Higher-order topological hyperbolic lattices

A hyperbolic lattice allows for any p-fold rotational symmetry, in stark contrast to a two-dimensional crystalline material, where only 2-fold, 3-fold, 4-fold, or 6-fold rotational symmetry is permitted. This unique feature motivates us to ask whether the enriched rotational symmetry in a hyperbolic lattice can lead to any new topological phases beyond a crystalline material. Here, by constructing and exploring tight-binding models in hyperbolic lattices, we theoretically demonstrate the existence of higher-order topological phases in hyperbolic lattices with 8-fold, 12-fold, 16-fold or 20-fold rotational symmetry, which is not allowed in a crystalline lattice. Since such models respect the combination of time-reversal symmetry and p-fold (p=8, 12, 16, or 20) rotational symmetry, p zero-energy corner modes are protected. For the hyperbolic 8, 3 lattice, we find a higher-order topological phase with a finite edge energy gap and a gapless phase. Our results thus open the door to studying higher-order topological phases in hyperbolic lattices.