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

2024/07/19 by Rossi, Vincenzo, Gianluca Panati, Panati, Gianluca
Computer Science · Mathematics · Engineering · #Polynomial and algebraic computation #Commutative Algebra and Its Applications #Advanced Numerical Analysis Techniques

paper · pdf · doi:10.48550/arxiv.2407.14235

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 K₀-theory of the Roe C^*-algebra of 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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