Topological models in rotationally symmetric quasicrystals

Abstract

We investigate the physics of quasicrystalline models in the presence of a uniform magnetic field, focusing on the presence and construction of topological states. This is done by using the Hofstadter model but with the sites and couplings denoted by the vertex model of the quasicrystal, giving the Hofstadter vertex model. We specifically consider two-dimensional quasicrystals made from tilings of two tiles with incommensurate areas, focusing on the five-fold Penrose and the eight-fold Ammann-Beenker tilings. This introduces two competing scales; the uniform magnetic field and the incommensurate scale of the cells of the tiling. Due to these competing scales the periodicity of the Hofstadter butterfly is destroyed. We observe the presence of topological edge states on the boundary of the system via the Bott index that exhibit two way transport along the edge. For the eight-fold tiling we also observe internal edge-like states with non-zero Bott index, which exhibit two way transport along this internal edge. The presence of these internal edge states is a new characteristic of quasicrystalline models in magnetic fields. We then move on to considering interacting systems. This is challenging, in part because exact diagonalization on a few tens of sites is not expected to be enough to accurately capture the physics of the quasicrystalline system, and in part because it is not clear how to construct topological flatbands having a large number of states. We show that these problems can be circumvented by building the models analytically, and in this way we construct models with Laughlin type fractional quantum Hall ground states.

arXiv comment13 pages, 16 figures, published PRB

References in corpus (45)

• Topological Insulators
• Discovery (theoretical prediction and experimental observation) of a large-gap topological-insulator class with spin-polarized single-Dirac-cone on the surface
• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Experimental realisation of the topological Haldane model
• Periodic table for topological insulators and superconductors
• Acoustic topological insulator and robust one-way sound transport
• Fractional quantum Hall states at zero magnetic field
• Direct Measurement of the Zak phase in Topological Bloch Bands
• Observation of phononic helical edge states in a mechanical 'topological insulator'
• Zoo of quantum-topological phases of matter
• Fractional Chern Insulator
• Observation of photonic anomalous Floquet Topological Insulators
• Observation of the topological soliton state in the Su-Schrieffer-Heeger model
• Experimental observation of anomalous topological edge modes in a slowly-driven photonic lattice
• Higher-Order Topological Insulators in Quasicrystals
• Topological Photonic Quasicrystals: Fractal Topological Spectrum and Protected Transport
• Topological Phases in Non-Hermitian Aubry-André-Harper Models
• Quantum Spin Hall Effect and Spin Bott Index in a Quasicrystal Lattice
• Topological phases without crystalline counterparts
• Topological Hofstadter Insulators in a Two-Dimensional Quasicrystal
• Theory of spin Bott index for quantum spin Hall states in non-periodic systems
• Adiabatic Preparation of Topological Order
• Aperiodic Weak Topological Superconductors
• Laughlin spin liquid states on lattices obtained from conformal field theory
• Topological states on fractal lattices
• Topology in the Sierpiński-Hofstadter problem
• Gap solitons in quasiperiodic optical lattices
• Hofstadter butterfly of a quasicrystal
• Lattice Laughlin States of Bosons and Fermions at Filling Fractions $1/q$
• Exact Eigenstates of Tight-Binding Hamiltonians on the Penrose Tiling
• Topological mechanics in quasicrystals
• Anyon braiding in semi-analytical fractional quantum Hall lattice models
• Anatomy of Topological Surface States: Exact Solutions from Destructive Interference on Frustrated Lattices
• Exact edge, bulk and bound states of finite topological systems
• Lattice effects on Laughlin wave functions and parent Hamiltonians
• Bulk Spectrum and K-theory for Infinite-Area Topological Quasicrystal
• Landau levels in quasicrystals
• Time-dependent topological systems: A study of the Bott index
• Quasielectrons as inverse quasiholes in lattice fractional quantum Hall models
• Generalized quasiperiodic Rauzy tilings
• Analytical study of the edge states in the bosonic Haldane model
• A higher-dimensional quasicrystalline approach to the Hofstadter and Fibonacci butterflies topological phase diagram and band conductance: symbolic sequences, Sturmian coding and self-similar rules at all magnetic fluxes
• Truncation of lattice fractional quantum Hall Hamiltonians derived from conformal field theory
• Fractalized Metals
• Super Hamiltonian in superspace for incommensurate superlattices and quasicrystals

Cited by in corpus (41)

• Higher-Order topological insulator in a dodecagonal quasicrystal
• Topological states in quasicrystals
• Adiabatic pumping via avoided crossings in stiffness modulated quasiperiodic beams
• Higher-order topological phases in crystalline and non-crystalline systems: a review
• On the Bott index of unitary matrices on a finite torus
• Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries
• Topological Gaps by Twisting
• Quasicrystalline second-order topological semimetals
• Physical properties of an Aperiodic monotile: Graphene-like features, chirality and zero-modes
• Topological Gaps in Quasi-Periodic Spin Chains: A Numerical and K-Theoretic Analysis
• Mechanics and dynamics of two-dimensional quasiperiodic composites
• The Mean-Field Bose Glass in Quasicrystalline Systems
• Topological charge pumping in quasiperiodic systems characterized by Bott index
• Effect of coordination on topological phases on self-similar structures
• Bulk Localised Transport States in Infinite and Finite Quasicrystals via Magnetic Aperiodicity
• Higher-dimensional Hofstadter butterfly on Penrose lattice
• Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing
• Confined states and topological phases in two-dimensional quasicrystalline $\pi$-flux model
• Confined states in the tight-binding model on the hexagonal golden-mean tiling
• Ferromagnetically ordered states in the Hubbard model on the $H_{00}$ hexagonal golden-mean tiling
• Weyl superconductivity and quasiperiodic Majorana arcs in quasicrystals
• Interlaced wire medium with quasicrystal lattice
• Electronic structure and transport in materials with flat bands: 2D materials and quasicrystals
• Holographic Foliations: Self-Similar Quasicrystals from Hyperbolic Honeycombs
• Quasicrystalline Spin Liquid
• Family of Aperiodic Tilings with Tunable Quantum Geometric Tensor
• Identification and properties of topological states in the bulk of quasicrystals
• Adiabatic pumping and transport in the Sierpinski-Hofstadter model
• Missing link between the 2D Quantum Hall problem and 1D quasicrystals
• Tensor network method for real-space topology in quasicrystal Chern mosaics
• Mott transition and correlation effects on strictly localized states in an octagonal quasicrystal
• Stripe order in quasicrystals
• Mapping Chern numbers in quasi-periodic interacting spin chains
• Disorder-induced chiral and helical Majorana edge modes in a two-dimensional Ammann-Beenker quasicrystal
• Site-selective correlations in interacting "flat-band" quasicrystals
• Exact Parent Hamiltonians for All Landau Level States in a Half-flux Lattice
• Identification of the Majorana edge modes in tight-binding systems based on the Krylov method
• Superconducting order parameter in aperiodic binary systems
• Topological states and flat bands in exactly solvable decorated Cayley trees
• The Bott index of two unitary operators and the integer quantum Hall effect
• One-dimensional topological quasiperiodic chain for directional wireless power transfer