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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 \(\mathbb {R}^d\) . 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’s sense. This threshold reduces, for \(d=2\) , to the almost-optimal threshold appearing in the Localization Dichotomy Conjecture.

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Algebraic Localization of Generalized Wannier Bases Implies Roe Triviality in Any Dimension

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  • 发表日期：

    2025年

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  • Vincenzo Rossi,
  • Gianluca Panati

摘要

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 \(\mathbb {R}^d\) . 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’s sense. This threshold reduces, for \(d=2\) , to the almost-optimal threshold appearing in the Localization Dichotomy Conjecture.