Position-Momentum Duality and Fractional Quantum Hall Effect in Chern Insulators

Abstract

We develop a first quantization description of fractional Chern insulators that is the dual of the conventional fractional quantum Hall (FQH) problem, with the roles of position and momentum interchanged. In this picture, FQH states are described by anisotropic FQH liquids forming in momentum-space Landau levels in a fluctuating magnetic field. The fundamental quantum geometry of the problem emerges from the interplay of single-body and interaction metrics, both of which act as momentum-space duals of the geometrical picture of the anisotropic FQH effect. We then present a novel broad class of ideal Chern insulator lattice models that act as duals of the isotropic FQH effect. The interacting problem is well-captured by Haldane pseudopotentials and affords a detailed microscopic understanding of the interplay of interactions and non-trivial quantum geometry.

arXiv comment10 pages, 2 figures. Accepted version

References in corpus (58)

• Topological Insulators
• Topological insulators and superconductors
• High temperature fractional quantum Hall states
• Fractional quantum Hall states at zero magnetic field
• Nearly-flat bands with nontrivial topology
• Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level
• Fractional quantum Hall effect in the absence of Landau levels
• Fractional Chern Insulator
• Topological Flat Band Models and Fractional Chern Insulators
• Interface engineering of quantum Hall effects in digital transition metal oxide heterostructures
• Geometrical Description of the Fractional Quantum Hall Effect
• Time-evolving a matrix product state with long-ranged interactions
• Fractional Quantum Hall Effect of Hard-Core Bosons in Topological Flat Bands
• Nearly flat band with Chern number C=2 on the dice lattice
• Band geometry of fractional topological insulators
• Floquet Fractional Chern Insulator in Doped Graphene
• Flat Chern Band in a Two-Dimensional Organometallic Framework
• Topological Nematic States and Non-Abelian Lattice Dislocations
• Generic Wavefunction Description of Fractional Quantum Anomalous Hall States and Fractional Topological Insulators
• Fractional Chern Insulators in Topological Flat bands with Higher Chern Number
• Zoology of Fractional Chern Insulators
• Topological flat band models with arbitrary Chern numbers
• New Class of Non-Abelian Spin-Singlet Quantum Hall States
• Emergent Many-Body Translational Symmetries of Abelian and Non-Abelian Fractionally Filled Topological Insulators
• Fractional Quantum Hall Effect in Topological Flat Bands with Chern Number Two
• Fractional Chern Insulators and the W-Infinity Algebra
• Bloch Model Wavefunctions and Pseudopotentials for All Fractional Chern Insulators
• Topological insulators and fractional quantum Hall effect on the ruby lattice
• Non-Abelian Quantum Hall Effect in Topological Flat Bands
• Series of Abelian and Non-Abelian States in C>1 Fractional Chern Insulators
• Flat bands with higher Chern number in pyrochlore slabs
• Band mass anisotropy and the intrinsic metric of fractional quantum Hall systems
• Hierarchy of fractional Chern insulators and competing compressible states
• Gauge-Fixed Wannier Wave-Functions for Fractional Topological Insulators
• Model anisotropic quantum Hall states
• Fractional quantum-Hall liquid spontaneously generated by strongly correlated t_2g electrons
• Adiabatic continuation of Fractional Chern Insulators to Fractional Quantum Hall States
• Enhancing the stability of a fractional Chern insulator against competing phases
• Pseudopotential Formalism for Fractional Chern Insulators
• Fractional Chern Insulators beyond Laughlin states
• Adiabatic continuity between Hofstadter and Chern insulator states
• From Fractional Chern Insulators to a Fractional Quantum Spin Hall Effect
• Hamiltonian theory of fractionally filled Chern bands
• Non-Abelian Fractional Chern Insulators from Long-Range Interactions
• Fractional Chern insulator on a triangular lattice of strongly correlated $t_{2g}$ electrons
• Fractional Chern insulators with strong interactions far exceeding bandgaps
• Lattice construction of pseudopotential Hamiltonians for Fractional Chern Insulators
• From fractional Chern insulators to Abelian and non-Abelian fractional quantum Hall states: adiabatic continuity and orbital entanglement spectrum
• Haldane Statistics for Fractional Chern Insulators with an Arbitrary Chern number
• The single-mode approximation for fractional Chern insulators and the fractional quantum Hall effect on the torus
• Designing Topological Bands in Reciprocal Space
• Repulsive Interactions in Quantum Hall Systems as a Pairing Problem
• Exact Solutions of Fractional Chern Insulators: Interacting Particles in the Hofstadter Model at Finite Size
• Bulk-edge correspondence in fractional Chern insulators
• D-Algebra Structure of Topological Insulators
• Edge Excitations in Fractional Chern Insulators
• Momentum-space instantons and maximally localized flat-band topological Hamiltonians
• Crystal-symmetry preserving Wannier states for fractional chern insulators

Cited by in corpus (112)

• Topological Photonics
• Topological Photonics
• Topological quantum matter in synthetic dimensions
• Fractional Chern Insulator States in Twisted Bilayer Graphene: An Analytical Approach
• Exact Landau Level Description of Geometry and Interaction in a Flatband
• Fractional quantum anomalous Hall states in twisted bilayer MoTe${}_2$ and WSe${}_2$
• Relations between topology and the quantum metric for Chern insulators
• Vortexability: A Unifying Criterion for Ideal Fractional Chern Insulators
• Extracting the quantum metric tensor through periodic driving
• Composite Fermi Liquid at Zero Magnetic Field in Twisted MoTe${}_2$
• Family of ideal Chern flat bands with arbitrary Chern number in chiral twisted graphene multilayers
• Anomalous Hall Crystals in Rhombohedral Multilayer Graphene I: Interaction-Driven Chern Bands and Fractional Quantum Hall States at Zero Magnetic Field
• Band structure engineering of ideal fractional Chern insulators
• Kähler geometry and Chern insulators: Relations between topology and the quantum metric
• Hierarchy of Ideal Flatbands in Chiral Twisted Multilayer Graphene Models
• Maximally Localized Wannier Orbitals, Interaction Models and Fractional Quantum Anomalous Hall Effect in Twisted Bilayer MoTe2
• Pressure--enhanced fractional Chern insulators in moiré transition metal dichalcogenides along a magic line
• Fundamental bound on topological gap
• Origin of Model Fractional Chern Insulators in All Topological Ideal Flatbands: Explicit Color-entangled Wavefunction and Exact Density Algebra
• Berry Curvature and Quantum Metric in $N$-band systems -- an Eigenprojector Approach
• Strong Coupling Theory of Magic-Angle Graphene: A Pedagogical Introduction
• Engineering geometrically flat Chern bands with Fubini-Study Kähler structure
• Quantum Metric Induced Phases in Moiré Materials
• Electromagnetic response of quantum Hall systems in dimensions five and six and beyond
• Floquet Mechanism for Non-Abelian Fractional Quantum Hall States
• Exact Many-Body Ground States from Decomposition of Ideal Higher Chern Bands: Applications to Chirally Twisted Graphene Multilayers
• Ideal Chern bands are Landau levels in curved space
• Stability of fractional Chern insulators in the effective continuum limit of Harper-Hofstadter bands with Chern number $|C|>1$
• Theory of fractional Chern insulator states in pentalayer graphene moiré superlattice
• Topological Lattice Models with Constant Berry Curvature
• Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers
• Generalized Pseudopotentials for the Anisotropic Fractional Quantum Hall Effect
• Transfer learning relaxation, electronic structure and continuum model for twisted bilayer MoTe${}_2$
• Band flatness optimization through complex analysis
• Model Fractional Chern Insulators
• Parent Berry curvature and the ideal anomalous Hall crystal
• Steady-state Hall response and quantum geometry of driven-dissipative lattices
• Tensor Berry connections and their topological invariants
• Adiabatic Approximation and Aharonov-Casher Bands in Twisted Homobilayer TMDs
• Geometric construction of Quantum Hall clustering Hamiltonians
• Ultra-Strong Spin-Orbit Coupling and Topological Moiré Engineering in Twisted ZrS2 Bilayers
• Quantum metric contribution to the pair mass in spin-orbit coupled Fermi superfluids
• Recent Developments in Fractional Chern Insulators
• Flat Topological Bands and Eigenstate Criticality in a Quasiperiodic Insulator
• Dual Haldane sphere and quantized band geometry in chiral multifold fermions
• Topological dynamics of gyroscopic and Floquet lattices from Newton's laws
• Origin of band flatness and constraints of higher Chern numbers
• Fractional Chern insulators with a non-Landau level continuum limit
• Theory of Generalized Landau Levels and Implication for non-Abelian States
• Variational Mapping of Chern Bands to Landau Levels: Application to Fractional Chern Insulators in Twisted MoTe${}_2$
• Revealing Chern number from quantum metric
• Higher vortexability: zero field realization of higher Landau levels
• Interacting topological Dirac magnons
• "Quantum Geometric Nesting" and Solvable Model Flat-Band Systems
• Self-consistent theory for the fractional quantum anomalous Hall effect in rhombohedral pentalayer graphene
• Fractional Chern Insulators in Singular Geometries
• Interplay of Band Geometry and Topology in Ideal Chern Insulators in Presence of External Electromagnetic Fields
• Extracting non-Abelian quantum metric tensor and its related Chern numbers
• Wave Functions for Fractional Chern Insulators on Disk Geometry
• Designing topology and fractionalization in narrow gap semiconductor films via electrostatic engineering
• Geometric mass acquisition via quantum metric: an effective band mass theorem for the helicity bands
• Time-Reversal Invariant Topological Moiré Flatband: A Platform for the Fractional Quantum Spin Hall Effect
• Competing orders in the Hofstadter t-J model
• Hyperbolic Fractional Chern insulators
• Corner Charge Fluctuation as an Observable for Quantum Geometry and Entanglement in Two-dimensional Insulators
• Nonlocal Moments in the Chern Bands of Twisted Bilayer Graphene
• Band geometry from position-momentum duality at topological band crossings
• Quantum Metric Nonlinear Spin-Orbit Torque Enhanced by Topological Bands
• Quantum anomalous Hall crystals in moiré bands with higher Chern number
• Quantum geometric bounds for observables: Linear responses, Drude weight, and orbital magnetization
• Non-Abelian Fractional Chern Insulator in Disk Geometry
• Quantum Hall effect in momentum space
• Broken Symmetry in Ideal Chern Bands
• Momentum-space Landau levels in driven-dissipative cavity arrays
• Programmable Hamiltonian engineering with quadratic quantum Fourier transform
• Particle-hole asymmetric ferromagnetism and spin textures in the triangular Hubbard-Hofstadter model
• Universal Wilson Loop Bound of Quantum Geometry
• Corner Charge Fluctuations and Many-Body Quantum Geometry
• Flat Bands and High Chern Numbers in Twisted Multilayer Graphene
• Information geometry of quantum critical submanifolds: relevant, marginal and irrelevant operators
• Geometric bound on structure factor
• Quantum response theory and momentum-space gravity
• Anyon superconductivity and plateau transitions in doped fractional quantum anomalous Hall insulators
• Chiral Wigner crystal phases induced by Berry curvature
• Anomalous fractional quantum Hall effect and multi-valued Hamiltonians
• Fractional Chern insulator states in an isolated flat band of zero Chern number
• Parafermions in Moiré Minibands
• Noncommutative Geometry and Deformation Quantization in the Quantum Hall Fluids with Inhomogeneous Magnetic Fields
• Mapping a fractional quantum Hall state to a fractional Chern insulator
• Isolated nearly flat higher Chern band in monolayer transition metal trihalides
• Constructing vortex functions and basis states of Chern insulators: ideal condition, inequality from index theorem, and coherent-like states on von Neumann lattice
• Ideal topological flat bands in chiral symmetric moiré systems from non-holomorphic functions
• Coexisting Kagome and Heavy Fermion Flat Bands in YbCr${}_6$Ge${}_6$
• Quantum-geometric dipole: a topological boost to flavor ferromagnetism in flat bands
• Family of Aperiodic Tilings with Tunable Quantum Geometric Tensor
• Symmetry-based classification of exact flat bands in single and bilayer moiré systems
• Ideal quantum geometry of the surface states of rhombohedral graphite and its effects on the surface superconductivity
• Quantum geometric tensors from sub-bundle geometry
• Normal state quantum geometry, non-locality and superconductivity
• How quantum fluctuations freeze a classical liquid and then melt it into a topological one
• Chiral spin liquid from magnetic Wannier states
• Elastic Response and Instabilities of Anomalous Hall Crystals
• Instability of Laughlin FQH liquids into gapless power-law correlated states with continuous exponents in ideal Chern bands: rigorous results from plasma mapping
• Topological two-body interaction obstructing trivial ground states: an indicator of fractional Chern insulators
• Exact Mappings in Condensed Matter Physics
• Exact Parent Hamiltonians for All Landau Level States in a Half-flux Lattice
• Probing the Quantum Geometry of Correlated Metals using Optical Conductivity
• Superfluid weight in disordered flat-band superconductors as a competition between localization functionals
• Orbital Description of Landau Levels
• Relaxation toward an Ideal Chern Band through Coupling to a Markovian Bath
• Abelian and non-Abelian fractionalized states in twisted MoTe${}_2$: A generalized Landau-level theory
• Flux attachment theory of fractional excitonic insulators