Higher Berry Phase of Fermions and Index Theorem

TL;DR

This paper explores the response of free Dirac fermions with spacetime-dependent mass parameters using index theory, revealing a higher Berry curvature and torsional Berry phases that characterize invertible field theories.

Contribution

It introduces a general formula for a higher Berry curvature in parameterized invertible field theories derived from Dirac fermions, extending the understanding of topological responses.

Findings

Derived a formula for higher Berry curvature in parameterized invertible theories.

Identified conditions where Berry curvature vanishes but torsional Berry phases remain.

Provided computable examples of torsional Berry phases.

Abstract

When a quantum field theory is trivially gapped, its infrared fixed point is an invertible field theory. The partition function of the invertible field theory records the response to various background fields in the long-distance limit. The set of background fields can include spacetime-dependent coupling constants, in which case we call the corresponding invertible theory a parameterized invertible field theory. We study such parameterized invertible field theories arising from free Dirac fermions with spacetime-dependent mass parameters using the Atiyah-Patodi-Singer index theorem for superconnections. In particular, the response to an infinitesimal modulation of the mass is encoded into a higher analog of the Berry curvature, for which we provide a general formula. When the Berry curvature vanishes, the invertible theory can still be nontrivial if there is a remaining torsional Berry phase, for which we list some computable examples.