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  • 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

  • c© Jacob Jordan, 2011.

    Produced in LATEX2ε.

  • 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.

    i

  • ii

  • Statements of Contributions

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

    sis

    • J. Jordan, R. Orús, 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. Orús 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

    iii

  • None.

    Published Works by the Author Incorporated into the Thesis

    • J. Jordan, R. Orús, 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. Orús 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).

    iv

  • 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.

    v

  • vi

  • 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 effort 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 efficient 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 difficulties.

    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 difficulty 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.

    vii

  • 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%).

    viii

  • 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

    ix

  • x

  • Contents

    List of Tables xvii

    List of Figures xix

    1 Introduction 1

    2 The Quantum-Classical Correspondence 5