Partition Function Zeros of an Ising Spin Glass

TL;DR

This paper investigates the distribution of partition function zeros in finite-size Ising spin glasses, exploring their connection to chaotic renormalization flows, Julia sets, and polynomial mappings, with implications for understanding spin glass behavior.

Contribution

It introduces a novel analysis of partition function zeros in spin glasses, linking mathematical polynomial mappings and chaotic dynamics to physical properties.

Findings

Zeros form complex patterns related to Julia sets

Connections between polynomial mappings and zero distributions are established

Discussion of potential limiting behaviors as system size grows

Abstract

We study the pattern of zeros emerging from exact partition function evaluations of Ising spin glasses on conventional finite lattices of varying sizes. A large number of random bond configurations are probed in the framework of quenched averages. This study is motivated by the relationship between hierarchical lattice models whose partition function zeros fall on Julia sets and chaotic renormalization flows in such models with frustration, and by the possible connection of the latter with spin glass behaviour. In any finite volume, the simultaneous distribution of the zeros of all partition functions can be viewed as part of the more general problem of finding the location of all the zeros of a certain class of random polynomials with positive integer coefficients. Some aspects of this problem have been studied in various branches of mathematics, and we show how polynomial mappings which are used in graph theory to classify graphs, may help in characterizing the distribution of zeros. We finally discuss the possible limiting set as the volume is sent to infinity.