Edge states and the bulk-boundary correspondence in Dirac Hamiltonians

TL;DR

This paper provides an analytic method to compute edge states and their dispersion in Dirac Hamiltonians, establishing a geometric bulk-boundary correspondence linking bulk properties to surface phenomena.

Contribution

It introduces a geometric formula for edge state dispersion and proves the bulk-boundary correspondence for Chern numbers in Dirac Hamiltonians.

Findings

Derived an explicit formula for edge state dispersion E(k).

Proved the bulk-boundary correspondence between Chern number and edge modes.

Applicable to both lattice and continuum Dirac models.

Abstract

We present an analytic prescription for computing the edge dispersion E(k) of a tight-binding Dirac Hamiltonian terminated at an abrupt crystalline edge. Specifically, we consider translationally invariant Dirac Hamiltonians with nearest-layer interaction. We present and prove a geometric formula that relates the existence of surface states as well as their energy dispersion to properties of the bulk Hamiltonian. We further prove the bulk-boundary correspondence between the Chern number and the chiral edge modes for quantum Hall systems within the class of Hamiltonians studied in the paper. Our results can be extended to the case of continuum theories which are quadratic in the momentum, as well as other symmetry classes.