Perturbative Approach to Flat Chern Bands in the Hofstadter Model

Abstract

We present a perturbative approach to the study of the Hofstadter model for when the amount of flux per plaquette is close to a rational fraction. Within this approximation certain eigenstates of the system are shown to be multi-component wavefunctions that connect smoothly to the Landau levels of the continuum. The perturbative corrections to these are higher Landau level contributions that break rotational invariance and allow the perturbed states to adopt the symmetry of the lattice. In the presence of interactions, this approach allows for the calculation of generalised Haldane pseudopotentials, and in turn, the many-body properties of the system. The method is sufficiently general that it can apply to a wide variety of lattices, interactions and magnetic field strengths.

arXiv comment40 pages, 15 figures; v2 includes minor changes, additional references and an expanded background section

References in corpus (46)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Berry Phase Effects on Electronic Properties
• Artificial gauge potentials for neutral atoms
• Hofstadter's butterfly in moire superlattices: A fractal quantum Hall effect
• Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure
• Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices
• Cloning of Dirac fermions in graphene superlattices
• Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices
• Fractional quantum Hall states at zero magnetic field
• Topological orders and Edge excitations in FQH states
• Nearly-flat bands with nontrivial topology
• Creation of effective magnetic fields in optical lattices: The Hofstadter butterfly for cold neutral atoms
• Fractional quantum Hall effect in the absence of Landau levels
• Rapidly Rotating Atomic Gases
• Synthetic gauge fields in synthetic dimensions
• Topological Flat Band Models and Fractional Chern Insulators
• Fractional quantum Hall states of atoms in optical Lattices
• Fractional Quantum Hall Physics in Topological Flat Bands
• Fractional Quantum Hall Effect in Optical Lattices
• Topological Nematic States and Non-Abelian Lattice Dislocations
• Generic Wavefunction Description of Fractional Quantum Anomalous Hall States and Fractional Topological Insulators
• Exact Parent Hamiltonian for the Quantum Hall States in a Optical Lattice
• Composite Fermion Theory for Bosonic Atoms in Optical Lattices
• High field fractional quantum Hall effect in optical lattices
• Fractional Chern Insulators and the W-Infinity Algebra
• Bloch Model Wavefunctions and Pseudopotentials for All Fractional Chern Insulators
• Critical and bicritical properties of Harper's equation with next nearest neighbor coupling
• Hierarchy of fractional Chern insulators and competing compressible states
• Pseudopotentials for Multi-particle Interactions in the Quantum Hall Regime
• Gauge-Fixed Wannier Wave-Functions for Fractional Topological Insulators
• Hofstadter butterfly as Quantum phase diagram
• Adiabatic continuation of Fractional Chern Insulators to Fractional Quantum Hall States
• SU(3) Spin-Orbit Coupling in Systems of Ultracold Atoms
• Adiabatic continuity between Hofstadter and Chern insulator states
• Bilayer quantum Hall phase transitions and the orbifold non-Abelian fractional quantum Hall states
• Entanglement spectrum and Wannier center flow of the Hofstadter problem
• Fractional Quantum Hall Effect of Lattice Bosons Near Commensurate Flux
• Fractional Chern insulators with strong interactions far exceeding bandgaps
• Semiclassical analysis and the magnetization of the Hofstadter model
• Edge excitations of bosonic fractional quantum Hall phases in optical lattices
• D-Algebra Structure of Topological Insulators
• Edge Excitations in Fractional Chern Insulators
• The Fractional Quantum Hall Effect on a Lattice
• Adiabatic preparation of vortex lattices
• Seeing Hofstadter's Butterfly in Atomic Fermi Gases

Cited by in corpus (37)

• Light-induced gauge fields for ultracold atoms
• Superfluidity in topologically nontrivial flat bands
• Observation of fractional Chern insulators in a van der Waals heterostructure
• Relations between topology and the quantum metric for Chern insulators
• Fractional Chern Insulators in Harper-Hofstadter Bands with Higher Chern Number
• Dynamical quantum phase transitions: Role of topological nodes in wavefunction overlaps
• Realizing and Adiabatically Preparing Bosonic Integer and Fractional Quantum Hall states in Optical Lattices
• Origin of Model Fractional Chern Insulators in All Topological Ideal Flatbands: Explicit Color-entangled Wavefunction and Exact Density Algebra
• Light-matter interactions in synthetic magnetic fields: Landau-photon polaritons
• Stability of fractional Chern insulators in the effective continuum limit of Harper-Hofstadter bands with Chern number $|C|>1$
• Topological multi-mode waveguide QED
• Geometric quench and nonequilibrium dynamics of fractional quantum Hall states
• Quantum Mechanics with a Momentum-Space Artificial Magnetic Field
• Exact Solutions of Fractional Chern Insulators: Interacting Particles in the Hofstadter Model at Finite Size
• Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model
• Wavefunctions of Symmetry Protected Topological Phases from Conformal Field Theories
• Quantum geometry and stability of the fractional quantum Hall effect in the Hofstadter model
• Bosonic Fractional Quantum Hall States on a Finite Cylinder
• Flat Topological Bands and Eigenstate Criticality in a Quasiperiodic Insulator
• Topologically quantized current in quasiperiodic Thouless pumps
• Fractional Chern insulators with a non-Landau level continuum limit
• Momentum-space Harper-Hofstadter model
• Josephson Coupled Moore-Read States
• Emergent Fermi sea in a system of interacting bosons
• Fractional Chern insulator edges and layer-resolved lattice contacts
• Bulk Localised Transport States in Infinite and Finite Quasicrystals via Magnetic Aperiodicity
• Hofstadter-Hubbard model with opposite magnetic fields: Bardeen-Cooper-Schrieffer pairing and superfluidity in the nearly flat butterfly bands
• Emergent Fracton Hydrodynamics of Ultracold Atoms in Partially Filled Landau Levels
• Finite-wavevector Electromagnetic Response in Lattice Quantum Hall Systems
• Superconductivity in a Topological Lattice Model with Strong Repulsion
• Chern insulator with large Chern numbers. Chiral Majorana fermion liquid
• Light-Matter interactions in Hofstadter lattice with the next-nearest neighbor couplings
• Topological phases of two-component bosons in species-dependent artificial gauge potentials
• Gauge fields induced by curved spacetime
• Topological quantum electrodynamics in synthetic non-Abelian gauge fields
• Critical magnetic flux for Weyl points in the three-dimensional Hofstadter model
• Theory of Hofstadter Superconductors