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.

arXiv comment16 pages, 18 figures

References in corpus (48)

• Quantized Electric Multipole Insulators
• Higher-Order Topological Insulators
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Topolectrical circuit realization of topological corner modes
• Reflection symmetric second-order topological insulators and superconductors
• Observation of a phononic quadrupole topological insulator
• An atom-by-atom assembler of defect-free arbitrary 2d atomic arrays
• Demonstration of a quantized microwave quadrupole insulator with topologically protected corner states
• Photonic quadrupole topological phases
• Two-dimensional second-order topological insulator in graphdiyne
• Higher-order bulk-boundary correspondence for topological crystalline phases
• Hyperbolic Lattices in Circuit Quantum Electrodynamics
• Surface State Magnetization and Chiral Edge States on Topological Insulators
• Higher-Order Topological Insulators in Quasicrystals
• Impurity Bound States and Greens Function Zeroes as Local Signatures of Topology
• Higher Order Topological Phases: A General Principle of Construction
• Topological Quadrupolar Semimetals
• Universal Approach to Magnetic Second-Order Topological Insulator
• Topological phases without crystalline counterparts
• Topological Phase Transitions in Disordered Electric Quadrupole Insulators
• Higher-order topological insulators in amorphous solids
• Many-Body Electric Multipole Operators in Extended Systems
• Many-Body Order Parameters for Multipoles in Solids
• Boundary-obstructed topological phases
• Topological insulators in magnetic fields: Quantum Hall effect and edge channels with non-quantized θ-term
• Higher-order Floquet topological phases with corner and bulk bound states
• Simulating hyperbolic space on a circuit board
• Crystallography of Hyperbolic Lattices
• Observation of novel topological states in hyperbolic lattices
• Quantum Simulation of Hyperbolic Space with Circuit Quantum Electrodynamics: From Graphs to Geometry
• Hyperbolic band theory
• Higher-order Topological Anderson Insulators
• Topological hyperbolic lattices
• Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics
• Automorphic Bloch theorems for hyperbolic lattices
• Hyperbolic Matter in Electrical Circuits with Tunable Complex Phases
• Structural Disorder Induced Second-order Topological Insulators in Three Dimensions
• Circuit Quantum Electrodynamics in Hyperbolic Space: From Photon Bound States to Frustrated Spin Models
• Higher-order topological insulators and semimetals in generalized Aubry-André-Harper models
• Hyperbolic Topological Band Insulators
• Universality of Hofstadter butterflies on hyperbolic lattices
• Chern insulator in a hyperbolic lattice
• Band theory and boundary modes of high-dimensional representations of infinite hyperbolic lattices
• Higher-order topological insulators in hyperbolic lattices
• Hyperbolic Band Theory under Magnetic Field and Dirac Cones on a Higher Genus Surface
• `Unhinging' the surfaces of higher-order topological insulators and superconductors
• Type-II quadrupole topological insulators
• Hinge solitons in three-dimensional second-order topological insulators

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• Hyperbolic non-Abelian semimetal
• Simulating Holographic Conformal Field Theories on Hyperbolic Lattices
• Converging Periodic Boundary Conditions and Detection of Topological Gaps on Regular Hyperbolic Tessellations
• 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
• Hyperbolic lattices and two-dimensional Yang-Mills theory
• 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