Replica Theory and Spin Glasses

TL;DR

This paper provides an overview of replica theory and spin glasses, including the replica method, solution of the SK model, replica symmetry breaking, and rigorous results, highlighting their significance in statistical physics.

Contribution

It introduces the replica method and its application to spin glasses, including the derivation of the SK model solution and the proof of hierarchical RSB's exactness.

Findings

Derivation of the eigenvalue distribution of GOE matrices

Solution of the SK model with replica symmetry breaking

Proof of the thermodynamic limit and RSB solution exactness

Abstract

These are notes from the lectures of Giorgio Parisi given at the autumn school "Statistical Physics, Optimization, Inference, and Message-Passing Algorithm", that took place in Les Houches, France from Monday September 30th, 2013, till Friday October 11th, 2013. The school was organized by Florent Krzakala from UPMC and ENS Paris, Federico Ricci-Tersenghi from "La Sapienza" Roma, Lenka Zdeborov\'a from CEA Saclay and CNRS, and Riccardo Zecchina from Politecnico Torino. The first lecture contains an introduction to the replica method, along with a concrete application to the computation of the eigenvalue distribution of random matrices in the GOE. In the second lecture, the solution of the SK model is derived, along with the phenomenon of replica symmetry breaking (RSB). In the third part, the physical meaning of the RSB is explained. The ultrametricity of the space of pure states emerges as a consequence of the hierarchical RSB scheme. Moreover, it is shown how some low temperature properties of physical observables can be derived by invoking the stochastic stability principle. Lecture four contains some rigorous results on the SK model: the existence of the thermodynamic limit, and the proof of the exactness of the hierarchical RSB solution.