Analytic Theory of Edge Modes in Topological Insulators

TL;DR

This paper analytically derives the spectrum and wave functions of edge modes in topological insulators, revealing how edge geometry and system parameters influence the presence and energy of these modes.

Contribution

It provides an analytical framework for understanding edge modes in topological insulators on square lattices, focusing on different edge geometries and their effects.

Findings

Edge modes depend on edge geometry and bulk gap location.

In (1,0) edges, modes appear at zone center or boundary.

In (1,1) edges, modes are always present but can be extremely weak.

Abstract

Spectrum and wave function of gapless edge modes are derived analytically for a tight-binding model of topological insulators on square lattice. Particular attention is paid to dependence on edge geometries such as the straight (1,0) and zigzag (1,1) edges in the thermodynamic limit. The key technique is to identify operators that combine to annihilate the edge state in the effective one-dimensional (1D) model with momentum along the edge. In the (1,0) edge, the edge mode is present either around the center of 1D Brillouin zone or its boundary, depending on location of the bulk excitation gap. In the (1,1) edge, the edge mode is always present both at the center and near the boundary. Depending on system parameters, however, the mode is absent in the middle of the Brillouin zone. In this case the binding energy of the edge mode near the boundary is extremely small; about $10^{-3}$ of the overall energy scale. Origin of this minute energy scale is discussed.