The Nature of the Probability Distribution Function of the Local Energy in Ising Spin Glass

TL;DR

This paper investigates the probability distribution of local energy in the $ ext{±}J$ Ising model, deriving relations at specific conditions and revealing multiple peaks related to frustration patterns through numerical analysis.

Contribution

It derives new relations for the local energy distribution at specific points and generalizes Nishimori's internal energy result, with numerical evidence of multiple peaks.

Findings

Relations for local energy distribution at $p=1/2$ and Nishimori-line.

Generalization of Nishimori's internal energy result.

Numerical evidence of multiple peaks related to frustration patterns.

Abstract

The nature of the probability distribution function of the local energy in the $\pm J$ Ising model has been investigated. At finite temperature, it has been derived that the probability distribution function must satisfy several relations at $p=1/2$ ($p$ is the concentrationof the ferromagnetic bond) and at Nishimori-line, respectively on any lattice in any dimension. They relate the probability distribution function corresponding to the local energy lower than $-\tanh (K)$ with that corresponding to the local enegy greater than $-\tanh (K)$. ($K$ is the inverse temperature.) The present results at Nishimori-line are, in a sense, generalization of Nishimori's result about the internal energy obtained by the local gauge transformation. Moreover, from the numerical calculation in the two-dimentional $\pm J$ Ising model, it is found that, in a certain temperature region, the probability distribution function of the local energy has several peaks which are related to the patterns of frustration around a bond of the lattice.