Introduction The Model and the algorithm Simulation of the system Results Conclusions

2D and 3D Antiferromagnetic Ising Model with”topological” term at θ = π

V. Azcoiti, E. Follana, G. Cortese, M. GiordanoDepartamento de Fisica Teorica, Universidad de Zaragoza

31st International Symposium on Lattice Field TheoryJuly 29, 2013, Mainz, Germany

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Outline

I Introduction

I The Model and the algorithm

I Simulation of the system

I Results

I Conclusions

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Introduction

Motivation of the work:

I Many interesting physical systems do not have an efficientnumerical algorithms yet.

I An example: QCD at finite density or with a non-vanishing θterm.

I It is thus of great interest to study novel simulationalgorithms.

In the present work we develop and test a geometric algorithmwhich is applicable to the two and three-dimensionalantiferromagnetic Ising model with an imaginary magnetic field iθ,and which solves the sign problem that this model has when usingstandard algorithms.

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

The Model and the algorithm

Reduced Hamiltonian of the system:

H[{sx},F , h] = −F∑

(x ,y)∈B

sxsy −h

2

∑x

sx ,

where

I F = J/(kT ) → coupling

I h = 2B/(kT ) → reduced magnetic field

I Q = 12

∑x sx (from −N2/2 and N2/2) → ”topological

charge”

It is then worth studying what happens for imaginary values of thereduced magnetic field h, i.e., for h = iθ.

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

weight of a configuration not a positive real number⇒ ”sign problem”

For θ = π we can circumvent this problem.

Z (F , θ = π) =∑

{sx}, sx=±1

eF∑

(x,y)∈B sx sy+i π2

∑z sz

=∑

{sx}, sx=±1

∏(x ,y)∈B

[cosh(F ) + sinh(F )sxsy ]∏z

sz ,

The terms that contribute to the partition function are those forwhich a given spin variable appears an odd number of times.

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Decomposing the lattice in two staggered sublattices we canrewrite our partition function as

Z (F , θ = π) =∑

{sx}, sx=±1

∏(x ,y)∈B

[cosh(F )− sinh(F )sxsy ]∏z

sz ;

⇒ Z (F , θ = π) = Z (−F , θ = π) at θ = π;

⇒ the terms that survive in this expansion are those with aneven number of terms sxsy .

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Considering the contributions following these rules:

I a factor 2N for the sum over the spins;

I a factor of sinh(|F |)N [b], where N [b] is the number of bondsfor a given configuration;

I a factor cosh(|F |)N [b] where N [b] = N [B]−N [b] is thenumber of inactive bonds;

the partition function becomes:

Z (F , θ = π) = 2N2

cosh(|F |)N [B]∑b∈B!

tanh(|F |)N [b] .

All configurations (”graphs”) positive weights ⇒ no sign problem !

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Simulation of the system

Number of links at any site → odd:

I 2D → 1, 3 bonds

I 3D → 1, 3, 5 bonds

The dual lattice sites:

u uuu

1

2

3

4 x∗

The state of bond i at the dual lattice site x∗ is denoted byAi (x

∗), and we set

Ai (x∗)→

{1 if active

0 if inactive

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

We can draw all the admissible configuration, specified by a vectorA(x∗).

I S(x∗) → graphs corresponding to these configurations.

I w(x∗) =∑4

i=1 Ai (x∗) → number of active bonds at site x∗ in

a given configuration.

The key observation is that, as the number of external bondstouching a vertex is fixed, and the total number of bonds touchinga vertex must be odd, when changing A(x∗) we must be sure thatthe number of internal bonds that change state, touching a givenvertex, is an even number, i.e., 0 or 2 (or 4 in 3D).

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

S(x∗) w(x∗) A(x∗)

r rrr0 (0, 0, 0, 0)

r rrr1 (1, 0, 0, 0)

r rrr1 (0, 1, 0, 0)

r rrr1 (0, 0, 1, 0)

r rrr1 (0, 0, 0, 1)

r rrr2 (1, 1, 0, 0)

r rrr2 (1, 0, 1, 0)

r rrr2 (1, 0, 0, 1)

Active bonds → solid line, inactive bonds → no line.w(x∗) =

∑4i=1 Ai (x

∗) → number of active bonds.10 / 22

Introduction The Model and the algorithm Simulation of the system Results Conclusions

Updating the configurations:

A(x∗)→

{IA(x∗) = A(x∗) ,

CA(x∗) = I − A(x∗)

where

I I → identity

I C → conjugation

The variation ∆w(x∗) of the number of active bonds is given by

∆w(x∗) =4∑

i=1

CAi (x∗)−Ai (x

∗) =4∑

i=1

Ii−2Ai (x∗) = 2[2−w(x∗)] .

To pass to another admissible configuration, we have only twopossibilities: either leave everything unchanged, or change the stateof all the bonds at the given square/dual lattice site (next table).

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

S(x∗) CS(x∗) ∆w(x∗)

r rrr r rrr4

r rrr r rrr2

r rrr r rrr2

r rrr r rrr2

r rrr r rrr2

r rrr r rrr0

r rrr r rrr0

r rrr r rrr0

Transformation under C12 / 22

Introduction The Model and the algorithm Simulation of the system Results Conclusions

Ergodicity

S(x∗) RS(x∗)

r rrr r rrrr rrr r rrrr rrr r rrrr rrr r rrrr rrr r rrrr rrr r rrr

Reduction: it coincides with the identity if A2 = 0, and withconjugation if A2 = 1

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Open boundary conditions: Ergodicity

r rrr rr. . .

rr rr rr

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r rrr rr. . .

rr rr rr

rrrrrr

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Figure: (Left) Right-most colum after the first step of reduction. (Right)Two right-most columns after the second step of reduction.

r rrr rr. . .

rr rr rr

rrrrrr

rrrrrr

. . .

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r rrr rr. . .

rr rr rr

rrrrrr

rrrrrr

Reduced configuration →

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Open boundary conditions: Ergodicity

r rrr rr. . .

rr rr rr

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r rrr rr. . .

rr rr rr

rrrrrr

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Figure: (Left) Right-most colum after the first step of reduction. (Right)Two right-most columns after the second step of reduction.

r rrr rr. . .

rr rr rr

rrrrrr

rrrrrr

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r rrr rr. . .

rr rr rr

rrrrrr

rrrrrr

Reduced configuration →

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Periodic boundary conditions: Ergodicity

r rrr rr

N−1 N 1

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rr rr rr

rrrrrr

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r rrr rr

N−1 N 1

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rr rr rr

rrrrrr

rrrrrr

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r rrr rr

N 1 2

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rr rr rr

rrrrrr

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r rrr rr

N 1 2

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Periodic boundary conditions: Ergodicity

r rrr rr

N−1 N 1

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rr rr rr

rrrrrr

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r rrr rr

N−1 N 1

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rr rr rr

rrrrrr

rrrrrr

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r rrr rr

N 1 2

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rr rr rr

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r rrr rr

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Periodic boundary conditions: Ergodicity

r rrr rr

N−1 N 1

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rr rr rr

rrrrrr

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r rrr rr

N−1 N 1

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rr rr rr

rrrrrr

rrrrrr

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r rrr rr

N 1 2

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r rrr rr

N 1 2

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15 / 22

Introduction The Model and the algorithm Simulation of the system Results Conclusions

Periodic boundary conditions: Ergodicity

r rrr rr

N−1 N 1

. . .

rr rr rr

rrrrrr

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r rrr rr

N−1 N 1

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rr rr rr

rrrrrr

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r rrr rr

N 1 2

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rr rr rr

rrrrrr

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15 / 22

Introduction The Model and the algorithm Simulation of the system Results Conclusions

Periodic boundary conditions: Ergodicity

r rrr rr

N 1 2

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rr rr rr

rrrrrr

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r rrr rr

N 1 2

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rrrrrr

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4 inequivalent reduced configurations in 2D

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Results: Correlation function in the 2D model

C (d ,F ) ≡ 〈sxsx+d 1〉

0 10 20 30 40d

-2

-1

0

1

2

corr

elat

ion

func

tion

s

L=256L=512L=1024

F=-0.4

0 10 20 30 40 50d

-2

-1

0

1

2

corr

elat

ion

func

tion

s

L=256L=512L=1024

F=-0.6

0 10 20 30 40d

-2

-1

0

1

2

corr

elat

ion

func

tion

s

L=256L=512L=1024

F=-0.8

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

0 20 40 60d

-1,5

-1

-0,5

0

0,5

1

1,5co

rrel

atio

n fu

ncti

ons

L=256L=512L=800L=1024

F=J/(kBT)=-1.0

0 10 20 30 40 50 60d

-1,5

-1

-0,5

0

0,5

1

1,5

corr

elat

ion

func

tion

s

L=64L=512L=1024

F=J/(kBT)=-2.0

Results is in agreement with B. M. McCoy and T. T. Wu, Phys.Rev. 155 (1967) 438 T. D. Lee and C. N. Yang, Phys. Rev. 87(1952) 410, V. Matveev and R. Shrock, J. Phys. A 28 (1995)4859 and with the mean-field calculation done by V. Azcoiti,E. Follana, and A. Vaquero, Nucl. Phys. B 851 (2011) 420.

I the apparent decrease of C (d ,F ) is probably due to theheavy-tailed probability distributions of the correlators

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

0 20 40 60d

-1,5

-1

-0,5

0

0,5

1

1,5co

rrel

atio

n fu

ncti

ons

L=256L=512L=800L=1024

F=J/(kBT)=-1.0

0 10 20 30 40 50 60d

-1,5

-1

-0,5

0

0,5

1

1,5

corr

elat

ion

func

tion

s

L=64L=512L=1024

F=J/(kBT)=-2.0

Results is in agreement with B. M. McCoy and T. T. Wu, Phys.Rev. 155 (1967) 438 T. D. Lee and C. N. Yang, Phys. Rev. 87(1952) 410, V. Matveev and R. Shrock, J. Phys. A 28 (1995)4859 and with the mean-field calculation done by V. Azcoiti,E. Follana, and A. Vaquero, Nucl. Phys. B 851 (2011) 420.

I the apparent decrease of C (d ,F ) is probably due to theheavy-tailed probability distributions of the correlators

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

0 20 40 60d

-1,5

-1

-0,5

0

0,5

1

1,5co

rrel

atio

n fu

ncti

ons

L=256L=512L=800L=1024

F=J/(kBT)=-1.0

0 10 20 30 40 50 60d

-1,5

-1

-0,5

0

0,5

1

1,5

corr

elat

ion

func

tion

s

L=64L=512L=1024

F=J/(kBT)=-2.0

Results is in agreement with B. M. McCoy and T. T. Wu, Phys.Rev. 155 (1967) 438 T. D. Lee and C. N. Yang, Phys. Rev. 87(1952) 410, V. Matveev and R. Shrock, J. Phys. A 28 (1995)4859 and with the mean-field calculation done by V. Azcoiti,E. Follana, and A. Vaquero, Nucl. Phys. B 851 (2011) 420.

I the apparent decrease of C (d ,F ) is probably due to theheavy-tailed probability distributions of the correlators

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Probability distributions of the logarithm of the correlators

-20 -10 0 10ln(C)

0

0,1

0,2

0,3

0,4

0,5

Fre

quen

cy

d=2d=8d=16d=32

2D, L=64, F=-0.4

-0,6 -0,4 -0,2 0 0,2 0,4 0,6ln(C)

0

0,1

0,2

0,3

0,4

0,5

Fre

quen

cy

d=2d=8d=16d=32

2D, L=64, F=-2.0

-0,4 -0,2 0 0,2 0,4C

0

50

100

150

200

Fre

quen

cy

d=2d=8d=16d=24d=32

3D, L=64, F=-2.0

For a low coupling |F | the values are spread in a wider range thanfor F = −2.0, and also that a long tail is developed for largedistances.

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Results: Correlation function in the 3D model

0 10 20 30 40 50d

-2

-1

0

1

2

C(d

,F) L=32

L=64L=128L=256

3D, F=J/(kBT)=-0.6

0 10 20 30 40 50 60 70d

-2

-1

0

1

2

C(d

,F) L=32

L=64L=128L=256

3D, F=J/(kBT)=-1.2

0 10 20 30 40 50 60 70d

-2

-1

0

1

2

C(d

,F) L=32

L=64L=128L=256

3D, F=J/(k|sBT)=-2.0

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Energy density and Specific Heat

0,0 0,5 1,0 1,5 2,0 2,5 3,0

|F|

-6

-5

-4

-3

-2

-1

0

Ene

rgy

dens

ity

L=16L=32L=64L=128L=256

2D

0,0 0,5 1,0 1,5 2,0

|F|

-4

-3

-2

-1

0

Ene

rgy

dens

ity

L=16L=32L=64L=128L=256

3D

0,0 0,5 1,0 1,5 2,0 2,5 3,0

|F|

0

0,02

0,04

0,06

0,08

0,1

Spec

ific

hea

t

L=16L=32L=64L=128L=256

2D

0,0 0,5 1,0 1,5 2,0

|F|

0,01

0,02

0,03

0,04

0,05

0,06

0,07

Spec

ific

hea

tL=16L=32L=64L=128L=256

3D

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Introduction The Model and the algorithm Simulation of the system Results Conclusions

Conclusions

I We developed, for a model not free from the sign problem, anew algorithm able to circumvent this obstacle

I We tested successfully for the 2D antiferromagnetic Isingmodel and we studied also the 3D version

I The behavior of the correlation function as well the specificheat are in agreement with the predictions

I No phase transitions for these models at θ = π

I This technique maybe can be applied to study otherinteresting models

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