Surface theorem for the Chern-Simons axion coupling

TL;DR

This paper presents a method to determine the quantized part of the Chern-Simons axion coupling in insulators using hybrid Wannier functions, revealing how surface termination affects the coupling and demonstrating Chern number transfer through a cyclic model.

Contribution

It introduces a procedure to extract the quantized axion coupling from hybrid Wannier functions and illustrates Chern number transfer in a layered model with surface effects.

Findings

Quantized axion coupling depends on surface termination.

Hybrid Wannier functions can determine the quantized part.

Chern number can be cyclically transferred between surfaces.

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.