The Noncommutative Index Theorem and the Periodic Table for Disordered Topological Insulators and Superconductors

TL;DR

This paper develops a noncommutative index theorem that generalizes the Atiyah-Singer theorem to classify and analyze disordered topological insulators and superconductors across all dimensions, confirming Bott periodicity and invariant robustness.

Contribution

It introduces a noncommutative index theorem providing a unified framework for topological classification in disordered systems, extending previous models to all symmetry classes and dimensions.

Findings

Reproduces Bott periodicity in the index formula

Classifies all topological invariants in the periodic table

Shows robustness of indices against symmetry-preserving perturbations

Abstract

We study a wide class of topological free-fermion systems on a hypercubic lattice in spatial dimensions $d\ge 1$. When the Fermi level lies in a spectral gap or a mobility gap, the topological properties, e.g., the integral quantization of the topological invariant, are protected by certain symmetries of the Hamiltonian against disorder. This generic feature is characterized by a generalized index theorem which is a noncommutative analogue of the Atiyah-Singer index theorem. The noncommutative index defined in terms of a pair of projections gives a precise formula for the topological invariant in each symmetry class in any dimension ($d \ge 1$). Under the assumption on the nonvanishing spectral or mobility gap, we prove that the index formula reproduces Bott periodicity and all of the possible values of topological invariants in the classification table of topological insulators and superconductors. We also prove that the indices are robust against perturbations that do not break the symmetry of the unperturbed Hamiltonian.