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

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.

18 pages, 37 epsfiles

References in corpus (3)

• Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension $d=1,2,3$ via a Monte-Carlo procedure in the disorder
• Probing tails of energy distributions using importance-sampling in the disorder with a guiding function
• A Monte Carlo Algorithm for Sampling Rare Events: Application to a Search for the Griffiths Singularity

Cited by in corpus (1)

• Testing Error Correcting Codes by Multicanonical Sampling of Rare Events