Numerically exact correlations and sampling in the two-dimensional Ising spin glass

TL;DR

This paper introduces a fast, exact computational method for calculating correlations and sampling in two-dimensional Ising spin glasses, extending existing techniques to arbitrary couplings and providing precise results and timings.

Contribution

It develops a new, efficient method and software for exact calculations and sampling in 2D Ising models with arbitrary couplings, improving upon previous approaches.

Findings

Accurate computation of partition functions and correlations

Effective exact sampling of spin configurations

Comparison with analytic results confirms accuracy

Abstract

A powerful existing technique for evaluating statistical mechanical quantities in two-dimensional Ising models is based on constructing a matrix representing the nearest neighbor spin couplings and then evaluating the Pfaffian of the matrix. Utilizing this technique and other more recent developments in evaluating elements of inverse matrices and exact sampling, a method and computer code for studying two-dimensional Ising models is developed. The formulation of this method is convenient and fast for computing the partition function and spin correlations. It is also useful for exact sampling, where configurations are directly generated with probability given by the Boltzmann distribution. These methods apply to Ising model samples with arbitrary nearest-neighbor couplings and can also be applied to general dimer models. Example results of computations are described, including comparisons with analytic results for the ferromagnetic Ising model, and timing information is provided.