Topological linear response of hyperbolic Chern insulators

Abstract

We establish a connection between the electromagnetic Hall response and band topological invariants in hyperbolic Chern insulators by deriving a hyperbolic analog of the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula. By generalizing the Kubo formula to hyperbolic lattices, we show that the Hall conductivity is quantized to $-e^2{C}_{ij}/h$, where $C_{ij}$ is the first Chern number. Through a flux-threading argument, we provide an interpretation of the Chern number as a topological invariant in hyperbolic band theory. We demonstrate that, although it receives contributions from both Abelian and non-Abelian Bloch states, the Chern number can be calculated solely from Abelian states, resulting in a tremendous simplification of the topological band theory. Finally, we verify our results numerically by computing various Chern numbers in the hyperbolic Haldane model.

main text: 17 pages 6 figures; supplemental: 12 pages and 1 figure

References in corpus (59)

• Gauge Theory Correlators from Non-Critical String Theory
• Quantum Spin Hall Effect in Graphene
• The Large N Limit of Superconformal Field Theories and Supergravity
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Classification of topological quantum matter with symmetries
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Periodic table for topological insulators and superconductors
• Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances
• Topological classification of crystalline insulators through band structure combinatorics
• Classification of Topological Insulators and Superconductors
• Hyperbolic Lattices in Circuit Quantum Electrodynamics
• Simulating hyperbolic space on a circuit board
• Crystallography of Hyperbolic Lattices
• Spin-charge Separated Solitons in a Topological Band Insulator
• Observation of novel topological states in hyperbolic lattices
• Quantum Simulation of Hyperbolic Space with Circuit Quantum Electrodynamics: From Graphs to Geometry
• Hyperbolic band theory
• Spin Charge Separation in the Quantum Spin Hall State
• Topological hyperbolic lattices
• Automorphic Bloch theorems for hyperbolic lattices
• Hyperbolic Matter in Electrical Circuits with Tunable Complex Phases
• Hyperbolic band topology with non-trivial second Chern numbers
• Quantum Hall Effect on the Hyperbolic Plane
• Circuit Quantum Electrodynamics in Hyperbolic Space: From Photon Bound States to Frustrated Spin Models
• Hyperbolic Topological Band Insulators
• Chern insulator in a hyperbolic lattice
• Universality of Hofstadter butterflies on hyperbolic lattices
• Band theory and boundary modes of high-dimensional representations of infinite hyperbolic lattices
• Many-body Chern number without integration
• Higher-order topological insulators in hyperbolic lattices
• Curving the space by non-Hermiticity
• Holography on tessellations of hyperbolic space
• Lattice Setup for Quantum Field Theory in AdS${}_2$
• Higher-order Topological Hyperbolic Lattices
• Conformal Quasicrystals and Holography
• Non-Abelian hyperbolic band theory from supercells
• Quantum phase transitions of interacting bosons on hyperbolic lattices
• Symmetry and topology of hyperbolic Haldane models
• 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
• Anderson localization transition in disordered hyperbolic lattices
• Hyperbolic non-Abelian semimetal
• Simulating Holographic Conformal Field Theories on Hyperbolic Lattices
• Central charges of aperiodic holographic tensor network models
• Breitenlohner-Freedman bound on hyperbolic tilings
• Twisted higher index theory on good orbifolds, II: fractional quantum numbers
• Equilibrium Formulae for Transverse Magneto-transport of Strongly Correlated Metals
• Converging Periodic Boundary Conditions and Detection of Topological Gaps on Regular Hyperbolic Tessellations
• Aperiodic spin chains at the boundary of hyperbolic tilings
• Hyperbolic Lattice for Scalar Field Theory in AdS${}_3$
• Spectral and Combinatorial Aspects of Cayley-Crystals
• On the hyperbolic Bloch transform
• Hyperbolic lattices and two-dimensional Yang-Mills theory
• Eigenmodes of the Laplacian on Hyperbolic Lattices
• Testing holographic duality in hyperbolic lattices
• Degeneracy-projected polarization formulas for Hall-type conductivities

Cited by in corpus (8)

• Hyperbolic Spin Liquids
• Wormhole-induced effective coupling in SYK chains
• Unveiling Non-Hermitian Spectral Topology in Hyperbolic Lattices with Non-Abelian Translation Symmetry
• Holographic Foliations: Self-Similar Quasicrystals from Hyperbolic Honeycombs
• Quantum geometric tensors from sub-bundle geometry
• Bose-Einstein condensation in exotic lattice geometries
• Entanglement scaling behaviors of free fermions on hyperbolic lattices
• Discrete holography and density of states in the crossover from hyperbolic to Euclidean lattices