Higher-order Topological Hyperbolic Lattices

TL;DR

This paper demonstrates the existence of higher-order topological phases in hyperbolic lattices with high rotational symmetry, revealing new topological phenomena beyond those possible in crystalline materials.

Contribution

The authors construct and analyze tight-binding models on hyperbolic lattices, showing for the first time that higher-order topological phases can exist with symmetries not allowed in crystalline lattices.

Findings

Existence of p-fold (p=8,12,16,20) protected zero-energy corner modes.

Identification of gapped, gapless, and reentrant topological phases in hyperbolic lattices.

Reentrant phase caused by finite-size effects affecting edge but not corner states.

Abstract

A hyperbolic lattice allows for any $p$-fold rotational symmetry, in stark contrast to a two-dimensional crystalline material, where only twofold, threefold, fourfold or sixfold 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 eight-fold, twelve-fold, sixteen-fold or twenty-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 gapped, a gapless and a reentrant gapped higher-order topological hyperbolic phases. The reentrant phase arises from finite-size effects, which open the gap of edge states while leave the gap of corner modes unchanged. Our results thus open the door to studying higher-order topological phases in hyperbolic lattices.