On topological obstructions to the existence of non-periodic Wannier bases
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908510676058112 |
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| author | Kordyukov, Yu. Manuilov, V. |
| author_facet | Kordyukov, Yu. Manuilov, V. |
| contents | Recently, M. Ludewig and G. C. Thiang introduced a notion of a uniformly localized Wannier basis with localization centers in an arbitrary uniformly discrete subset $D$ in a complete Riemannian manifold $X$. They show that, under certain geometric conditions on $X$, the class of the orthogonal projection onto the span of such a Wannier basis in the $K$-theory of the Roe algebra $C^*(X)$ is trivial. In this paper, we clarify the geometric conditions on $X$, which guarantee triviality of the $K$-theory class of any Wannier projection. We show that this property is equivalent to triviality of the unit of the uniform Roe algebra of $D$ in the $K$-theory of its Roe algebra, and provide a geometric criterion for that. As a consequence, we prove triviality of the $K$-theory class of any Wannier projection on a connected proper measure space $X$ of bounded geometry with a uniformly discrete set of localization centers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_00125 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On topological obstructions to the existence of non-periodic Wannier bases Kordyukov, Yu. Manuilov, V. Operator Algebras Mathematical Physics Metric Geometry Recently, M. Ludewig and G. C. Thiang introduced a notion of a uniformly localized Wannier basis with localization centers in an arbitrary uniformly discrete subset $D$ in a complete Riemannian manifold $X$. They show that, under certain geometric conditions on $X$, the class of the orthogonal projection onto the span of such a Wannier basis in the $K$-theory of the Roe algebra $C^*(X)$ is trivial. In this paper, we clarify the geometric conditions on $X$, which guarantee triviality of the $K$-theory class of any Wannier projection. We show that this property is equivalent to triviality of the unit of the uniform Roe algebra of $D$ in the $K$-theory of its Roe algebra, and provide a geometric criterion for that. As a consequence, we prove triviality of the $K$-theory class of any Wannier projection on a connected proper measure space $X$ of bounded geometry with a uniformly discrete set of localization centers. |
| title | On topological obstructions to the existence of non-periodic Wannier bases |
| topic | Operator Algebras Mathematical Physics Metric Geometry |
| url | https://arxiv.org/abs/2304.00125 |