Unified Topological Response Theory for Gapped and Gapless Free Fermions

TL;DR

This paper develops a unified topological response theory for both gapped and gapless free fermion systems, linking bulk responses and edge anomalies through a common Dirac fermion framework in higher dimensions.

Contribution

It introduces a systematic scheme to enumerate electromagnetic and strain responses of free fermions using a parent Dirac theory, unifying responses of gapped and gapless phases.

Findings

Derives quantized insulator responses from Chern-Simons theory.

Predicts spectral flow along dislocations in Weyl semimetals.

Reveals relationships between different responses via position-momentum interchange.

Abstract

We derive a scheme for systematically enumerating the responses of gapped as well as gapless systems of free fermions to electromagnetic and strain fields starting from a common parent theory. Using the fact that position operators in the lowest Landau level of a quantum Hall state are canonically conjugate, we consider a massive Dirac fermion in $2n$ spatial dimensions under $n$ mutually orthogonal magnetic fields and reinterpret physical space in the resulting zeroth Landau level as phase space in $n$ spatial dimensions. The bulk topological responses of the parent Dirac fermion, given by a Chern-Simons theory, translate into quantized insulator responses, while its edge anomalies characterize the response of gapless systems. Moreover, various physically different responses are seen to be related by the interchange of position and momentum variables. We derive many well-known responses, and demonstrate the utility of our theory by predicting spectral flow along dislocations in Weyl semimetals.