An Introduction to the Theory of Spin Glasses

TL;DR

This paper reviews key theoretical and numerical methods used to study spin glasses, covering fully connected models, tree-like systems, and low-dimensional cases, highlighting open problems in the field.

Contribution

It provides a comprehensive overview of analytical and numerical techniques for spin glasses, emphasizing recent developments and open challenges.

Findings

Comparison of methods for different models

Insights into low-dimensional spin glass behavior

Identification of key open problems

Abstract

We review the main methods used to study spin glasses. In the first part, we focus on methods for fully connected models and systems defined on a tree, such as the replica method, the Thouless-Anderson-Palmer formalism, the cavity method, and the dynamical mean-field theory. In the second part, we deal with the description of low-dimensional systems, mostly in three spatial dimensions, which are mostly studied through numerical simulations. We conclude by mentioning some of the main open problems in the field.