Unified description of Dirac electrons on a curved surface of topological insulators

TL;DR

This paper develops a general geometric framework to describe Dirac electrons on curved surfaces of topological insulators, extending the understanding from flat to curved geometries and clarifying the role of spin connection.

Contribution

It introduces a unified method to derive the surface Dirac equation on curved topological insulator surfaces using differential geometry.

Findings

Unified description of Dirac electrons on curved surfaces.

Clarification of the physical meaning of spin connection.

Framework applicable to various surface geometries.

Abstract

Existence of a protected surface state described by a massless Dirac equation is a defining property of the topological insulator. Though this statement can be explicitly verified on an idealized flat surface, it remains to be addressed to what extent it could be general. On a curved surface, the surface Dirac equation is modified by the spin connection terms. Here, in the light of the differential geometry, we give a general framework for constructing the surface Dirac equation starting from the Hamiltonian for bulk topological insulators. The obtained unified description clarifies the physical meaning of the spin connection.