Nearest-Neighbor Tight-Binding Realization of Hyperbolic Lattices with $\mathbb{Z}_2$ Gauge Structures

TL;DR

Realizing $\mathbb{Z}_2$ gauge extensions of hyperbolic lattices is challenging due to non-Abelian translation groups; this work classifies projective representations of the triangle group $\Delta(2,8,8)$ via the second cohomology $H^2(\Delta(2,8,8),\mathbb{Z}_2)$, yielding six independent invariants that generate $64$ symmetry classes. It then constructs simple nearest-neighbor tight-binding models for all classes, associating invariants to flux patterns in rotation and mirror sectors, and analyzes spectra using the Abelian hyperbolic band theory, revealing all-flat bands along specific momentum directions and van Hove singularities aligned with discrete eigenenergies. The results establish a general route to gauge-extended hyperbolic lattices and provide a foundation for exploring symmetry fractionalization and spin-liquid physics in non-Euclidean geometries, with potential extensions to $\mathbb{Z}_n$ and $U(1)$ gauge structures.

Abstract

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.

Figures (13)

• Figure 1: Symmetries of different triangle groups. The primitive cells are plotted by blue polygons.(a) Triangle group $\Delta(2,8,8)$ presented by alternating grey and white triangles. The reflection lines $a, b, c$ are the generators shown in Eq. (\ref{['eq:triangle']}). (b) Triangle group $\Delta(2,4,4)$ (square lattice) with its reflection lines $a, b, c$.
• Figure 2: (a) The $\pi$-flux of rotation operator $\tilde{R}^4=-1$. (b) Conventional mirror symmetry $M$ is replaced by (c) a $C_2$ rotation around a horizontal axis. Red edges represent hopping coefficients $t_{ij}=-1$ while black edges represent $t_{ij}=1$.
• Figure 3: Flux distributions with different cohomology invariants of the triangle group $\Delta(2,8,8)$.(a) The fluxes $\Phi_i, i=1, 2, 3$ correspond to cohomology invariants $\alpha_i, i=1, 2, 3$ as shown in Eq. (\ref{['eq:288relation']}). (b) The fluxes $\Phi_i, i=4, 5, 6$ correspond to cohomology invariants $\beta_i, i=1, 2, 3$ as shown in Eq. (\ref{['eq:288relation']}).
• Figure 4: Top-down projection schematic of the tight-binding model with flux distribution $(\alpha_1, \alpha_2, \alpha_3, \beta_1, \beta_2, \beta_3)=(-1, 1, 1, 1, 1, 1)$. The black/red lines represent hopping coefficients $t_{ij}=1/-1$ for both top and bottom layers. The blue dots represent the hopping coefficients $t_{ij}=1$ between the top and bottom layers. The primitive cell is shown by blue curved lines and the periodic boundaries $\bar{i}$ and $i$ are related by the translation operator $\gamma_i$.
• Figure 5: Top-down projection schematic of the tight-binding model with flux distribution $(\alpha_1, \alpha_2, \alpha_3, \beta_1, \beta_2, \beta_3)=(1, -1, 1, 1, 1, 1)$. The black/red lines represent hopping coefficients $t_{ij}=1/-1$ for both top and bottom layers. The blue dots represent the hopping coefficients $t_{ij}=1$ between the top and bottom layers. The supercell is shown by blue curved lines and the periodic boundaries $\bar{i}$ and $i$ are related by the translation operator $\tilde{\gamma}_i$.