Entanglement spectrum and Wannier center flow of the Hofstadter problem

Abstract

We examine the quantum entanglement spectra and Wannier functions of the square lattice Hofstadter model. Consistent with previous work on entanglement spectra of topological band structures, we find that the entanglement levels exhibit a spectral flow similar to that of the full system's energy spectrum. While the energy spectra are continuous, with open boundary conditions the entanglement spectra exhibit discontinuities associated with the passage of an energy edge state through the Fermi level. We show how the entanglement spectrum can be understood by examining the band projectors of the full system and their behavior under adiabatic pumping. In so doing we make connections with the original TKNN work on topological two-dimensional band structures and their Chern numbers. Finally we consider Wannier states and their adiabatic flows, and draw connections to the entanglement properties.

14 + 4 pages, 12 figures. Introductory material expanded. Figures explained in more detail. New appendix added. Minor typographical errors corrected. Published version

References in corpus (6)

• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Quantum critical scaling of the geometric tensors
• Wannier representation of Z_2 topological insulators
• Exponential localization of Wannier functions in insulators
• A General Theorem Relating the Bulk Topological Number to Edge States in Two-dimensional Insulators
• Trace Index and Spectral Flow in the Entanglement Spectrum

Cited by in corpus (41)

• Hourglass Fermions
• Topological Insulators from Group Cohomology
• Topological indices for open and thermal systems via Uhlmann's phase
• Dynamical quantum phase transitions: Role of topological nodes in wavefunction overlaps
• Berry-phase description of Topological Crystalline Insulators
• Entanglement Entropy Scaling Laws and Eigenstate Typicality in Free Fermion Systems
• Direct measurement of topological invariants in optical lattices
• Crystallographic splitting theorem for band representations and fragile topological photonic crystals
• Electromagnetic response of quantum Hall systems in dimensions five and six and beyond
• Free-fermion Entanglement Spectrum through Wannier Interpolation
• Free-Fermion entanglement and orthogonal polynomials
• Perturbative Approach to Flat Chern Bands in the Hofstadter Model
• Entanglement polarization for the topological quadrupole phase
• Imaging topology of Hofstadter ribbons
• Relating the entanglement spectrum of noninteracting band insulators to their quantum geometry and topology
• Position momentum Duality in the Entanglement Spectrum of Free Fermions
• Entanglement-spectrum characterization of ground-state nonanalyticities in coupled excitation-phonon models
• Measuring Chern numbers in Hofstadter strips
• Entanglement Chern number for an extensive partition of a topological ground state
• Entanglement Spectrum and Entanglement Hamiltonian of a Chern insulator with open boundaries
• Entanglement in Fermionic Chains and Bispectrality
• Entanglement Spectra and Entanglement Thermodynamics of Hofstadter Bilayers
• Entangled multi-component 4D quantum Hall states from photonic crystal defects
• Entanglement in a second order topological insulator on a square lattice
• Quantum entanglement and non-Hermiticity in free-fermion systems
• Entanglement spectrum and entropy in Floquet topological matter
• Disentangled topological numbers by a purification of entangled mixed states for non-interacting fermion systems
• Entanglement entropy of inhomogeneous XX spin chains with algebraic interactions
• Entanglement of Free Fermions on Hadamard Graphs
• On the relation of the entanglement spectrum to the bulk polarization
• Entanglement spectrum of fermionic bilayer honeycomb lattice: Hofstadter butterfly
• Random Fractal Ansatz for the configurations of Two-Dimensional Critical Systems
• BaHgSn: A Dirac semimetal with surface hourglass fermions
• Invariance of topological indices under Hilbert space truncation
• Entanglement entropy and entanglement spectrum of Bi${}_{1-x}$Sb${}_x$ (111) bilayers
• Manifestations of topological band crossings in bulk entanglement spectrum: An analytical study for integer quantum Hall states
• Interacting fermions in two dimension in simultaneous presence of disorder and magnetic field
• Refined Characterization of Lattice Chern Insulators by Bulk Entanglement Spectrum
• Dynamical Geometry of the Haldane Model under a Quantum Quench
• Exact Mappings in Condensed Matter Physics
• Glide-resolved photoemission spectroscopy: Measuring topological invariants in nonsymmorphic space groups