Tunable cornerlike states in topological type-II hyperbolic lattices

Abstract

Type-II hyperbolic lattices constitute a new class of hyperbolic structures that are projected onto the Poincaré ring and possess both an inner and an outer boundary. In this work, we reveal the higher-order topological phases in type-II hyperbolic lattices, characterized by the generalized quadrupole moment. Unlike the type-I hyperbolic lattices where zero-energy cornerlike states exist on a single boundary, the higher-order topological phases in type-II hyperbolic lattices possess zero-energy cornerlike states localized on both the inner and outer boundaries. These findings are verified within both the modified Bernevig-Hughes-Zhang model and the Benalcazar-Bernevig-Hughes model. Furthermore, we demonstrate that the higher-order topological phase remains robust against weak disorder in type-II hyperbolic lattices. Our work provides a route for realizing and controlling higher-order topological states in type-II hyperbolic lattices.

Figures (8)

• Figure 1: In the Poincaré ring, the vertices of the polygons correspond to the sites of a type-II hyperbolic lattice. The symbol $\{p=8, q=3, k=6\}$ denotes a tiling by regular $p$-sided polygons on the Poincaré ring, where $q$ polygons meet at each vertex in the bulk. Here, $r_{h}=e^{-2\pi/kP}$ represents the characteristic radius of the type-II hyperbolic lattice, where the structural parameter $k=6$ is an integer and $P=1.559$ is a geometry constant arXiv2305.04862PhysRevLett.133.06160310.1038/s42005-025-01990-w. The parameter $k$ denotes the number of repeating units (yellow sector) in the lattice. A rotation by $2\pi/k$ of the sites within one unit brings them into coincidence with the sites of an adjacent repeating unit.
• Figure 2: (a) Energy spectrum of the Hamiltonian $H$ in the $\{8, 3, 4\}$ lattice when $g=0$. (b) The probability distribution of the boundary states marked with blue dots in (a). (c) Energy spectrum of the Hamiltonian $H$ in the $\{8, 3, 4\}$ lattice when $g=0.5$. (d) The probability distribution of the eight zero-energy eigenstates marked with blue dots in (c). (e) Energy of the Hamiltonian $H$ as a function of the Wilson mass $g$. (f) The quadrupole moment $Q_{xy}$ as a function of the Wilson mass $g$. Here, we take the parameters $M=-1$, $t_{1}=t_{2}=1$, and $\eta=2$.
• Figure 3: (a) Energy of the Hamiltonian $H+H_{W1}$ as a function of the disorder strength $W_{1}$ when $g=0.5$. (b) The quadrupole moment $Q_{xy}$ as a function of the disorder strength $W_{1}$ when $g=0.5$. The error bar represents the standard deviation of 100 samples. (c) Energy spectrum of the Hamiltonian $H+H_{W1}$ when $g=0.5$ and $W_{1}=0.2$. (d) The probability distribution of the eight zero-energy eigenstates marked with blue dots in (c). Here, we take the parameters $M=-1$, $t_{1}=t_{2}=1$, $\eta=2$, and $k=4$.
• Figure 4: (a) Energy spectrum of the Hamiltonian $H$ in the $\{8, 3, 8\}$ lattice when $g=0.5$ and $\eta=4$. (b) The probability distribution of the sixteen zero-energy eigenstates marked with blue dots in (a). (c) Energy of the Hamiltonian $H$ for different $k$ when $g=0.5$ and $\eta=4$. (d) Energy of the Hamiltonian $H+H_{W1}$ as a function of the disorder strength $W_{1}$ when $k=8$. (e) Comparison of structures between the $\{8,3,4\}$ and $\{8,3,8\}$ lattices. The yellow sectors mark the repeating units. Here, we take the parameters $M=-1$, $t_{1}=t_{2}=1$, $g=0.5$, and $\eta=4$.
• Figure 5: (a) Schematics of how to change site positions in the $\{8, 3, 8\}$ lattice. Left panel: A type-II hyperbolic lattice with sixteen zero-energy cornerlike states. The coordinate $(x_{n}, y_{n})$ of the $n$th site is expressed on the complex plane as $r_{n} e^{i\theta_{n}}$ with $-\pi \le \theta_{n} < \pi$. To correctly compute the generalized quadrupole moment $Q_{x^{\prime}y^{\prime}}$, we apply a coordinate transformation defined piecewise: for $0 \le \theta_n < 3\pi/4$, $\theta_{n}^{\prime} = \frac{2}{3}\theta_n$; for $3\pi/4 \le \theta_n < \pi$, $\theta_{n}^{\prime} = 2\theta_n - \pi$; for $-\pi \le \theta_n < -\pi/4$, $\theta_{n}^{\prime} = \frac{2}{3}\theta_n - \pi/3$; for $-\pi/4 \le \theta_n < 0$, $\theta_{n}^{\prime} = 2\theta_n$. The transformed coordinate is $r_n e^{i\theta_n^{\prime}}$ or $(x^{\prime}, y^{\prime})$. Right panel: The type-II hyperbolic lattice after the coordinate transformation. (b) Energy of the Hamiltonian $H$ as a function of the Wilson mass $g$ in the $\{8, 3, 6\}$ lattice. (c) The generalized quadrupole moment $Q_{x^{\prime}y^{\prime}}$ as a function of the Wilson mass $g$ in the $\{8, 3, 6\}$ lattice. Here, we take the parameters $M=-1$, $t_{1}=t_{2}=1$, and $\eta=4$.