A New Method to Calculate the Spin-Glass Order Parameter of the Two-Dimensional +/-J Ising Model

TL;DR

This paper introduces a new numerical method for calculating the spin-glass order parameter in the 2D +/-J Ising model, enabling more accurate analysis of phase transitions.

Contribution

A novel method using an identity to compute moments of the spin overlap as simple averages, improving accuracy and efficiency over traditional transfer matrix approaches.

Findings

Evidence suggests a finite-temperature spin-glass phase transition.

Scaling plots indicate possible phase transition, but results are affected by corrections.

Uncertainty remains whether the transition temperature is zero or finite.

Abstract

A new method to numerically calculate the $n$th moment of the spin overlap of the two-dimensional $\pm J$ Ising model is developed using the identity derived by one of the authors (HK) several years ago. By using the method, the $n$th moment of the spin overlap can be calculated as a simple average of the $n$th moment of the total spins with a modified bond probability distribution. The values of the Binder parameter etc have been extensively calculated with the linear size, $L$, up to L=23. The accuracy of the calculations in the present method is similar to that in the conventional transfer matrix method with about $10^{5}$ bond samples. The simple scaling plots of the Binder parameter and the spin-glass susceptibility indicate the existence of a finite-temperature spin-glass phase transition. We find, however, that the estimation of $T_{\rm c}$ is strongly affected by the corrections to scaling within the present data ($L\leq 23$). Thus, there still remains the possibility that $T_{\rm c}=0$, contrary to the recent results which suggest the existence of a finite-temperature spin-glass phase transition.