Equality of the bulk and edge Hall conductances in a mobility gap

TL;DR

This paper proves the equality of bulk and edge Hall conductances in 2D quantum Hall systems with localized bulk states, using appropriate definitions and averaging procedures, including for non-ergodic systems.

Contribution

It establishes the equivalence of bulk and edge conductances in localized regimes and introduces a suitable definition for edge conductance involving time averaging and localized states.

Findings

Bulk and edge conductances are equal in localized regimes.

A proper definition of edge conductance requires time averaging or localized state contributions.

Quantized conductance plateaus are established for non-ergodic systems.

Abstract

We consider the edge and bulk conductances for 2D quantum Hall systems in which the Fermi energy falls in a band where bulk states are localized. We show that the resulting quantities are equal, when appropriately defined. An appropriate definition of the edge conductance may be obtained through a suitable time averaging procedure or by including a contribution from states in the localized band. In a further result on the Harper Hamiltonian, we show that this contribution is essential. In an appendix we establish quantized plateaus for the conductance of systems which need not be translation ergodic.