Abstract
The Chern-Simons axion coupling of a bulk insulator is only defined modulo a quantum of e^2/h. The quantized part of the coupling is uniquely defined for a bounded insulating sample, but it depends on the specific surface termination. Working in a slab geometry and representing the valence bands in terms of hybrid Wannier functions, we show how to determine that quantized part from the excess Chern number of the hybrid Wannier sheets located near the surface of the slab. The procedure is illustrated for a tight-binding model consisting of coupled quantum anomalous Hall layers. By slowly modulating the model parameters, it is possible to transfer one unit of Chern number from the bottom to the top surface over the course of a cyclic evolution of the bulk Hamiltonian. When the evolution of the surface Hamiltonian is also cyclic, the Chern pumping is obstructed by chiral touchings between valence and conduction surface bands.
Highlights
The axion field was originally introduced as a strategy for resolving the non-violation of time reversal (T ) and spatial inversion (P) symmetry in quantum chromodynamics (QCD).[1]
Working in a slab geometry and representing the valence bands in terms of hybrid Wannier functions, we show how to determine that quantized part from the excess Chern number of the hybrid Wannier sheets located near the surface of the slab
In the basis of hybrid Wannier functions (HWFs) maximally-localized along the surface-normal direction, the Chern-Simons axion (CSA) coupling of a thick slab becomes a sum of two terms: (i) a nonquantized contribution associated with the bulklike HWFs far from the surfaces, and (ii) a quantized contribution given by the excess Chern number of the Wannier sheets near the surface
Summary
The axion field was originally introduced as a strategy for resolving the non-violation of time reversal (T ) and spatial inversion (P) symmetry in quantum chromodynamics (QCD).[1]. This is another manifestation of Eq (2): changing θCS by a multiple of 2π amounts to adding quantum anomalous Hall layers at the surface, without modifying the bulk.[17] Another property of band insulators that behaves in this way is the bulk polarization P , whose electronic part can be expressed as a Berry phase.[18] The Berry phase may change by an integer multiple of 2π under unitary transformations, and the quantum of ambiguity in the bulk definition of P · ncan be resolved by taking into account quantized contributions of e/Acell to the surface charge density associated with an insulating surface with orientation n .19. Appendix A deals with the quantization of the CSA coupling by mirror symmetry at isolated points along the pumping cycle, and in Appendix B we map the second Chern number over an augmented parameter space
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