Topological effective field theories for Dirac fermions from index theorem

TL;DR

This paper develops a method to derive topological effective actions for (3+1) Dirac fermions on general backgrounds, extending existing techniques to non-Hermitian cases, enabling topological classification of quantum systems with relativistic fermions.

Contribution

It introduces a novel extension of the index theorem approach to non-Hermitian Dirac operators, allowing topological effective actions to be derived in broader physical contexts.

Findings

Derived topological actions for non-Hermitian Dirac fermions

Extended index theorem methods to non-Hermitian operators

Provided a framework for topological classification of quantum systems

Abstract

Dirac fermions have a central role in high energy physics but it is well known that they emerge also as quasiparticles in several condensed matter systems supporting topological order. We present a general method for deriving the topological effective actions of (3+1) massless Dirac fermions living on general backgrounds and coupled with vector and axial-vector gauge fields. The first step of our strategy is standard (in the Hermitian case) and consists in connecting the determinants of Dirac operators with the corresponding analytical indices through the zeta-function regularization. Then, we introduce a suitable splitting of the heat kernel that naturally selects the purely topological part of the determinant (i.e. the topological effective action). This topological effective action is expressed in terms of gauge fields using the Atiyah-Singer index theorem which computes the analytical index in topological terms. The main new result of this paper is to provide a consistent extension of this method to the non Hermitian case where a well-defined determinant does not exist. Quantum systems supporting relativistic fermions can thus be topologically classified on the basis of their response to the presence of (external or emergent) gauge fields through the corresponding topological effective field theories.