Topological equivalence of crystal and quasicrystal band structures

Abstract

A number of recent articles have reported the existence of topologically non-trivial states and associated end states in one-dimensional incommensurate lattice models that would usually only be expected in higher dimensions. Using an explicit construction, we here argue that the end states have precisely the same origin as their counterparts in commensurate models and that incommensurability does not in fact provide a meaningful connection to the topological classification of systems in higher dimensions. In particular, we show that it is possible to smoothly interpolate between states with commensurate and incommensurate modulation parameters without closing the band gap and without states crossing the band gap.

7 pages, 9 figures. Editors' Suggestion

References in corpus (7)

• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Classification of topological insulators and superconductors in three spatial dimensions
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• The Quantum Spin Hall Effect: Theory and Experiment
• Observation of Topological Phase Transitions in Photonic Quasicrystals
• Majorana fermions in density modulated p-wave superconducting wires

Cited by in corpus (57)

• Topological phase transition in non-Hermitian quasicrystals
• Topological phases of generalized Su-Schrieffer-Heeger models
• Localization and adiabatic pumping in a generalized Aubry-André-Harper model
• Edge states in trimer lattices
• Topological phase in a non-Hermitian PT symmetric system
• Topological Hofstadter Insulators in a Two-Dimensional Quasicrystal
• Localization and topological transitions in non-Hermitian quasiperiodic lattices
• Radiative topological states in resonant photonic crystals
• The Aubry-André model as the hobbyhorse for understanding localization phenomenon
• Virtual Topological Insulators with Real Quantized Physics
• Constructing a Weyl semimetal by stacking one dimensional topological phases
• Kaleidoscope of symmetry protected topological phases in one-dimensional periodically modulated lattices
• Fibonacci optical lattices for tunable quantum quasicrystals
• Electromagnetic response of quantum Hall systems in dimensions five and six and beyond
• Topological Edge Modes by Smart Patterning
• Topological superconductivity in quasicrystals
• Topological Anderson insulators with different bulk states in quasiperiodic chains
• Boundary spin polarization as robust signature of topological phase transition in Majorana nanowires
• Topological properties of the long-range Kitaev chain with Aubry-André-Harper modulation
• Fractional boundary charges in quantum dot arrays with density modulation
• Topological mechanics in quasicrystals
• Breakdown of the correspondence between the real-complex and delocalization-localization transitions in non-Hermitian quasicrystals
• Theory of a 3+1D fractional chiral metal: interacting variant of the Weyl semimetal
• From fractional boundary charges to quantized Hall conductance
• Realizing the Harper model with Ultracold Atoms in a Ring Lattice
• Topological Anderson insulators in an Ammann-Beenker quasicrystal and a snub-square crystal
• The K-Theoretic Bulk-Boundary Principle for Dynamically Patterned Resonators
• Classical Dimers on Penrose Tilings
• Topological Properties of Ultracold Bosons in One-Dimensional Quasiperiodic Optical Lattice
• Phasonic Spectroscopy of a Quantum Gas in a Quasicrystalline Lattice
• Anomalous charge pumping in a one-dimensional optical superlattice
• Time Quasilattices in Dissipative Dynamical Systems
• Rational Approximations of Quasi-Periodic Problems via Projected Green's Functions
• Critical integer quantum Hall topology and the integrable Maryland model as a topological quantum critical point
• Josephson effect in a Fibonacci quasicrystal
• Quantum Emulation of Extreme Non-equilibrium Phenomena with Trapped Atoms
• Entangled multi-component 4D quantum Hall states from photonic crystal defects
• Quasiperiodicity and 2D Topology in 1D Charge Ordered Materials
• Emergence of quasiperiodic Bloch wave functions in quasicrystals
• Phononic Topological States in 1D Quasicrystals
• Colloquium: Synthetic quantum matter in non-standard geometries
• Bulk-boundary correspondance for Sturmian Kohmoto like models
• Topological superconductivity in Fibonacci quasicrystals
• Structural spillage: an efficient method to identify non-crystalline topological materials
• Quantum transport in a one-dimensional quasicrystal with mobility edges
• Topological Phase Transitions from Harper to Fibonacci Crystals
• Self-duality of One-dimensional Quasicrystals with Spin-Orbit Interaction
• Unraveling Multifractality and Mobility Edges in Quasiperiodic Aubry-André-Harper Chains through High-Harmonic Generation
• Superconductivity and charge-density-wave in the Holstein model on the Penrose Lattice
• Compressibility enhancement in an almost staggered interacting Harper model
• Hamiltonian Cycles on Ammann-Beenker Tilings
• Nonlinear topological Toda quasicrystal
• Topological end states in two-orbital double-exchange model for colossal magnetoresistive manganites
• Two-strand ladder network variants: localization, multifractality, and quantum dynamics under an Aubry-André-Harper kind of quasiperiodicity
• Localization and multifractal properties of the long-range Kitaev chain in the presence of an Aubry-André-Harper modulation
• Disorder-induced chiral and helical Majorana edge modes in a two-dimensional Ammann-Beenker quasicrystal
• Quantum state transfer and maximal entanglement between distant qubits using a minimal quasicrystal pump