An introduction to spin systems for mathematicians

TL;DR

This paper provides a gentle introduction to spin systems and their connection to quantum field theory, aimed at mathematicians, highlighting key concepts and recent advances in classifying topological phases of matter.

Contribution

It offers a mathematical perspective on spin systems and topological quantum field theory, bridging condensed matter physics and advanced mathematical ideas for the first time.

Findings

Overview of quantum field theory and spin systems

Discussion of topological quantum field theories in condensed matter

Connection between nonlocal operators and TQFTs

Abstract

We give a leisurely, albeit woefully incomplete, overview of quantum field theory, its relevance to condensed matter systems, and spin systems, which proceeds via a series of illustrative examples. The goal is to provide readers from the mathematics community a swift route into recent condensed matter literature that makes use of topological quantum field theory and ideas from stable homotopy theory to attack the problem of classification of topological (or SPT) phases of matter. The toric code and Heisenberg spin chain are briefly discussed; important conceptual ideas in physics, that may have somehow evaded discussion for those with purely mathematical training, are also reviewed. Emphasis is placed on the connection between (algebras of) nonlocal operators and the appearance of nontrivial TQFTs in the infrared.