Index theorems, generalized Hall currents and topology for gapless defect fermions

TL;DR

This paper demonstrates how the index of the fermion operator can identify gapless modes on defects in topological materials, revealing a generalized Hall current and linking topology in momentum and real space.

Contribution

It introduces a novel use of the fermion operator index to detect defect-bound gapless modes and elucidates the topological origin of generalized Hall currents in such systems.

Findings

Index of fermion operator signals gapless defect modes

Quantum Hall-like currents exist without conserved charges or anomalies

Explicit examples illustrate the theoretical framework

Abstract

We show how the index of the fermion operator from the Euclidean action can be used to uncover the existence of gapless modes living on defects (such as edges and vortices) in topological insulators and superconductors. The 1-loop Feynman diagram that computes the index reveals an analog of the Quantum Hall current flowing on and off the defect -- even in systems without conserved currents or chiral anomalies -- and makes explicit the interplay between topology in momentum and coordinate space. We provide several explicit examples.