Symmetry and topology of hyperbolic Haldane models

Abstract

Particles hopping on a two-dimensional hyperbolic lattice feature unconventional energy spectra and wave functions that provide a largely uncharted platform for topological phases of matter beyond the Euclidean paradigm. Using real-space topological markers as well as Chern numbers defined in the higher-dimensional momentum space of hyperbolic band theory, we construct and investigate hyperbolic Haldane models, which are generalizations of Haldane's honeycomb-lattice model to various hyperbolic lattices. We present a general framework to characterize point-group symmetries in hyperbolic tight-binding models, and use this framework to constrain the multiple first and second Chern numbers in momentum space. We observe several topological gaps characterized by first Chern numbers of value $1$ and $2$. The momentum-space Chern numbers respect the predicted symmetry constraints and agree with real-space topological markers, indicating a direct connection to observables such as the number of chiral edge modes. With our large repertoire of models, we further demonstrate that the topology of hyperbolic Haldane models is trivialized for lattices with strong negative curvature.

main text (14 pages with 7 figures and 2 tables) + appendices (28 pages with 10 figures and 2 tables) + bibliography (2 pages)

References in corpus (27)

• Photonic topological pumping through the edges of a dynamical four-dimensional quantum Hall system
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Exploring 4D Quantum Hall Physics with a 2D Topological Charge Pump
• Simulating hyperbolic space on a circuit board
• Crystallography of Hyperbolic Lattices
• Observation of novel topological states in hyperbolic lattices
• Automorphic Bloch theorems for hyperbolic lattices
• Hyperbolic Matter in Electrical Circuits with Tunable Complex Phases
• A Fermi-Hubbard Optical Tweezer Array
• Hyperbolic band topology with non-trivial second Chern numbers
• Circuit Quantum Electrodynamics in Hyperbolic Space: From Photon Bound States to Frustrated Spin 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
• Periodic boundary conditions on the pseudosphere
• Higher-order topological insulators in hyperbolic lattices
• Holography on tessellations of hyperbolic space
• Higher-order Topological Hyperbolic Lattices
• Quantum phase transitions of interacting bosons on hyperbolic lattices
• Density of states of tight-binding models in the hyperbolic plane
• Selberg trace formula in hyperbolic band theory
• Towards Explicit Discrete Holography: Aperiodic Spin Chains from Hyperbolic Tilings
• Breitenlohner-Freedman bound on hyperbolic tilings
• Topological Sliding Moiré Heterostructure
• On the hyperbolic Bloch transform
• Properties of Eigenvalues on Riemann Surfaces with Large Symmetry Groups

Cited by in corpus (20)

• Non-Abelian hyperbolic band theory from supercells
• Anderson localization transition in disordered hyperbolic lattices
• Hyperbolic non-Abelian semimetal
• Simulating Holographic Conformal Field Theories on Hyperbolic Lattices
• Topological linear response of hyperbolic Chern insulators
• Engineering topological states and quantum-inspired information processing using classical circuits
• 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
• Four-dimensional Floquet topological insulator with an emergent second Chern number
• Five-dimensional Floquet topological semimetals with emergent Yang monopoles and linked Weyl surfaces
• Wormhole-induced effective coupling in SYK chains
• Non-Hermitian catalysis of spontaneous symmetry breaking on Euclidean and hyperbolic lattices
• Quantum geometric tensors from sub-bundle geometry
• Bose-Einstein condensation in exotic lattice geometries
• Real-space representation of the second Chern number
• Entanglement scaling behaviors of free fermions on hyperbolic lattices
• Topological states and flat bands in exactly solvable decorated Cayley trees
• Discrete holography and density of states in the crossover from hyperbolic to Euclidean lattices