Higher-order Topological Hyperbolic Lattices

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.

16 pages, 18 figures

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• Quasicrystalline second-order topological semimetals
• Topological linear response of hyperbolic Chern insulators
• Colloquium: Synthetic quantum matter in non-standard geometries
• Higher-order Topological Insulators and Semimetals in Three Dimensions without Crystalline Counterparts
• Hyperbolic Fractional Chern insulators
• Anderson transition and mobility edges on hyperbolic lattices with randomly connected boundaries
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• Hyperbolic Spin Liquids
• Higher-order topology in twisted multilayer systems: a review
• Uncovering Exceptional Contours in non-Hermitian Hyperbolic Matter
• Entanglement scaling behaviors of free fermions on hyperbolic lattices
• Tunable cornerlike states in topological type-II hyperbolic lattices
• Real-space formulation of topology for disordered Rice-Mele chains without chiral symmetry
• Discrete holography and density of states in the crossover from hyperbolic to Euclidean lattices
• Topological states and flat bands in exactly solvable decorated Cayley trees