Homotopy in statistical physics

TL;DR

This paper reviews how homotopy theory helps classify topological defects in condensed matter physics, discussing recent developments and future research directions in low-dimensional systems.

Contribution

It provides a pedagogic introduction to homotopy theory and reviews its application to classifying topological defects in statistical-mechanical systems.

Findings

Homotopy theory effectively classifies topological defects.

Recent research advances in low-dimensional systems are summarized.

Future directions include exploring new topological phenomena.

Abstract

In condensed matter physics and related areas, topological defects play important roles in phase transitions and critical phenomena. Homotopy theory facilitates the classification of such topological defects. After a pedagogic introduction to the mathematical methods involved in topology and homotopy theory, the role of the latter in a number of mainly low-dimensional statistical-mechanical systems is outlined. Some recent activities in this area are reviewed and some possible future directions are discussed.