Antiferromagnetic Ising model in an imaginary magnetic field

TL;DR

This paper investigates the two-dimensional antiferromagnetic Ising model with an imaginary magnetic field, providing exact analytical results and benchmarks for Monte Carlo methods in systems with a sign problem.

Contribution

It offers exact cumulant calculations and physical observable results for a model with a sign problem, serving as a benchmark for Monte Carlo techniques.

Findings

Confirmed the mean-field qualitative picture of the model.

Provided accurate high-temperature regime results including the critical line.

Validated the reliability of analytical methods in complex sign problem systems.

Abstract

We study the two-dimensional antiferromagnetic Ising model with a purely imaginary magnetic field, which can be thought of as a toy model for the usual $\theta$ physics. Our motivation is to have a benchmark calculation in a system which suffers from a strong sign problem, so that our results can be used to test Monte Carlo methods developed to tackle such problems. We analyze here this model by means of analytical techniques, computing exactly the first eight cumulants of the expansion of the effective Hamiltonian in powers of the inverse temperature, and calculating physical observables for a large number of degrees of freedom with the help of standard multi-precision algorithms. We report accurate results for the free energy density, internal energy, standard and staggered magnetization, and the position and nature of the critical line, which confirm the mean-field qualitative picture, and which should be quantitatively reliable, at least in the high-temperature regime, including the entire critical line.