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arXiv · 2407.14235

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

Vincenzo Rossi·Gianluca Panati
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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.

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Vincenzo Rossi, Gianluca Panati. 2024-07-19. Algebraic localization of generalized Wannier bases implies Roe triviality in any dimension. https://arxiv.org/abs/2407.14235

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