Hybrid higher-order skin-topological effect in hyperbolic lattices

We investigate the non-Hermitian Haldane model on hyperbolic $\{8,3\}$ and $\{12,3\}$ lattices and showcase its intriguing topological properties in the simultaneous presence of non-Hermitian effect and hyperbolic geometry. From bulk descriptions of the system, we calculate the real-space non-Hermitian Chern numbers by generalizing the method from its Hermitian counterpart and present a corresponding phase diagram of the model. For boundaries, we find that skin-topological modes appear in the range of the bulk energy gap under certain boundary conditions, which can be explained by an effective one-dimensional zigzag chain model mapped from a hyperbolic lattice boundary. Remarkably, these skin-topological modes are localized at specific corners of the boundary, constituting a hybrid higher-order skin-topological effect on hyperbolic lattices.

Figure 1

(a) Schematic illustration of the Haldane model on a hyperbolic $\{8,3\}$ lattice. The black dashed lines indicate the next-nearest-neighbor hoppings with amplitudes $t_2{e}^{\pm i\varphi }$. The sign of $\varphi$ is positive (negative) for the (opposite) direction marked by the arrow, which arises due to an alternating magnetic flux. (b) Energy spectrum of the Hamiltonian in Eq. (1) on a hyperbolic $\{8,3\}$ lattice with open boundary condition. (c) The corresponding density of states of bulk (blue) and edge (green) states. The red dotted lines in (b) and (c) estimate the topological gap above the energy $\mathrm{Re}\left(E\right)=0$ (the lower one is symmetric). Here the parameters in (b) and (c) are $t_1=1, t_2=0.2, m=0$, and $\gamma =0.3$.

Figure 2

(a) Convergence of the real-space Chern number for the two non-Hermitian topological phases as the lattice size increases. The blue (red) symbol represents the case that the chemical potential is located in the lower (upper) topological gap. (b) $C_{\text{RS}}$ as a function of chemical potential $\mu$ on a hyperbolic $\{8,3\}$ lattice with 500 sites. IPR for various disorder strengths without (c) and with (d) the non-Hermitian potential. In (c), and (a), (b), (d), the values of the imaginary potential are $\gamma =0$ and 0.3, respectively. The other parameters are the same as those in Fig. 1.

Figure 3

(a) Energy spectrum and (b) the corresponding density of states of the Hamiltonian in Eq. (1) on a hyperbolic $\{8,3\}$ lattice obtained by HBT (red dots or line) with periodic boundary conditions. In (a) and (b), the same quantities calculated by directly diagonalizing the Hamiltonian of an open hyperbolic flake are also plotted for comparison. The phase diagrams in the $(\gamma ,m)$ plane: (c) at half filling; (d) below or above half filling. CI (TI) represents conventional (topological) insulator. Here the parameters not specified are the same as those used in Fig. 1.

Figure 4

Demonstration of the skin-topological corner modes in terms of the sum of the distributions of the edge states in the energy range $1-1.4t_1$ for (a) $\gamma =0.3$ and (b) $\gamma =-0.3$. The other parameters are the same as those in Fig. 1.

Figure 5

(a) Illustration of the inequivalent one-fourth of the studied open hyperbolic lattice. Zoomed details of the edge geometries in the dotted line boxes located at (b) the top (red) and (d) the diagonal (green). (c) and (e) show the outermost sites arranged in zigzag shapes for the boundary fragments (b) and (d), respectively. The arrows in (c) and (e) denote the directions along which the next-nearest-neighbor hopping phase $\varphi$ is positive.

Figure 6

The distributions of the eigenstates of the one-dimensional zigzag chain model mapped from the boundary of the inequivalent one-fourth geometry in Fig. 5 at (a) $\gamma =0.8$ and (b) $\gamma =-0.8$. (c) and (d) are the distributions on a closed one-dimensional chain corresponding to the boundary of the whole hyperbolic disk with $\gamma =0.8$ and $\gamma =0$, respectively. The eigenstates are distinguished by different colors in (a)–(d).

Figure 7

(a) A smoother boundary created by cutting the hyperbolic lattice with a fixed-radius circle. (b) The distributions of the edge states in the topological energy gap at $\gamma =0.3$, and no skin-topological corner modes are visible. The other parameters are the same as those in Fig. 1.

Figure 8

(a) Schematic illustration of the Haldane model on a hyperbolic $\{12,3\}$ lattice. (b) Energy spectrum of the non-Hermitian Haldane Hamiltonian on an open hyperbolic $\{12,3\}$ lattice. (c) The corresponding density of states of bulk (blue) and edge (green) states. The red dotted lines in (b) and (c) estimate the topological gap above the $\mathrm{Re}\left(E\right)=0$ (the lower one is symmetric). Here the parameters in (b) and (c) are $t_1=1, t_2=0.4, m=0$, and $\gamma =0.1$.

Figure 9

(a) The inequivalent one-sixth of the open hyperbolic $\{12,3\}$ lattice. (b) The skin-topological modes appearing symmetrically at six corners of an open hyperbolic $\{12,3\}$ lattice.

Figure 10

Sketch of hyperbolic $\{8,3\}$ (a) and $\{12,3\}$ (b) lattices with marked unit cell (red dots) and fundamental domain of the Bravais lattice (red polygon), respectively. ${\gamma }_j$ in (a) and (b) are the generators of their lattice.