Nearest-Neighbor Tight-Binding Realization of Hyperbolic Lattices with $\mathbb{Z}_2$ Gauge Structures
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909881654575104 |
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| author | Kong, Xianghong Liu, Xingsi Yang, Shuihua Yan, Zhiyuan Chen, Weijin Xu, Zhixia Qiu, Cheng-Wei |
| author_facet | Kong, Xianghong Liu, Xingsi Yang, Shuihua Yan, Zhiyuan Chen, Weijin Xu, Zhixia Qiu, Cheng-Wei |
| contents | A systematic framework for realizing $\mathbb{Z}_2$ gauge extensions of hyperbolic lattices within the nearest-neighbor tight-binding formalism is developed. Using the triangle group $Δ(2,8,8)$ as an example, we classify all inequivalent projective symmetry groups by computing the second cohomology group $H^2(Δ(2,8,8),\mathbb{Z}_2)$. Each class corresponds to a distinct flux configuration and can be constructed by tight-binding models to verify the symmetry relations of the extended group. The translation subgroups of the $\mathbb{Z}_2$ extended lattices are associated with high genus surfaces, which follows the Riemann-Hurwitz formula. By applying the Abelian hyperbolic band theory, we find the all-flat dispersions along specific directions in momentum space and van Hove singularities correlated with discrete eigenenergies. Our results establish a general route to investigate gauge-extended hyperbolic lattices and provide a foundation for further studying symmetry fractionalization and spin liquid phases in non-Euclidean geometries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00380 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nearest-Neighbor Tight-Binding Realization of Hyperbolic Lattices with $\mathbb{Z}_2$ Gauge Structures Kong, Xianghong Liu, Xingsi Yang, Shuihua Yan, Zhiyuan Chen, Weijin Xu, Zhixia Qiu, Cheng-Wei Optics A systematic framework for realizing $\mathbb{Z}_2$ gauge extensions of hyperbolic lattices within the nearest-neighbor tight-binding formalism is developed. Using the triangle group $Δ(2,8,8)$ as an example, we classify all inequivalent projective symmetry groups by computing the second cohomology group $H^2(Δ(2,8,8),\mathbb{Z}_2)$. Each class corresponds to a distinct flux configuration and can be constructed by tight-binding models to verify the symmetry relations of the extended group. The translation subgroups of the $\mathbb{Z}_2$ extended lattices are associated with high genus surfaces, which follows the Riemann-Hurwitz formula. By applying the Abelian hyperbolic band theory, we find the all-flat dispersions along specific directions in momentum space and van Hove singularities correlated with discrete eigenenergies. Our results establish a general route to investigate gauge-extended hyperbolic lattices and provide a foundation for further studying symmetry fractionalization and spin liquid phases in non-Euclidean geometries. |
| title | Nearest-Neighbor Tight-Binding Realization of Hyperbolic Lattices with $\mathbb{Z}_2$ Gauge Structures |
| topic | Optics |
| url | https://arxiv.org/abs/2511.00380 |