Two-State Spin Systems with Negative Interactions

TL;DR

This paper investigates the computational complexity of approximating partition functions in two-state spin systems with arbitrary interactions, revealing new phase transitions between hard and easy regimes.

Contribution

It extends previous work by analyzing general 2x2 interaction matrices, identifying regions with different computational complexities and phase transitions.

Findings

Determines regions where approximation is -hard

Identifies regions with polynomial-time approximation schemes

Reveals new computational phase transitions in spin systems

Abstract

We study the approximability of computing the partition functions of two-state spin systems. The problem is parameterized by a $2\times 2$ symmetric matrix. Previous results on this problem were restricted either to the case where the matrix has non-negative entries, or to the case where the diagonal entries are equal, i.e. Ising models. In this paper, we study the generalization to arbitrary $2\times 2$ interaction matrices with real entries. We show that in some regions of the parameter space, it's \#P-hard to even determine the sign of the partition function, while in other regions there are fully polynomial approximation schemes for the partition function. Our results reveal several new computational phase transitions.