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

TL;DR

This work provides a π0‑level, constructive classification of topological insulators across all Altland‑Zirnbauer classes and dimensions by formulating a locality framework based on spherically‑local operators and a bulk‑non‑triviality condition. It lifts traditional K‑theory invariants to path‑connected components of symmetry‑constrained, gapped Hamiltonians via a bulk‑localization and compression scheme, showing that the strong invariants are complete and reproduce the Kitaev periodic table. The analysis covers both complex and real symmetry classes, employing Dirac phase/projection index pairings and van Daele K‑theory to obtain a full π0 correspondence with the table entries. The results imply robust phase classifications in strongly disordered settings and offer a rigorous, constructive route to realize all topological phases as path components in the appropriate operator algebras. Overall, the paper unifies K‑theoretic indices with deformation‑theoretic, locality‑driven methods to achieve π0‑level completeness of topological phase 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 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 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. We do this by providing the appropriate notion of locality, as well as a novel, so-called bulk non-triviality, which together reproduce the Kitaev table. Once the natural definitions are identified, the main technical achievement is lifting $K$-theory calculations to $π_0$ of unitaries and projections.

Figures (3)

• Figure 1: Illustration for \ref{['lem:contained in cone']} (cone decoupling). Given a closed subset $J\subset S^{d-1}$, the complement cone $C_{J^{c}}$ is partitioned into $E\sqcup F$ so that the "bulk" part $E$ has uniformly small coupling to $C_J$ (i.e. $\|\Lambda_E A \Lambda_J\|$ is small), while the remainder $F$ is directionally thin: for any closed cone $C_I$ disjoint from $C_J$, the intersection $F\cap C_I$ is finite.
• Figure 2: Schematic for \ref{['lem:contained in annulus']}. Points $x_k$ are chosen in disjoint annuli $B_{r_k}\setminus B_{r_{k-1}}$ so that the operator $A$ couples $\delta_{x_k}$ essentially only inside its own annulus: the complement $\mathbb{Z}^d\setminus(B_{r_k}\setminus B_{r_{k-1}})$ is $\varepsilon_k$-weakly coupled to $\{x_k\}$.
• Figure 3: Illustration for \ref{['lem:localized centers']}. The finite sets $R_k$ (support islands for $B\delta_{x_k}$) lie in disjoint shells and become asymptotically localized in a single direction on $S^{d-1}$. Consequently, for disjoint dyadic cones $C_I$ and $C_J$, only finitely many islands can intersect both cones.

Theorems & Definitions (119)

• Definition 1.1: spherical locality
• Definition 1.2: bulk non-triviality
• Theorem 1.3: Kitaev table agrees with path-connected components of non-trivial insulators
• Definition 2.1: exponential locality
• Proposition 2.2
• proof
• Definition 2.3: Uniform Roe algebra
• Definition 2.4: Non-Uniform Roe algebra
• Definition 2.5: $\Lambda$-locality
• Definition 2.6: Dirac locality