Topological Classification of Insulators: III. Non-interacting Spectrally-Gapped Systems in All Dimensions

TL;DR

This paper classifies non-interacting, spectrally-gapped electronic systems across all dimensions and symmetry classes, confirming the topological classification aligns with the Kitaev periodic table via a novel topological approach.

Contribution

It introduces a new topological framework that makes the strong invariants complete and confirms the correspondence between topological phases and Abelian groups for all symmetry classes.

Findings

Confirmed the conjecture relating topological phases to Abelian groups.

Derived the Kitaev periodic table as path-connected components of Hamiltonian space.

Identified natural notions of locality and bulk non-triviality for classification.

Abstract

We study non-interacting electrons in disordered materials which exhibit a spectral gap, in each of the ten Altland-Zirnbauer symmetry classes, in all space dimensions. We define an appropriate space of Hamiltonians and a topology on it so that the so-called strong topological invariants become \emph{complete} invariants yielding the Kitaev periodic table, but now derived as the set of path-connected components of the space of Hamiltonians, rather than as $K$-theory groups. We thus confirm the conjecture (phrased e.g. in \cite{KatsuraKoma2018}) regarding a one-to-one correspondence between topological phases of gapped non-interacting systems and the respective Abelian groups $\{0\},\ZZ,2\ZZ,\ZZ_2$ in the spectral gap regime. A central conceptual achievement of the paper is the identification of the natural notions of locality and bulk non-triviality for this classification problem. Once these are in place, the main technical step is to lift the relevant $K$-theory calculations to $\pi_0$ of unitaries and projections.