Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Habib Ammari, Jiayu Qiu

TL;DR

This work proves bulk-edge correspondence for finite two-dimensional ergodic disordered systems by introducing a refined bulk index $\mathcal{E}_{bulk,\omega,\mathcal{I},\rho}^{ref}$ that accounts for bulk-localized states in a mobility gap $\mathcal{I}$ and an edge index $\mathcal{E}_{edge,\omega,\mathcal{I}',\rho,L}$ defined on finite samples. The authors show that, under an Aizenman–Molchanov mobility-gap hypothesis, the edge index converges almost surely to the refined bulk index as the domain size grows: $\lim_{L\to\infty}\mathcal{E}_{edge,\omega,\mathcal{I}',\rho,L}=\mathcal{E}_{bulk,\omega,\mathcal{I}',\rho}^{ref}$. They establish the mobility gap via geometric decoupling with a rational-distribution assumption for the random potential and validate the framework on the Qi–Wu–Zhang model, including an appendix demonstration of nontrivial bulk-topology in the disordered setting. The results extend rigorous BEC from periodic to ergodic disordered finite systems, providing a solid mathematical foundation for two-dimensional topological alloy-type materials and the robustness of edge modes in realistic, finite samples.

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.

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

TL;DR

This work proves bulk-edge correspondence for finite two-dimensional ergodic disordered systems by introducing a refined bulk index that accounts for bulk-localized states in a mobility gap and an edge index defined on finite samples. The authors show that, under an Aizenman–Molchanov mobility-gap hypothesis, the edge index converges almost surely to the refined bulk index as the domain size grows: . They establish the mobility gap via geometric decoupling with a rational-distribution assumption for the random potential and validate the framework on the Qi–Wu–Zhang model, including an appendix demonstration of nontrivial bulk-topology in the disordered setting. The results extend rigorous BEC from periodic to ergodic disordered finite systems, providing a solid mathematical foundation for two-dimensional topological alloy-type materials and the robustness of edge modes in realistic, finite samples.

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.

Paper Structure

This paper contains 15 sections, 24 theorems, 181 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Set-up and Main Results
  3. Notation
  4. Preliminaries
  5. Some Algebraic Properties of Exponentially Localized Operators
  6. Covariant Operators and Trace Per Unit of Volume
  7. Bulk-edge Correspondence for Finite Disordered Systems: Proof of Theorem \ref{['thm_bec_disorder']}
  8. Step 1: Equivalent Formulation of the Edge Index
  9. Step 2: Changing Hamiltonians from Box to Bulk
  10. Step 3: Link to the Refined Bulk Index
  11. Aizenman-Molchanov Localization Under Random Gap Opening Potential
  12. Lifshitz Tail and Initial Length Estimate Near Gap Opening Point
  13. Geometric Decoupling: Proof of Theorem \ref{['thm_anderson_localization']}
  14. The Qi-Wu-Zhang Model
  15. Nontriviality of the Bulk Index

Key Result

Proposition 2.5

There exists $\Sigma\subset \mathcal{I}$ such that $\mathbb{P}-$almost surely.

Figures (1)

  • Figure 1: A finite sample $\Lambda_L$ (dashed square with black sites).

Theorems & Definitions (37)

  • Remark 2.2
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Theorem 2.8: BEC principle for finite systems qiu2025bec_finite
  • Proposition 2.9
  • Theorem 2.10: BEC for finite disordered systems
  • Theorem 2.13
  • Remark 2.14
  • Remark 2.15
  • ...and 27 more