Continuum Theory of Edge States of Topological Insulators: Variational Principle and Boundary Conditions

TL;DR

This paper develops a continuum theoretical framework with natural boundary conditions to accurately model edge states in topological insulators, validated by comparison with tight-binding calculations, revealing non-monotonic gap behavior at specific widths.

Contribution

It introduces a variational energy functional and boundary conditions for continuum modeling of topological insulator edge states, bridging analytical and numerical approaches.

Findings

Continuum theory accurately describes low energy edge states.

Natural boundary conditions are derived for such systems.

Edge state hybridization gap varies non-monotonically with ribbon width.

Abstract

We develop a continuum theory to model low energy excitations of a generic four-band time reversal invariant electronic system with boundaries. We propose a variational energy functional for the wavefunctions which allows us derive natural boundary conditions valid for such systems. Our formulation is particularly suited to develop a continuum theory of the protected edge/surface excitations of topological insulators both in two and three dimensions. By a detailed comparison of our analytical formulation with tight binding calculations of ribbons of topological insulators modeled by the Bernevig-Hughes-Zhang (BHZ) hamiltonian, we show that the continuum theory with the natural boundary condition provides an appropriate description of the low energy physics. As a spin-off, we find that in a certain parameter regime, the gap that arises in topological insulator ribbons of finite width due to the hybridization of edges states from opposite edges, depends non-monotonically on the ribbon width and can nearly vanish at certain "magic widths".