Algebraic localization of generalized Wannier bases implies Roe triviality in any dimension
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929427554762752 |
|---|---|
| author | Rossi, Vincenzo Panati, Gianluca |
| author_facet | Rossi, Vincenzo Panati, Gianluca |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14235 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algebraic localization of generalized Wannier bases implies Roe triviality in any dimension Rossi, Vincenzo Panati, Gianluca Mathematical Physics Mesoscale and Nanoscale Physics Operator Algebras 81V70, 81Q70 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. |
| title | Algebraic localization of generalized Wannier bases implies Roe triviality in any dimension |
| topic | Mathematical Physics Mesoscale and Nanoscale Physics Operator Algebras 81V70, 81Q70 |
| url | https://arxiv.org/abs/2407.14235 |