Homological bulk-edge correspondence for Weyl semimetal

TL;DR

This paper establishes a homological bulk-edge correspondence for 3D Weyl semimetals, linking spectral edge data to bulk topological invariants via relative homology classes, and characterizes Fermi arcs as homology cycles.

Contribution

It introduces a homological framework for bulk-edge correspondence in Weyl semimetals, connecting spectral and topological data through relative homology classes.

Findings

Homology classes on surface and bulk momentum spaces are shown to correspond.

Fermi arcs are represented by homology cycles derived from the edge Hamiltonian.

The framework confirms the topological nature of Fermi arcs in Weyl semimetals.

Abstract

For a certain translation invariant tight-binding model of three-dimensional Weyl semimetals, we establish a bulk-edge correspondence as an equality of two relative homology classes, based on an idea of Mathai and Thiang: From spectral information on the edge Hamiltonian, we construct a relative homology class on the surface momentum space. This class agrees with the image under the surface projection of a homology class on the bulk momentum space relative to the Weyl points, constructed from the bulk Hamiltonian. Furthermore, the relative homology class on the surface momentum space can be represented by homology cycles whose images constitute the Fermi arc, the locus where the edge Hamiltonian admits zero spectrum.