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

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 $\{0\},\mathbb{Z},2\mathbb{Z},\mathbb{Z}_2$ in the spectral gap regime. The central conceptual point is that spherical locality and bulk non-triviality are the two structural hypotheses which make this non-stable statement true. Spherical locality provides the real-space asymptotic locality needed for the strong index pairings, while bulk non-triviality removes lower-dimensional or edge-type configurations which would otherwise create extra path-components. Once this phase space has been identified, the algebraic input is the standard $K$-theory of the associated Paschke-dual picture, and the remaining technical task is to lift that information to $\pi_0$ of symmetry-constrained projections and unitaries. These definitions of locality and bulk-non-triviality are expected to be the portable part of the argument in regimes, such as mobility gaps and interacting systems, where ordinary stabilized $K$-theory is not by itself the right formulation of the physical classification problem.