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A spectral localizer approach to strong topological invariants in the mobility gap regime

Tom Stoiber

Abstract

Topological phases of gapped one-particle Hamiltonians with (anti)-unitary symmetries are classified by strong topological invariants according to the Altland-Zirnbauer table. Those indices are still well-defined in the regime of strong disorder when the spectral gap is replaced by a mobility gap, however, many questions regarding their robustness and existence of topological boundary states are wide open. We apply the recently developed spectral localizer method to prove results on the stability of strong topological invariants under a notion of continuous homotopy that preserves a mobility gap condition. Using the local computability afforded by the spectral localizer we show that for parametrized random families that satisfy a fractional moments bound the probability distribution of the strong topological invariant changes continuously. In particular, for ergodic families the almost sure index must be constant on any path which preserves the mobility gap. Using similar methods, we also prove a result on the delocalization of interface states between two mobility-gapped systems which have differing strong invariants.

A spectral localizer approach to strong topological invariants in the mobility gap regime

Abstract

Topological phases of gapped one-particle Hamiltonians with (anti)-unitary symmetries are classified by strong topological invariants according to the Altland-Zirnbauer table. Those indices are still well-defined in the regime of strong disorder when the spectral gap is replaced by a mobility gap, however, many questions regarding their robustness and existence of topological boundary states are wide open. We apply the recently developed spectral localizer method to prove results on the stability of strong topological invariants under a notion of continuous homotopy that preserves a mobility gap condition. Using the local computability afforded by the spectral localizer we show that for parametrized random families that satisfy a fractional moments bound the probability distribution of the strong topological invariant changes continuously. In particular, for ergodic families the almost sure index must be constant on any path which preserves the mobility gap. Using similar methods, we also prove a result on the delocalization of interface states between two mobility-gapped systems which have differing strong invariants.

Paper Structure

This paper contains 9 sections, 17 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Index pairings with a Dirac operator
  3. The spectral localizer
  4. Continuity of the index pairing for random families
  5. Mobility-gapped Hamiltonians
  6. Translation-invariance and ergodicity
  7. Delocalization of interfaces
  8. Clifford algebras and their symmetries
  9. Regularization of Dirac operators

Key Result

Theorem 1.1

Let $T$ be a metric space, ${\cal L}\subset {\mathbb R}^d$ a Delone set and $(\Omega,{\mathbb P})$ a probability space as in Definition def:mbg. We assume that $t \in T \mapsto h(t)$ is a parametrized family of random Hamiltonians on $\ell^2({\cal L})\otimes {\mathbb C}^n$, i.e. for each $t$ one has Then the probability that the index takes any particular value $z\in {\mathbb Z}$ respectively $z\

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1: LS2LS3
  • Theorem 3.2: DS21
  • ...and 16 more