Exact Algorithm for Sampling the 2D Ising Spin Glass

TL;DR

This paper introduces an exact, efficient sampling algorithm for 2D Ising spin glasses at finite temperature, enabling the study of complex configurations and correlations beyond traditional simulation methods.

Contribution

The paper presents a novel recursive algorithm that uses Pfaffian elimination to efficiently sample spin configurations of 2D Ising spin glasses with improved runtime and precision handling.

Findings

Runs in O(n^{3/2}) time for n spins

Handles both planar and toroidal samples

Effective for system sizes up to 128^2

Abstract

A sampling algorithm is presented that generates spin glass configurations of the 2D Edwards-Anderson Ising spin glass at finite temperature, with probabilities proportional to their Boltzmann weights. Such an algorithm overcomes the slow dynamics of direct simulation and can be used to study long-range correlation functions and coarse-grained dynamics. The algorithm uses a correspondence between spin configurations on a regular lattice and dimer (edge) coverings of a related graph: Wilson's algorithm [D. B. Wilson, Proc. 8th Symp. Discrete Algorithms 258, (1997)] for sampling dimer coverings on a planar lattice is adapted to generate samplings for the dimer problem corresponding to both planar and toroidal spin glass samples. This algorithm is recursive: it computes probabilities for spins along a "separator" that divides the sample in half. Given the spins on the separator, sample configurations for the two separated halves are generated by further division and assignment. The algorithm is simplified by using Pfaffian elimination, rather than Gaussian elimination, for sampling dimer configurations. For n spins and given floating point precision, the algorithm has an asymptotic run-time of O(n^{3/2}); it is found that the required precision scales as inverse temperature and grows only slowly with system size. Sample applications and benchmarking results are presented for samples of size up to n=128^2, with fixed and periodic boundary conditions.