Robustness of topological phases on aperiodic lattices
Yuezhao Li
TL;DR
This work develops a rigorous framework to study the robustness of topological phases on aperiodic lattices by comparing two observable C*-algebras: the groupoid C*-algebra of a Delone set and the Roe (coarse) C*-algebra. By constructing *-homomorphisms between these models via regular representations and exploiting position spectral triples, the authors show that strong topological phases are detected by index pairings in the Roe setting, while phases produced by stacking are forced to be weak and vanish in Roe K-theory. The key technical advance is showing that the bulk cycle paired with localized spectral triples reproduces the integer invariants in KK-theory and that the groupoid-model invariants pull back to the coarse geometry in a way that preserves robustness under short-range perturbations and symmetry constraints. The results unify groupoid and coarse-geometric approaches to aperiodic topological insulators, providing concrete tools to assess which invariants survive under perturbations and how stacking phenomena fail to yield robust coarse-scale indices.
Abstract
We study the robustness of topological phases on aperiodic lattices by constructing *-homomorphisms from the groupoid model to the coarse-geometric model of observable C*-algebras. These *-homomorphisms induce maps in K-theory and Kasparov theory. We show that the strong topological phases in the groupoid model are detected by position spectral triples. We show that topological phases coming from stacking along another Delone set are always weak in the coarse-geometric sense.