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

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 usual Hall conductance, which is a well-studied quantized topological number. On the other hand, the edge index, which characterizes the averaged angular momentum of edge modes 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.