Unified Topological Response Theory for Gapped and Gapless Free Fermions

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.

19 pages, 7 figures

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• Dislocation defect as a bulk probe of monopole charge of multi-Weyl semimetals
• Proximity-induced equilibrium supercurrent and perfect superconducting diode effect due to band asymmetry
• A Comment On Berry Connections
• Definition and Classification of Fermi Surface Anomalies
• Effective field theory of Berry Fermi liquid from the coadjoint orbit method
• Gapless Fermionic Systems as Phase-space Topological Insulators: Non-perturbative Results from Anomalies
• Topological Phases for Extended Objects: Semiclassical Phase-Space Approach with Tensorial Coordinates
• Fragile dislocation modes in obstructed atomic topological phases
• Fermion liquids as quantum Hall liquids in phase space: A unified approach for anomalies and responses