Critical Point Scaling of Ising Spin Glasses in a Magnetic Field

TL;DR

This paper investigates the critical point scaling behavior of Ising spin glasses in a magnetic field, analyzing the crossover scaling function and examining the existence of the de Almeida-Thouless line in various dimensions.

Contribution

The study provides analytical and numerical calculations of the crossover scaling function for spin glasses, and offers evidence against the existence of the de Almeida-Thouless line in dimensions less than or equal to six.

Findings

Analytical form of the scaling function in the mean-field limit.

Numerical estimates of the scaling function in three dimensions.

No evidence of the de Almeida-Thouless line for d ≤ 6.

Abstract

Critical point scaling in a field $H$ applies for the limits $t\to 0$, (where $t=T/T_c-1$) and $H\to 0$ but with the ratio $R=t/H^{2/\Delta}$ finite. $\Delta$ is a critical exponent of the zero-field transition. We study the replicon correlation length $\xi$ and from it the crossover scaling function $f(R)$ defined via $1/(\xi H^{4/(d+2-\eta)}) \sim f(R)$. We have calculated analytically $f(R)$ for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension d=3 we have determined the exponents and the critical scaling function $f(R)$ within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for $f(R)$ in d=3 in reasonable agreement with those of the Monte Carlo simulations at the values of R for which they can be compared. If there were a de Almeida-Thouless (AT) line for $d \le 6$ it would appear as a zero of the function $f(R)$ at some negative value of R, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for $d \le 6$, which we review.