Algebraic localization of generalized Wannier bases implies Roe triviality in any dimension

TL;DR

This paper establishes a link between algebraic localization of generalized Wannier bases and topological triviality in non-periodic quantum systems, providing a decay rate threshold that guarantees triviality in Roe $K$-theory.

Contribution

It introduces a decay rate threshold for generalized Wannier functions that ensures topological triviality in Roe $K$-theory across any dimension.

Findings

Derived a decay threshold depending on dimension for triviality

Connected localization properties with Roe $K$-theory triviality

Reduced the threshold to near-optimal in 2D case

Abstract

With the aim of understanding the localization topology correspondence for non periodic gapped quantum systems, we investigate the relation between the existence of an algebraically well-localized generalized Wannier basis and the topological triviality of the corresponding projection operator. Inspired by the work of M. Ludewig and G.C. Thiang, we consider the triviality of a projection in the sense of coarse geometry, i.e. as triviality in the $K_0$-theory of the Roe $C^*$-algebra of $\mathrm{R}^d$. We obtain in Theorem 2.8 a threshold, depending on the dimension, for the decay rate of the generalized Wannier functions which implies topological triviality in Roe sense. This threshold reduces, for $d = 2$, to the almost optimal threshold appearing in the Localization Dichotomy Conjecture.