Higher-order topological insulators in hyperbolic lattices

To explore the non-Euclidean generalization of higher-order topological phenomena, we construct a higher-order topological insulator model in hyperbolic lattices by breaking the time-reversal symmetry (TRS) of quantum spin Hall insulators. We investigate three kinds of hyperbolic lattices, i.e., hyperbolic ${4,5}, {8,3}$, and ${12,3}$ lattices, respectively. The non-Euclidean higher-order topological behavior is characterized by zero-energy effective corner states appearing in hyperbolic lattices. By adjusting the variation period of the TRS breaking term, we obtain 4, 8, and 12 zero-energy effective corner states in these three different hyperbolic lattices, respectively. It is found that the number of zero-energy effective corner states of a hyperbolic lattice depends on the variation period of the TRS breaking term. The real-space quadrupole moment is employed to characterize the higher-order topology of the hyperbolic lattice with four zero-energy effective corner states. Via symmetry analysis, it is confirmed that the hyperbolic zero-energy effective corner states are protected by the particle-hole symmetry $P$, the effective chiral symmetry $S{m}_{z}$, and combined symmetries ${C}_{p}T$ and ${C}_{p}{m}_{z}$. The hyperbolic zero-energy effective corner states remain stable unless these four symmetries are broken simultaneously. The topological nature of hyperbolic zero-energy effective corner states is further confirmed by checking the robustness of the zero-energy modes in the hyperbolic lattices in the presence of disorder. Our paper provides a route for research on hyperbolic higher-order topological insulators in non-Euclidean geometric systems.