Bulk-Edge Correspondence for Finite Two-dimensional Ergodic Disordered Systems

TL;DR

This paper rigorously establishes the bulk-edge correspondence in finite two-dimensional ergodic disordered systems, linking topological invariants of the bulk to measurable edge properties, with implications for quantum Hall physics.

Contribution

It provides a rigorous proof of the bulk-edge correspondence for finite disordered systems, connecting bulk topological invariants to edge phenomena under ergodic conditions.

Findings

Edge index converges to bulk index as system size increases

Bulk index includes Hall conductance and localization contributions

Existence of the mobility gap is proven using geometric decoupling

Abstract

In this paper, we rigorously prove the bulk-edge correspondence for finite two-dimensional ergodic disordered systems. Specifically, we focus on the short-range Hamiltonians with ergodic disordered on-site potentials. We first introduce the bulk and edge indices, which are both well-defined within the Aizenman-Molchanov mobility gap. On the one hand, the bulk index is the sum of the Hall conductance, which is a well-studied quantized topological number, and an additional contribution from the bulk-localized modes as a consequence of the Anderson localization. On the other hand, the edge index, which characterizes the averaged angular momentum of waves in the mobility gap, is uniquely associated with finite systems. Our main result proves that as the sample size tends to infinity, the edge index converges to the bulk index almost surely. Our findings provide a rigorous foundation for the bulk-edge correspondence principle for finite disordered systems. The existence of the Aizenman-Molchanov mobility gap is proved by the geometric decoupling method, introduced by Aizenman and Molchanov [Comm. Math. Phys., 1993], under a rational assumption on the distribution of the random potential. For completeness, all assumptions are checked on a prototypical model for (quantum) anomalous Hall physics.