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

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