Effective model for massless Dirac electrons on a surface of weak topological insulators

TL;DR

This paper derives an explicit 2D model for massless Dirac electrons on the side surfaces of weak topological insulators from a 3D bulk Hamiltonian, confirming its accuracy and applicability to complex surface geometries.

Contribution

It provides the first explicit derivation of a 2D surface state model from a 3D bulk Hamiltonian for WTIs, validating the model's accuracy.

Findings

The 2D model reproduces the 3D excitation spectrum accurately.

The model is applicable to surfaces with atomic steps.

The derivation establishes a firm theoretical basis for the 2D surface state model.

Abstract

In a typical situation, gapless surface states of a three-dimensional (3D) weak topological insulator (WTI) appear only on sides, leaving the top and bottom surfaces gapped. To describe massless Dirac electrons emergent on such side surfaces of a WTI, a two-dimensional (2D) model consisting of a series of one-dimensional helical channels is usually employed. Yet, explicit derivation of such a model from a 3D bulk Hamiltonian has been lacking. Here, we explicitly derive an effective 2D model for the WTI surface states starting from a Wilson-Dirac Hamiltonian for the bulk WTI and establish a firm basis to the hitherto hypothesized 2D model. We show that the resulting 2D model accurately reproduces the excitation spectrum of surface Dirac electrons determined by the 3D model. We also show that the 2D model is applicable to a side surface with atomic steps.