Bulk-boundary correspondence in three dimensional topological insulators

TL;DR

This paper explores the relationship between bulk topological invariants and surface states in 3D topological insulators, revealing that only strong topological phases have robust surface state parity, aligning with a $ ext{ extbf{Z}}$ classification.

Contribution

It demonstrates that surface state parity, not the number, is the key robust feature distinguishing strong topological insulators, supporting a $ ext{ extbf{Z}}$ bulk-boundary correspondence.

Findings

Surface state parity is robust in strong topological insulators.

Weak topological insulators are indistinguishable from trivial insulators via surface states.

The results support a $ ext{ extbf{Z}}$ classification of topological phases.

Abstract

We discuss the relation between bulk topological invariants and the spectrum of surface states in three dimensional non-interacting topological insulators. By studying particular models, and considering general boundary conditions for the electron wavefunction on the crystal surface, we demonstrate that using experimental techniques that probe surface states, only strong topological and trivial insulating phases can be distinguished; the latter state being equivalent to a weak topological insulator. In a strong topological insulator, only the {\it parity} of the number of surface states, but not the number itself, is robust against time-reversal invariant boundary perturbations. Our results suggest a $\z$ definition of the bulk-boundary correspondence, compatible with the $\z$ classification of topological insulators.