Finite size scaling of the de Almeida-Thouless instability in random sparse networks

TL;DR

This paper investigates the finite size effects on the de Almeida-Thouless instability in spin glasses on sparse networks, revealing scaling behaviors and the impact of external fields through numerical and theoretical analysis.

Contribution

It introduces a phenomenological approach to evaluate the spin glass susceptibility in finite sparse networks and examines the validity of the known scaling relation under external fields.

Findings

Good agreement with theory in high temperature region

Scaling relation holds without external fields

Finite size corrections are significant with external fields

Abstract

We study, in random sparse networks, finite size scaling of the spin glass susceptibility $\chi_{\rm SG}$, which is a proper measure of the de Almeida-Thouless (AT) instability of spin glass systems. Using a phenomenological argument regarding the band edge behavior of the Hessian eigenvalue distribution, we discuss how $\chi_{\rm SG}$ is evaluated in infinitely large random sparse networks, which are usually identified with Bethe trees, and how it should be corrected in finite systems. In the high temperature region, data of extensive numerical experiments are generally in good agreement with the theoretical values of $\chi_{\rm SG}$ determined from the Bethe tree. In the absence of external fields, the data also show a scaling relation $\chi_{\rm SG}=N^{1/3}F(N^{1/3}|T-T_c|/T_c)$, which has been conjectured in the literature, where $T_c$ is the critical temperature. In the presence of external fields, on the other hand, the numerical data are not consistent with this scaling relation. A numerical analysis of Hessian eigenvalues implies that strong finite size corrections of the lower band edge of the eigenvalue distribution, which seem relevant only in the presence of the fields, are a major source of inconsistency. This may be related to the known difficulty in using only numerical methods to detect the AT instability.