Distribution of Lee-Yang zeros and Griffiths singularities in the $\pm J$ model of spin glasses

TL;DR

This paper analyzes the distribution of Lee-Yang zeros in 2D and 3D $\

Contribution

It provides the first evidence of Griffiths singularities in equilibrium spin glass systems through analysis of partition function zeros.

Findings

Zeros are distributed widely in the complex field plane.

Zeros on the imaginary axis dominate critical behavior.

Density of zeros on the imaginary axis shows an essential singularity.

Abstract

We investigate the distribution of zeros of the partition function of the two- and three-dimensional symmetric $\pm J$ Ising spin glasses on the complex field plane. We use the method to analytically implement the idea of numerical transfer matrix which provides us with the exact expression of the partition function as a polynomial of fugacity. The results show that zeros are distributed in a wide region in the complex field plane. Nevertheless we observe that zeros on the imaginary axis play dominant roles in the critical behaviour since zeros on the imaginary axis are in closer proximity to the real axis. We estimate the density of zeros on the imaginary axis by an importance-sampling Monte Carlo algorithm, which enables us to sample very rare events. Our result suggests that the density has an essential singularity at the origin. This observation is consistent with the existence of Griffiths singularities in the present systems. This is the first evidence for Griffiths singularities in spin glass systems in equilibrium.