Exact Ground States of Large Two-Dimensional Planar Ising Spin Glasses

TL;DR

This paper introduces an efficient algorithm based on Kasteleyn's method for computing exact ground states of large two-dimensional planar Ising spin glasses, enabling analysis of much larger systems than previously possible.

Contribution

The authors develop a novel algorithm leveraging Kasteleyn cities to compute exact ground states for 2D Ising spin glasses up to 3000x3000 spins, surpassing prior size limitations.

Findings

Successfully computed ground states for 3000x3000 spin lattices.

Verified the correctness of heuristic ground state computations.

Evaluated and improved heuristic solution quality.

Abstract

Studying spin-glass physics through analyzing their ground-state properties has a long history. Although there exist polynomial-time algorithms for the two-dimensional planar case, where the problem of finding ground states is transformed to a minimum-weight perfect matching problem, the reachable system sizes have been limited both by the needed CPU time and by memory requirements. In this work, we present an algorithm for the calculation of exact ground states for two-dimensional Ising spin glasses with free boundary conditions in at least one direction. The algorithmic foundations of the method date back to the work of Kasteleyn from the 1960s for computing the complete partition function of the Ising model. Using Kasteleyn cities, we calculate exact ground states for huge two-dimensional planar Ising spin-glass lattices (up to 3000x3000 spins) within reasonable time. According to our knowledge, these are the largest sizes currently available. Kasteleyn cities were recently also used by Thomas and Middleton in the context of extended ground states on the torus. Moreover, they show that the method can also be used for computing ground states of planar graphs. Furthermore, we point out that the correctness of heuristically computed ground states can easily be verified. Finally, we evaluate the solution quality of heuristic variants of the Bieche et al. approach.