

>

= TL

!

!$left$

"

and!

!

!$right"

>

= TR

!

!

!$right"

>

. The inverse matrices T"1L and T"1

R have been inserted to resolve

the identity. ii) The contraction of T"1L , %!!

C and T"1R into the matrix Q, which is then

split by SVD. The determination of 'C and 'C . iii) The new state |)# is in the canonicalform about the operator %C

.

Lemma II. Making the replacements 'C & '!!C , %C & %!!

C and 'D & '!!D. Our updated

state contains the tensors '!!C , %!!

C , '!!D and %D.

The two dominant computational costs of this algorithm are:

1. The computational cost of determining the scalar product matrices L and R. This

is done by evolving the random vectors 0L and 0R as in fig. A.12.

2. The computational cost of performing the local decomposition (SVD).

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154 Infinite MPS Methods for Computing the Environment

Figure A.11: The evolution of an MPS under non-unitary gates. i) The state TA |)#.ii) The computation of the left scalar product matrix, L. The computation of the rightscalar product matrix follows by analogy. iii) The transformations 'C & 'C , 'D & 'D

and %D & %D iv) The state is unchanged, but is orthonormalised such that the sectionsto the left and the right of the orange box can be approximated by the identity matrix.The local decomposition and truncation can then find the new state.

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A.3 MPS-based Contraction Schemes for PEPS 155

Figure A.12: The left and right scalar product matrices are commonly found by beginningwith some random left and right vectors, 0L and 0R and converging them under the actionof repeated transfer matrices.

A.3 MPS-based Contraction Schemes for PEPS

In this section we describe how the scheme outlined in the previous section can be used

to compute the environment of the iPEPS for various lattice geometries, and compare

the computational cost of these implementations. In this discussion, we assume that our

states have been computed by the iPEPS algorithm of Chapter 5. This introduces the

concept of a minimal covering PEPS representation of the lattice. This is the minimum

number of distinct PEPS tensors that are needed to perform an iPEPS evolution, and

is a function of both the Hamiltonian and the lattice geometry. For instance, consider a

Hamiltonian with identical two-body terms describing a system on the square lattice. The

physical ground state might be invariant under any lattice translation (e.g. a spin liquid),

but since the iPEPS algorithm update breaks translational invariance (see Appendix C

for more details), it turns out that the minimal covering for the PEPS is invariant under

shifts by two lattice sites, giving the A/B pattern seen throughout this thesis. Since

we don’t know if and by how much the ground state breaks translational invariance of

the Hamiltonian, there is no guarantee that this representation represents the ground

state well. Nevertheless, it is the minimal set of tensors with which we can represent a

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156 Infinite MPS Methods for Computing the Environment

Figure A.13: The gates ’a’ and ’b’, formed by contracting the PEPS tensors with theirconjugate versions along the physical index.

ground state for this Hamiltonian and lattice geometry in the iPEPS algorithm. For this

chapter, we assume that our Hamiltonian contains identical terms, and proceed using the

minimal covering for each of the geometries we consider. However, it is possible to scale

the approach we describe to compute the environment for PEPS networks with any type

of regular periodicity.

A.3.1 The Square Lattice

A Diagonal Scheme

Consider the task of computing the environment, as depicted in fig. 5.4. Around our

region of interest we have PEPS tensors contracted with their conjugate pair along the

physical index. Making the substitutions A-A* & a and B 'B* & b as in fig. A.13, this

structure becomes a 2D tensor network. We can then use an infinite MPS to describe

the boundary state and express a and b as gates operating on this state (see fig. A.14i).

Starting with random initial states (open boundary conditions) we iterate the procedure

outlined in the previous sections, to obtain the evolved states:

$)L| = $)L| limk#$

(TATB)k , |)R# = limk#$

(TATB)k |)R# (A.6)

This leaves us with the structure shown in fig. A.14ii. Next, we evolve the finite states

|vR# and $vL| in directions perpendicular to the infinite MPS states. This is a 1D problem

that is easily solved on a classical computer. Having done this, we are left with the

structure shown in fig. A.14iii.

Before advancing, it is worth stating that although we depict the gates a and b in fig. A.14,

it is usually worth keeping these structures in the uncontracted A-A* and B-B* forms.

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A.3 MPS-based Contraction Schemes for PEPS 157

Figure A.14: The diagonal contraction scheme for the square lattice

That is, it is beneficial on the grounds of computational cost to leave the d-dimensional

physical index uncontracted. We favoured the gates a and b for reasons of schematic

clarity and to fully make the connection with the discussion of section A.1.

A Parallel Contraction Scheme

An alternative scheme for contracting the environment is to use for the boundary state

an MPS that traverses the lattice in a direction parallel to the PEPS bonds. This process

is summarised in figure A.15. Here, we have chosen to compute the environment around

a horizontal link between B and A, by starting with infinite horizontal MPSes (see fig.

A.15i). The evolution of this MPS is a di!erent procedure to that for the diagonal scheme,

where the gate operations almost directly mapped to those in the discussion in section

A.1. However, the algorithm for the parallel scheme follows the same general principles.

Firstly, we find attach a row of gates to the state. Secondly, we find the left and right

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158 Infinite MPS Methods for Computing the Environment

Figure A.15: The parallel contraction scheme for the square lattice

scalar product matrices (see A.16) about a gate operation. Then, we transform the state

into the canonical form, and perform the gate operation and truncation. Once we have

determined $)L| and |)R#, we converge the finite-dimensional left and right eigenvectors,

$vL| and |vR# to complete the description of the environment.

A.3.2 Beyond the Square Lattice - the Hexagonal, Kagome and

Triangular Lattices

In this section, we briefly introduce schemes for contracting 2D networks of other geome-

tries. Like the square lattice, the hexagonal lattice can be covered with just two tensors A

and B. In fact, the similarity extends further. Problems on the hexagonal lattice can be

solved by applying the square iPEPS algorithm with a simple modification. Starting with

the four-legged tensors A and B and removing every instance of one of the four unique

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A.3 MPS-based Contraction Schemes for PEPS 159

Figure A.16: The computation of the scalar product matrices in the parallel scheme.

links (up, down, left or right) we are left with an infinite hexagonal lattice. However, we

are curious about whether a dedicated algorithm for the hexagonal lattice captures corre-

lations more e"ciently and how its computational complexity compares with the solution

for the square lattice.

One possible contraction scheme for the hexagonal lattice is shown in fig. A.17. The

stages of this algorithm are by now familiar - the evolution of some infinite MPSes,

$)L| & $)L| , |)R# & |)R# under the action of infinite rows of gates, followed by the

convergence of the left and right vectors $vL|& $vL| and |vR# & |vR#. The key operations,

such as computing the scalar product matrices should follow by analogy with the square

schemes.

In fig. A.18 we depict a possible contraction scheme for the Kagome lattice. The minimal

covering consists of three distinct tensors, A, B and C. In the dotted red box, we show

the gate operations on the infinite MPS. Comparison with the square schemes in figs.

A.14 and A.15 shows that the update to the infinite MPS is of one of two types. The first

stage is maps exactly to the square parallel scheme with gates a and b, and the second

stage maps directly to the square diagonal scheme with a single gate c. Following this,

vectors $vL| and |vR# are converged by exactly the same algorithm used in the square

parallel scheme.

In fig. A.19, we show a scheme for contracting a PEPS for the triangular lattice. Each

lattice site has six nearest neighbours, and this increased connectivity leads to a more

complex algorithm. For example, even before considering the computational cost of basic

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160 Infinite MPS Methods for Computing the Environment

Figure A.17: A contraction scheme for the hexagonal lattice

stages, it can be seen that the infinite boundary MPS contains 12 repeating tensors.

A.3.3 Computational Cost Comparison

To conclude, we compare the leading order computational cost of each of these contrac-

tion schemes (see Table A.3.3). The schemes we have implemented do not necessarily

represent an exhaustive or optimal set of schemes, but the computational costs are inter-

esting for a couple of reasons. Firstly, as determining the environment is the dominant

computational task in most iPEPS simulations, it puts in perspective the computational

resources required to perform iPEPS simulations and justifies the limits to which we could

provide results (e.g. the exclusion of triangular lattice Ising model results in Chapter 11).

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A.3 MPS-based Contraction Schemes for PEPS 161

Figure A.18: A contraction scheme for the Kagome lattice

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162 Infinite MPS Methods for Computing the Environment

Figure A.19: A contraction scheme for the triangular lattice

Secondly, it gives a rough idea of the relationship between the connectivity of a PEPS

and the computational complexity of its associated iPEPS implementation.

The computational costs are based on the following assumptions.

1. Each link of the PEPS has the same bond dimension D and each infinite MPS has

a bond dimension of &.

2. The local dimension, d, is the same at every lattice site.

3. The convergence of the infinite MPS involves the S* steps. The determination of

the scalar product matrices (see fig. A.12 and A.16) within each step involves SLR

iterations.

4. The convergence of the vectors |vR# and $vL| involves Sv iterations.

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A.3 MPS-based Contraction Schemes for PEPS 163

Computation of Orthonormalization & Computation ofL & R local decomposition |vR# & $vL|

(S* iterations) (Sv iterations)

Square Diagonal &3D4 + &2D6d &3D6 &3D4 + &2D6d+ SLR (&3D4)

Square Horizontal &3D6 &3D6 &3D4 + &2D6d+ SLR (&2D8d)

Hexagonal &2D6d2+ &3D6 &3D4 + &2D5d+ SLR (&3D4)

Kagome &3D6 &3D6 &3D4 + &2D6d+ SLR (&2D8d)

Triangular SLR(&3D8 &3D12 &3D6 + &2D9d+&2D11d)

Table A.1: The leading order computational costs for the square, hexagonal, Kagome andTriangular lattice infinite-MPS based contraction schemes.

The leading order costs are shown in table A.3.3. There are two comments. Firstly,

comparing the square diagonal and horizontal computational costs, one can see that it

is possible to have di!erent computational cost scaling for the same lattice geometry. In

practice, the di!erence in the cost of computing L and R may allow access to higher D

and & in the diagonal scheme than in the horizontal scheme. A separate issue is whether

one scheme or the other is naturally better at representing the correlations in the square

lattice. We have not investigated this extensively and any relation could be heavily model

dependent, but what empirical evidence we have seen seems to indicate that the horizontal

scheme performs very slightly better for the same D and &. The second point to make

is that there does not appear to be a simple relation between the coordination number

of the lattice, z, and the computational complexity of contracting the PEPS representing

the lattice. For instance, from our results there is very little di!erence between the cost

of contracting a hexagonal PEPS (z = 3) and a square PEPS (z = 4) with the diagonal

scheme. However there is a vast increase in computational complexity when contracting

the triangular PEPS (z = 6). Together, these suggest that some of the ongoing challenges

in this area are to find the optimal contraction scheme for each lattice geometry, and

then to investigate whether additional approximations can be made to further reduce the

computational cost.

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164 Infinite MPS Methods for Computing the Environment

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Appendix B

Corner Transfer Matrix Renormalization

Group Algorithms for the Square Lattice

B.1 Problem Overview

As explained in Chapter 4, the corner transfer matrix (CTM) approach consists of so-

called corner transfer matrices and edge tensors surrounding a unit cell. The idea of

the corner transfer matrix renormalization group (CTMRG) algorithm is that the by

operating on these tensors in certain ways, we can find instances of them that collectively

approximate the environment. In this appendix, we describe several variations of an

algorithm for finding such tensors.

We restrict ourselves to a 2" 2 unit-cell containing tensors TA, TB, TC and TD and their

conjugate versions. Our CTM is described by the corner tensors Cm for m = 1...4 and

the edge tensors En for n = 1...8. This structure is shown in fig. B.1.

Each of the CTM tensors is a reduced-rank approximation to a particular section of the

environment. The bonds between CTM tensors, each of dimension &, allow for correlations

to flow between these sections. A higher & captures more correlations and hence gives a

better approximation to the actual environment. The numerical task solved by Nishino

[NO96] was to determine an e"cient (polynomial time) means of evolving to such an

environment, keeping the rank bounded at each step. In this appendix, we discuss two of

our own implementations, describing our reasoning behind them and the computational

cost of each implementation.

165

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166Corner Transfer Matrix Renormalization Group Algorithms for the

Square Lattice

Figure B.1: The basic CTM structure for a PEPS with 2" 2 periodicity

B.2 The Corner Transfer Matrix Renormalization Group

for iPEPS

In the CTMRG algorithms we will describe, each step consists of i) insertion, ii) absorption

and iii) renormalization.

The insertion process involves replicating part of basic CTM unit cell along side the

existing unit cell, creating an excess of tensors in the unit-cell. The tensors are inserted

in such a way that they do not disrupt the regular pattern of the lattice. For example, we

may make any of the insertions shown in fig. B.2. This can be interpreted as i) placing a

block to the left, ii) placing a block to the right or iii) parting the CTM down the middle

and inserting a 2 " 2 block. The interpretation is not particularly important, but may

preempt the way in which we absorb the CTM tensors.

The absorption process contracts some of the excess unit-cell tensors along their physical

dimension and then into the environment tensors (see figure B.3. Here, we make one step

to the left, and one step to the right, creating the tensors C1 & C4 and E1, E2, E5 and E6.

Each absorption step increases some bond dimensions between the environment tensors,

in this case from & to &D2.

To keep the CTM representation compact, we must find renormalization tensors Q and W

that reduce the dimension of the bonds between the environment tensors, whilst keeping as

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B.2 The Corner Transfer Matrix Renormalization Group for iPEPS 167

Figure B.2: Insertion of a 2" 2 block of sites i) to the left of the existing unit-cell ii) tothe right of the existing unit-cell iii) in the middle of the existing unit-cell

much information about the environment as possible (see figure B.4). On computational

cost grounds we prefer to absorb and renormalise alternately rather than make multiple

absorptions and then a more severe renormalization. For each of the interpretations of

the insertions in figure B.2, we can choose a di!erent absorption scheme. For i), we may

make two absorption and renormalization steps to the right, so that our unit cell is again

A-B-C-D. Analogously, we can for ii) make two steps to the left. For iii), we take the left

hand column and move it to the left, and the right hand column and move it to the right.

Our unit cell is now B-A-D-C.

A complete cycle of absorptions and renormalizations has been completed when we have

made an equal number of steps in each direction and we have returned to the A-B-C-D

unit-cell structure. For i) this means making, for example, two steps right, two steps

down, two steps left and finally two steps up. For ii) it is similarly simple. It is easily

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168Corner Transfer Matrix Renormalization Group Algorithms for the

Square Lattice

Figure B.3: The absorption process in a horizontal direction, increasing the dimension ofthe bonds.

seen that for iii) we can achieve a full cycle by alternating horizontal and vertical steps.

For an infinite system, this process of insertion, absorption and renormalization continues

until there is convergence of the environment, as may be determined, for example, by

examining the eigenvalues of the corner matrices.

The idea is that the Q and W tensors e!ectively keep the most important correlations in

the environment and that they resolve the identity in some reduced subspace. The absorp-

tion and renormalization steps in CTMRG are similar to the coarse-graining procedure

of DMRG, and so this process is also called coarse-graining.

B.3 Coarse-Graining Approaches

The most physically motivated choice of Q and W would be that for which the renor-

malized density matrix 'R (shown in figure B.4) is nearest in some sense to the density

matrix after absorption, 'A (shown on the RHS of figure B.3). That is, we directly ap-

ply the principles of DMRG to the CTM. A variational solution to this problem seems

computationally expensive for a number of reasons. Firstly, it would add another layer of

iteration, and secondly it scales exponentially in the unit-cell size.

The first example of an e"cient CTMRG algorithm for computing the environment of

PEPS/TPS states was developed by Nishino and Okunishi [NO96]. Here, the authors

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B.4 The Directional CTMRG Approach 169

Figure B.4: Renormalization of the vertical bonds by the renormalization operators Qand W .

describe a technique where at each point a unit-cell tensor is contracted diagonally into

the corner transfer matrix. The renormalization tensors are determined by firstly ’cutting’

the index that needs to be renormalized, secondly contracting the rest of the network

leaving the cut indices open, and thirdly decomposing this ’density matrix’ to obtain the

unitary renormalization operators.

For the iPEPS algorithm, it was unclear whether this algorithm could be directly applied.

Firstly, the procedure of [NO96] was developed for a isotropic, translationally invariant

system. Secondly, the act of contracting the entire network other than a ’cut’ link seemed

expensive and di"cult to scale to larger unit-cells. Thirdly, there was concern that choos-

ing the renormalization operators based on the present environment might cause the CTM

to very quickly converge to some local stationary state.

B.4 The Directional CTMRG Approach

Motivated by the algorithm in [NO96], Orus and Vidal described a directional variant

of this algorithm [OV09]. A slightly modified version of this is shown in fig. B.5. This

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170Corner Transfer Matrix Renormalization Group Algorithms for the

Square Lattice

diagram details the stages to renormalize the tensors C1, C4, E2 and E1 after a step to

the left.

The steps of this algorithm are as follows. Starting with the corner tensors C [0]1 and C [0]

4 ,

in fig. B.5i, we form the tensors C [0]1 and C [0]

4 by moving the edge tensors E3 and E8 to

the left. Next, we assert that as the links between C1 and E2, and E1 and C4 correspond

to the accumulation of the same correlations, the same renormalization tensors should

be applied. To determine these tensors, we firstly mix the subspaces by computing the

sum T = C [0]1 C [0]†

1 +)

C [0]4 C [0]†

4

**. The conjugation of the second term is an important

modification to the algorithm in [OV09], and helps to stabilise the algorithm. From this

sum, the renormalization tensor Q is determined by singular value decomposition.

In step iv), we form the matrix R[0]1 by contracting C [0]

1 , E[0]2 ,E[0]

3 and ta. Likewise, we form

the matrix R[0]4 from the bottom four tensors. In step v), we determine the renormalization

tensor W by the same reasoning as step iii). Lastly, we apply the renormalization tensors

to obtain the new tensors E[1]2 and E[1]

1 .

The beauty of this algorithm is its simplicity. In practice it is very fast, as it involves only

one full-rank SVD per step (in the determination of W and W †). Unfortunately, it was

observed in some cases that the eigenvalues of the corner matrices did not converge but

oscillated, and as a result the physical quantities of the PEPS also oscillated.

B.5 An Improved Directional Algorithm

A concern with the above approach is the way in which the two subspaces are combined.

Deriving the renormalization operators from the matrix sum C1C†1 + (C4C

†4)

* seems an

ad-hoc choice. An approach more motivated from experience with tensor networks is to

treat the four tensors of the CTM as a finite MPS.

The steps of this algorithm are shown in fig. B.6. It begins in the same way as the

previous algorithm, absorbing E[0]3 into C [0]

1 and E[0]8 into C [0]

4 . In step ii), we assume the

legs marked black correspond to local Hilbert space. We assume the indices of these legs

run over a set of vectors that together form an orthonormal basis. Then, we carry out

the orthonomalization of the finite MPS, determining the operators Q1 and Q4 that when

acting on C [0]1 and C [0]

4 return tensors C [0]1 and C [0]

4 . The ’scalar product’ matrix of these

tensors, defined in the same way as for a partition of an infinite MPS (see Appendix A) is

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B.5 An Improved Directional Algorithm 171

Figure B.5: The stages of the first directional CTMRG algorithm.

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172Corner Transfer Matrix Renormalization Group Algorithms for the

Square Lattice

Figure B.6: The stages of a more stable, but computationally more expensive directionalCTMRG algorithm.

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B.6 Recent Developments 173

the identity matrix. In step iii), we mix the subspaces by contracting the inverse matrices,

Q"11 and Q"1

4 , and determine the singular value decomposition. We then truncate to retain

only the & largest singular values, and contract to form the new tensors C [1]1 and C [1]

4 .

Following this, we perform the same operations on the upper and lower matrices R[0]1 and

R[0]4 , which are defined in fig. B.5, to find the updated tensors E[1]

2 and E[1]1 .

This algorithm is more stable than the first directional scheme. It is also more e!ective.

In computing the magnetization of the 2D classical Ising model, this method performed

slightly better than the earlier version. However, each step involves three as many SVD

operations, and as a result the overall cost is roughly two to three times higher.

B.6 Recent Developments

In the appendix of [CJV10] we describe a CTMRG algorithm that extends Nishino’s orig-

inal algorithm to an anisotropic lattice with an A-B-C-D unit cell. It has been observed

that this scheme shows particular promise. It is extremely stable and converges to solu-

tions quicker than either of the methods described above. For a study of fermionic 2D

lattice systems where the PEPS were converged via the simplified update, this CTMRG

scheme produced lowest energy results than other proposed schemes. However, a detailed

analysis of all three CTMRG schemes against a system with an exact solution, or a com-

plete simulation of each method using the full environment to update the PEPS has not

been carried out.

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174Corner Transfer Matrix Renormalization Group Algorithms for the

Square Lattice

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Appendix C

Update Schemes for PEPS tensors

This appendix describes techniques developed to update the PEPS tensors during the

iPEPS algorithm. Our discussion is based on an infinite square PEPS covered by tensors

A and B, with bond dimensions D. We also assume that the Hamiltonian contains

nearest neighbour terms, and by virtue of this that the Suzuki-Trotter decomposition of

the imaginary-time evolution operator yields two-body gates. So whilst in this appendix

we only deal with the update of a pair of tensors sharing a single link, the approaches

developed here can be extended to models with more complex interactions.

C.1 Problem Overview

Following the reasoning of section 5.3, the iPEPS algorithm proceeds by applying a single

gate g to a particular link of the PEPS. We call this state |$g# * g |$#. Then, we want to

find the new state with modified tensors A! and B! called |$A"B"#, such that we minimise

the square distance:

dse = ||$A"B"# ' |$g#|2 * minA",B"

($$g | $g# ' $$A"B" | $g# ' $$g | $A"B"#+ $$A"B" | $A"B"#)

(C.1)

This quantity is depicted as a tensor network in fig. C.1. Here, we use the approximate

six-tensor environment (see fig. 5.4), which can computed by either the infinite MPS

approach (Appendix A) or the corner transfer matrix (Appendix B). A single update

amounts to finding the tensors A! and B! that minimise dse.

The problem as stated is an unconstrained quadratic optimization and is easy to solve.

The tensors A! and B! are determined by contracting A, B and the gate and splitting the

175

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176 Update Schemes for PEPS tensors

Figure C.1: The expression of the distance metric in terms of tensor contractions

resulting tensor. However, the dimension of the interjoining bond may then be as high as

Dd2. Continuing with such an approach, the dimensions of the PEPS bonds may continue

to grow exponentially over subsequent iterations, causing the computational cost of basic

operations to inflate and rendering the PEPS a computationally ine"cient representation

of the state. As a result, a basic requirement of the update is that it must find the tensors

A! and B! that minimise the square error, subject to the restriction that the dimension

of any PEPS bond is limited by D.

This is the same problem that was encountered with 1D systems and the MPS (see Ap-

pendix A for a discussion). In the iTEBD algorithm, it was seen that a very good approx-

imation to the optimal updated MPS tensors could be determined by simply truncating

the tensors returned by the local split operation. However, the basis for this algorithm

lay in the fact that the MPS can be expressed in a canonical form. In 2D, there exists

no known canonical form. A PEPS cannot be partitioned into two subsystems by cutting

a single bond, and the 2D analogy to the bipartite Schmidt representation about a link

seems elusive. However, since enforcing a restriction on the bond dimension does not

change the convex nature of the optimisation surface, there are many established numer-

ical algorithms that can be employed to minimise dse and determine A! and B!. In the

next sections, we describe two such algorithms.

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C.2 A Variational Algorithm 177

Figure C.2: The tensor network representation of equation C.2.

C.2 A Variational Algorithm

Here, we consider the A! and B! tensors as separate subspaces within to minimise the

cost function. The process is to alternate between the subspaces, locally minimising the

cost function at each step, in the hope of approaching the global minimum. In essence,

this means that for each iteration and each subsystem, we are solving an unconstrained

quadratic optimization problem. In order to do this, we must firstly define how to compute

the gradient of dse with respect to A! or B!. For a small deviation, ,!A and ignoring second

and higher-order terms, the change in the cost function ,dse,Amay be represented as a

tensor network as in figure C.2. Making the substitutions in figure C.3 and expressing

the tensor A! as a vector a!, the change in the cost function may be written:

,dse,A = ',a†NA 'N †A,a + ,a†MAa + a†MA,a (C.2)

It is easily seen that solving the linear equation MAa! = NA gives the ,dse,A = 0 solution.

Since MA is positive defined, it is possible to show that the second-order derivative is al-

ways positive, and hence the zero-gradient solution corresponds to a global minima within

the space of a! vectors. We then reshape a! into a tensor A! and make the replacement

A& A!.

After updating A, we find the analogous tensors NB and MB and solve for the vector b!

and the tensor B! accordingly before updating B.

The solution of MAa! = NA is obtained by inverting the matrix MA, at a cost of D12.

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178 Update Schemes for PEPS tensors

Figure C.3: The matrices NA and MA, which are required to calculate the derivative ofthe distance function with respect to the elements of A.

A More e!cient implementation

A more e"cient implementation of this approach can be achieved by updating the minimal

subspace of the tensors A! and B! a!ected by the link involved in the update. To do this,

we split the tensors A! and B! according to figure C.4. Then, we only need to update the

tensors X and Y . Assuming that d + D2, X and Y contain D2d2 coe"cients and MX

and MY are now a square matrices with D2d rows. The cost of inverting the matrices

MX and MY is D6d3.

A good initial starting point for the tensors X ! and Y ! may be obtained by contracting

the initial X and Y with the gate, appropriately splitting and then truncating the new

shared index to D.

C.3 Conjugate Gradient Algorithm

The well-known conjugate gradient (CG) algorithm o!ers an alternative means of finding

the PEPS tensors that minimise the distance cost function. It is similar to a steepest

descent algorithm, in that it uses gradient information to travel towards the minimum.

In this way we can think of CG optimisation as a directed walk in the space of bounded

dimension PEPS tensors, whereas the variational approach is a series of hops in this space.

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C.3 Conjugate Gradient Algorithm 179

Figure C.4: The splitting of A and B into W, X, Y and Z. Our new optimization domainis the elements of X and Y, reducing the number of coe"cients in our search from 2D4dto 2D2d2.

The conjugate gradient algorithm aims to solve an optimization where the objective takes

a quadratic form,

f(v) =1

2v† · T · v ' r · v, (C.3)

and we wish to solve for the v that minimises f(v). If we know T and r, the problem

is solved trivially. But assume that we do not know T and r. All that we know is that

the problem is approximately quadratic (and convex), and how to compute f(v) and the

gradient3f . The PEPS update problem can be cast in this form. Consider if we vectorize

the components of A! and B! into vectors a! and b! that are then concatenated to form

v. We know that the problem we are solving is quadratic and convex, and we can easily

evaluate the function f(v) = dse. The missing piece of the puzzle is being able to compute

the derivative of dse with respect to the individual elements of the vector v.

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180 Update Schemes for PEPS tensors

Since the vector v is a concatenation of a! and b!, we can construct the vector 3dse from

the concatenation of the vectors 3dse,A and 3dse,B. We have already seen that for a small

change, ,a! in the tensor A!, the cost function changes as

,dse,A = ',a!†NA" 'N †A",a! + ,a!†MA"a! + a!†MA",a! (C.4)

Consider that ,a! = ,x + j,y. In the purely real case,

,dse,A = ',xNA 'N †A,x + ,xMAa! + a!†MA,x

= 24(,x(MAa! 'NA))(C.5)

In the purely imaginary case, ,a! = j,y,

,dse,A = j,yNA ' jN †A,y ' j,yMAa! + ja!†MA,y

= 25(,y(MAa! 'NA))(C.6)

Realising that the gradient is the direction of greatest increase, it is easy to see that the

gradient for the elements of A! is then given by,

3dse,A = 4(MAa! 'NA) + j5(MAa! 'NA) (C.7)

This now defines the way in which we should modify the coe"cients of A!, in order to

move in the direction of the greatest increase in dse,A. The direction g0,A = '3dse,A is

then the direction of maximal decrease. Similarly,

3dse,B = 4(MBb! 'NB) + j5(MBb! 'NB) (C.8)

Concatenating 3dse,A and 3dse,B gives the gradient 3dse and the direction of greatest

decrease g0 = '3dse

The CG algorithm, like steepest descent, works an an iterative sense. The key di!erence

between the CG algorithm and steepest descent methods is what happens after the first

iteration. Labeling our choice of direction in the nth step as hn, in the first iteration, we

choose the direction of steepest descent

h0 = g0 (C.9)

In following iterations, we take the direction as

hn = gn + .nhn"1 (C.10)

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C.4 Comparison of the Methods 181

where in the standard Fletcher-Reeves form,

.n =gn · gn

gn"1 · gn"1, (C.11)

in Polak-Ribiere form,

.n =gn · (gn ' gn"1)

gn"1 · gn"1, (C.12)

and in Hestenes-Stiefel form,

.n =gn · (gn ' gn"1)

gn"1 · (gn ' gn"1), (C.13)

As explained in [PTVF92], CG techniques avoid the problem of the steepest descent

method getting stuck in narrow valleys by using a combination of the current gradient

information and past gradient information. Theoretically, for an exact quadratic min-

imization with N2 free parameters, the conjugate gradient scheme finds the minima in

N line minimizations. Even when the quadratic form is only approximate, a significant

speedup over generic steepest descent methods is observed.

Line Minimization

Once the gradient gn and direction of optimization hn have been calculated, we undertake

a line minimization by stepping along the line in the direction hn looking for a turning

point. This is an iterative process itself. The technique we use to do this borrows heavily

from [PTVF92] and [Mac04]. At each step along the line, q, we need to calculate the

instantaneous gradient 3dse,n,q, and the scalar product, lq = '3dse,n,q · hn. We call this

the line gradient. Whilst it is di"cult to exactly determine the turning point, we can

approximate its position by detecting two close points with di!ering sign of the gradient

and interpolating between them.

C.4 Comparison of the Methods

C.4.1 Stability and E"ectiveness

The choice of which method to use is a trade-o! between stability and e!ectiveness. In

general, the variational update performs better (i.e. returns a slightly lower energy PEPS

approximation) where it is stable. But our empirical observation is that its stability is

only ensured for D = 2.

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182 Update Schemes for PEPS tensors

We suggest that the variational solution has three drawbacks.

• The updates of tensors A! and B! are performed separately. As a result, it is possible

we will preferentially optimise in one subspace and end up at a local minima.

• The inversion of the matrix M introduces spurious correlations in the system.

• It is more susceptible to the so-called positive definite problem

The first concern is greatly mitigated by finding good initial starting points, which as

explained above is simple in the case of a two-body nearest-neighbour gate. However,

when we have more exotic update terms and more tensors to update - such as the four-

body plaquette terms considered in Chapter 10 - this can be a problem.

The second and third problems are of major concern and interrelated. To understand

their origin and implications we must consider the way in which the environment is an

approximation to the actual environment. As a simple model, consider that we are up-

dating A and our representation of MX is diagonal. Also, consider that the environment

is contracted with finite &, and that this approximation introduces an error. In our simple

model, this error is represented as the nth eigenvalue %n of MX being o!set a small amount

(n from the exact (infinite &) value. When we invert MX , this small error can become very

significant. Given an eigenvalue of MX , %n = %n,exact + (n, under inversion this becomes

%"1n = %"1

n,exact ' (n+2

n,exact. For large %n,exact the e!ect of !n will be negligible. For small

%n,exact, however, the displacement of %"1n from %"1

n,exact can be very large. This does not

e!ect our ability to minimise dse, however it can mean that the tensors we find are far

from those which would be obtained if the environment were computed exactly. We may

find large amounts of numerical noise in the tensors, and when the replacement of the

new tensors globally is enforced, this means that we may introduce spurious correlations

to the system. This makes it progressively harder to converge an environment with finite

&.

Consider now that one of the eigenvalues of MX is a small negative value. An exact

representation of MX is positive-semidefinite, but the finite-& approximation can destroy

this symmetry. The e!ect of this is that the convexity of the problem is destroyed. There

exists a direction along which the cost function decreases uniformly, where in the infinite-&

representation, no such direction exists. During inversion, this small negative component

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C.4 Comparison of the Methods 183

becomes a large negative component and greatly changes the coe"cients of the new tensor.

This is known as the positive definite problem.

The problems in the above discussion are problems related to the approximate nature of

the environment, and simply surface during the variational update. In light of this, it is

imperative to consider the relative influence of these problems in the CG update. Firstly,

there is no inversion in the CG update. The environment is used to compute the gradient,

but the gradient is linear in the environment and so small errors to small components do

not have a great influence on determining the new tensors. Secondly, the CG update is in

essence a walk rather than a series of jumps. So even though there may exist a direction

in which the cost function always decreases, since this is most likely associated with a

small negative eigenvalue, the slope in this direction will be very slight. It is likely that

the CG algorithm will preference directions in which the descent is more severe and for

which the line minimization encounters a turning point. Even if it does choose to move

in this uniformly decreasing direction, since there are only a finite number of steps in any

stage of the algorithm it is likely that the resulting state will not be displaced too far.

In our iPEPS simulations, we have only found the variational procedure stable for D = 2

PEPS with nearest neighbour updates. For higher D, the finite-& e!ects become too

severe, and the CG algorithm must be used. Also, for the four-site update used in Chapter

10, it was seen that the variational update too heavily preferences the first tensor of the

update, causing the algorithm to find a solution that did not optimise the cost function

well in a finite number of steps. It should also be noted that by-in-large, when both

approaches work well, the observed physical properties agree, but the characteristics of

the tensor coe"cients may not. This means that, as an example, a D = 2 PEPS converged

with the variational update is not necessarily a good starting point for a D = 3 simulation

with the CG algorithm.

C.4.2 Computational Cost of the Link Update Schemes

Computational Cost of the Variational Update

The cost of the variational update depends on two main computations:

1. The cost of computing MX and NX (and MY and NY ).

2. The inversion of the MX (and MY ) operator.

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184 Update Schemes for PEPS tensors

We assume that there are a total of SV iterations of the variational update. For the first

component, we incur an out-of-loop cost scaling as &2D6d2 + &3D4d2 + &2D4d4, and a

per-iteration cost scaling as D5d5. The inversion in the most aggressive implementation

scales as D6d3. Thus, the total cost scales as &2D6d2+&3D4d2+&2D4d4+SV(D5d5+D6d3)

For the CG update, we assume SCG iterations of the update, each containing a line

minimization of SLM steps. In practice, only the maximum number of line minimization

steps is specified, as it is possible for the turning point to be detected quickly. We need

to once again calculate MX , NX , MY and NY to compute the gradient at a given point.

Thus, the CG algorithm has an out-of-loop cost scaling as &2D6d2 + &3D4d2 + &2D4d4

and an in-loop cost scaling as SLMD5d5. The total cost scales as &2D6d2 + &3D4d2 +

&2D4d4 +SCGSLMD5d5. As mentioned earlier, SCG itself scales with the dimension of the

vector space in which the solution vector, v, exists. Thus, for the update of a single link,

SCG % D2d2.

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