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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.
Highlights
Spurred by the discovery of topological insulators, topological phases have become a vital part of condensed-matter physics over the last decade [1,2,3,4]
II, we review the key property of the zeroth Landau level (ZLL), which provides the physical justification for our construction
We have shown that the responses and anomalies of a gapped or gapless system living in n spatial dimensions can be described by a single response theory of a gapped system living in 2n spatial dimensions
Summary
Spurred by the discovery of topological insulators, topological phases have become a vital part of condensed-matter physics over the last decade [1,2,3,4]. We interpret additional perturbations in the phase-space gauge fields as physical quantities such as the n-dimensional system’s electromagnetic (EM) field, strain field, Berry curvatures, and Hamiltonian Topological defects, such as monopoles, in the phase-space gauge fields allow us to generalize to systems with dislocations and with point Fermi surfaces such as graphene and Weyl semimetals. One has to write down the ChernSimons action in phase space, vary it with respect to each gauge field, and consider each boundary to obtain all the bulk, boundary, and gapless responses in real space Following this procedure, we show carefully that screw dislocations in Weyl semimetals trap chiral modes which are well localized around the dislocation at momentum values away from the Weyl nodes. VII, we summarize our work and suggest extensions of our theory to more nontrivial systems
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