A General Theorem Relating the Bulk Topological Number to Edge States in Two-dimensional Insulators

TL;DR

This paper proves a universal theorem linking bulk topological invariants, like the Chern number, to the existence of edge states in two-dimensional insulators, unifying quantum Hall effects and their robustness.

Contribution

It establishes a general, robust relation between bulk topological order and edge states, applicable even with disorder and interactions, for both charge and spin topological insulators.

Findings

Bulk topological order guarantees gapless edge states under certain conditions.

The relation holds despite disorder and interactions.

Reconciles bulk topological stability with edge state instability in open systems.

Abstract

We prove a general theorem on the relation between the bulk topological quantum number and the edge states in two dimensional insulators. It is shown that whenever there is a topological order in bulk, characterized by a non-vanishing Chern number, even if it is defined for a non-conserved quantity such as spin in the case of the spin Hall effect, one can always infer the existence of gapless edge states under certain twisted boundary conditions that allow tunneling between edges. This relation is robust against disorder and interactions, and it provides a unified topological classification of both the quantum (charge) Hall effect and the quantum spin Hall effect. In addition, it reconciles the apparent conflict between the stability of bulk topological order and the instability of gapless edge states in systems with open boundaries (as known happening in the spin Hall case). The consequences of time reversal invariance for bulk topological order and edge state dynamics are further studied in the present framework.