Replica symmetry breaking, complexity and spin representation in the generalized random energy model

TL;DR

This paper analyzes the generalized random energy model (GREM) using the replica method, exploring replica symmetry breaking patterns, generalized complexity, and proposing a new hierarchical spin-glass model with multiple phase transitions.

Contribution

It extends the analysis of GREM by employing the replica method to include higher step RSB and introduces a generalized complexity concept and a new hierarchical spin-glass model.

Findings

Higher step RSB is effective for describing spin-glass phases.

Generalized complexity characterizes the hierarchical structure.

Hierarchical models exhibit multiple-step phase transitions.

Abstract

We study the random energy model with a hierarchical structure known as the generalized random energy model (GREM). In contrast to the original analysis by the microcanonical ensemble formalism, we investigate the GREM by the canonical ensemble formalism in conjunction with the replica method. In this analysis, spin-glass-order parameters are defined for respective hierarchy level, and all possible patterns of replica symmetry breaking (RSB) are taken into account. As a result, we find that the higher step RSB ansatz is useful for describing spin-glass phases in this system. For investigating the nature of the higher step RSB, we generalize the notion of complexity developed for the one-step RSB to the higher step and demonstrate how the GREM is characterized by the generalized complexity. In addition, we propose a novel mean-field spin-glass model with a hierarchical structure, which is equivalent to the GREM at a certain limit. We also show that the same hierarchical structure can be implemented to other mean-field spin models than the GREM. Such models with hierarchy exhibit phase transitions of multiple steps in common.