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

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.