Spin-Glass Theory for Pedestrians

TL;DR

This paper provides a comprehensive, pedagogical review of mean-field spin-glass theory, focusing on the p-spin spherical model through thermodynamics, dynamics, and complexity analyses, emphasizing methodological consistency.

Contribution

It offers an accessible, unified overview of multiple approaches to spin-glass models, highlighting their mutual consistency and pedagogical clarity.

Findings

Analysis of the p-spin spherical model using three approaches

Demonstration of the consistency between thermodynamics, dynamics, and complexity results

Clarification of the static and dynamic transitions in spin-glasses

Abstract

In these notes the main theoretical concepts and techniques in the field of mean-field spin-glasses are reviewed in a compact and pedagogical way, for the benefit of the graduate and undergraduate student. One particular spin-glass model is analyzed (the p-spin spherical model) by using three different approaches. Thermodynamics, covering pure states, overlaps, overlap distribution, replica symmetry breaking, and the static transition. Dynamics, covering the generating functional method, generalized Langevin equation, equations for the correlation and the response, the Mode Coupling approximation, and the dynamical transition. And finally complexity, covering the mean-field (TAP) free energy, metastable states, entropy crisis, threshold energy, and saddles. Particular attention has been paid on the mutual consistency of the results obtained from the different methods.