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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 883 (2014) 656–684

www.elsevier.com/locate/nuclphysb

A geometric Monte Carlo algorithm for

the antiferromagnetic Ising model with “topological”

term at θ = π

V. Azcoiti

,G.Cortese

a,b

, E. Follana

a,∗

, M. Giordano

Departamento de Física Teórica, Universidad de Zaragoza, Calle Pedro Cerbuna 12, E-50009 Zaragoza, Spain

Instituto de Física Teórica, UAM/CSIC, Cantoblanco, E-28049 Madrid, Spain

Institute for Nuclear Research of the Hungarian Academy of Sciences (ATOMKI), Bem tér 18/c,

H-4026 Debrecen, Hungary

Received 13 January 2014; received in revised form 4 April 2014; accepted 5 April 2014

Available online 13 April 2014

Editor: Hubert Saleur

Abstract

In this work we study the two and three-dimensional antiferromagnetic Ising model with an imaginary

magnetic

ﬁeld iθ at θ = π. In order to perform numerical simulations of the system we introduce a new

geometric algorithm not affected by the sign problem. Our results for the 2D model are in agreement with

the analytical solution. We also present new results for the 3D model which are qualitatively in agreement

with mean-ﬁeld predictions.

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

1. Introduction

Since its introduction many years ago, the Ising model has been a prototype statistical sys-

tem for studying phase transitions and critical phenomena [1]. With the advent of the epoch of

computer numerical simulations to study statistical systems, this model has become even more

Corresponding author.

E-mail addresses: azcoiti@azcoiti.unizar

.es (V. Azcoiti), cortese@unizar.es (G. Cortese), efollana@unizar.es

(E. Follana), giordano@atomki.mta.hu (M. Giordano).

http://dx.doi.org/10.1016/j.nuclphysb.2014.04.004

The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

V. Azcoiti et al. / Nuclear Physics B 883 (2014) 656–684 657

important as a test bench to develop new algorithms. There are many interesting physical systems

for which, due to the sign problem, we do not have efﬁcient numerical algorithms. Some exam-

ples include QCD at ﬁnite density or with a non-vanishing θ term. This situation has hindered

progress in such ﬁelds for a long time, and it is thus of great interest to study novel simulation

algorithms. In the present work we develop and test a new algorithm, which belongs to a class of

“geometric” algorithms [2–20], and which is applicable to the D  2 dimensional antiferromag-

netic Ising model with an imaginary magnetic ﬁeld iθ (see [21] and [22])atθ = π , with which

we are able to solve the sign problem that afﬂicts this model when using standard algorithms.

Preliminary results were presented in [24].

This paper is organized as follows. In Section 2 we

introduce the antiferromagnetic Ising

model. In Section 3 we derive our geometric representation of the model. Section 4 is devoted

to the construction of the numerical algorithm. In Section 5 we present the numerical results

and ﬁnally Section 6 contains our conclusions. Some technical details on the ergodicity of the

algorithm as well as on the numerical analysis are contained in Appendices A–D.

2. The antiferromagnetic Ising model with a topological term

We consider the Ising model in D  2 dimensions, deﬁned on a hypercubic lattice Λ with

an even number of sites L = 2n in each direction, and with either open or periodic boundary

conditions. The Hamiltonian of the model is



},J,B



=−J



(x,y)∈B

− B



. (1)

Here the spin variables are s

=±1, and the sum



(x,y)∈B

is over the pairs of sites (x, y) that are

nearest neighbors; we denote the set of all such pairs by B. Moreover, J is the coupling between

nearest neighbors, and B is an external magnetic ﬁeld. The reduced Hamiltonian H = H/(k

T),

where T is the temperature and k

the Boltzmann constant, is written as



},F,h



=−F



(x,y)∈B

−



, (2)

with F = J/(k

T), h = 2B/(k

T). As the total number of spins is L

= (2n)

, and therefore

even, the quantity Q =



is an integer number, taking values between −L

/2 and L

/2.

Q can then be thought of as playing the role of a topological charge. It is then worth studying

what happens for imaginary values of the reduced magnetic ﬁeld h, i.e., for h = iθ. The topo-

logical charge Q is odd under the Z

transformation s

→−s

∀x: while at θ = 0 the system is

symmetric under this transformation, for θ = 0,π the Z

symmetry of the system is explicitly

broken. At θ = π the contribution of the topological charge to the Boltzmann factor amounts to

iπQ

= (−1)

−Q

, (3)

i.e., this contribution is Z

invariant, and therefore the Z

symmetry is restored; it has to be

checked if it is spontaneously broken or not.

3. Geometric representation of the model

3.1. Partition function

We will now introduce a geometric representation for the model at h = iθ = iπ. Let us rewrite

the partition function of the system,

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