Topology, geometry and quantum interference in condensed matter physics

TL;DR

This paper explores the role of topological and geometric terms in effective actions derived from quantum field theory, highlighting their implications in condensed matter phenomena like quantum Hall effects and topological insulators.

Contribution

It provides a systematic discussion of topological terms in effective actions, their origins, and their applications across various condensed matter systems.

Findings

Topological terms arise as phases of fermionic determinants.

These terms encode quantum anomalies in fermionic models.

Applications include charge density waves, topological insulators, and superconductors.

Abstract

The methods of quantum field theory are widely used in condensed matter physics. In particular, the concept of an effective action was proven useful when studying low temperature and long distance behavior of condensed matter systems. Often the degrees of freedom which appear due to spontaneous symmetry breaking or an emergent gauge symmetry, have non-trivial topology. In those cases, the terms in the effective action describing low energy degrees of freedom can be metric independent (topological). We consider a few examples of topological terms of different types and discuss some of their consequences. We will also discuss the origin of these terms and calculate effective actions for several fermionic models. In this approach, topological terms appear as phases of fermionic determinants and represent quantum anomalies of fermionic models. In addition to the wide use of topological terms in high energy physics, they appeared to be useful in studies of charge and spin density waves, Quantum Hall Effect, spin chains, frustrated magnets, topological insulators and superconductors, and some models of high-temperature superconductivity.