Boundary Condition Analysis of First and Second Order Topological Insulators

TL;DR

This paper analytically investigates boundary conditions in lattice Dirac fermion models for first and second order topological insulators, revealing how symmetry constrains boundary states and confirming the bulk-edge correspondence.

Contribution

It provides an analytical framework for boundary conditions in topological insulators, linking symmetry constraints to edge and hinge state properties.

Findings

Derived dispersion relations for edge and hinge states.

Identified symmetry constraints on boundary conditions.

Confirmed the bulk-edge correspondence for hinge states.

Abstract

We analytically study boundary conditions of the Dirac fermion models on a lattice, which describe the first and second order topological insulators. We obtain the dispersion relations of the edge and hinge states by solving these boundary conditions, and clarify that the Hamiltonian symmetry may provide a constraint on the boundary condition. We also demonstrate the edgehinge analog of the bulk-edge correspondence, in which the nontrivial topology of the gapped edge state ensures gaplessness of the hinge state.