Haake-Lewenstein-Wilkens approach to spin-glasses revisited

TL;DR

This paper revisits the Haake-Lewenstein-Wilkens approach to the Edwards-Anderson spin glass model, providing new insights into the order parameter and phase transitions across different dimensions using saddle point methods.

Contribution

It applies saddle point analysis to the HLW approach, offering new predictions on spin glass order and transitions in 2D, 3D, and 4D systems, challenging existing beliefs.

Findings

Predicts non-zero Edwards-Anderson order parameter in 3D and 4D below critical temperature.

Suggests possible spin glass order in 2D at low temperatures.

Provides analytical support for or against the existence of spin glass phases in various dimensions.

Abstract

We revisit the Haake-Lewenstein-Wilkens (HLW) approach to Edwards-Anderson (EA) model of Ising spin glass [Phys. Rev. Lett. 55, 2606 (1985)]. This approach consists in evaluation and analysis of the probability distribution of configurations of two replicas of the system, averaged over quenched disorder. This probability distribution generates squares of thermal copies of spin variables from the two copies of the systems, averaged over disorder, that is the terms that enter the standard definition of the original EA order parameter, qEA. We use saddle point/steepest descent method to calculate the average of the Gaussian disorder in higher dimensions. This approximate result suggest that qEA >0 at 0 <T <Tc in 3D and 4D. The case of 2D seems to be a little more subtle, since in the present approach energy increase for a domain wall competes with boundary/edge effects more strongly in 2D; still our approach predicts spin glass order at sufficiently low temperature. We speculate, how these predictions confirm/contradict widely spread opinions that: i) There exist only one (up to the spin flip) ground state in EA model in 2D, 3D and 4D; ii) There is (no) spin glass transition in 3D and 4D (2D). This paper is dedicated to the memories of Fritz Haake and Marek Cieplak.