Fractional Quantum Hall Physics in Topological Flat Bands

Abstract

We present a pedagogical review of the physics of fractional Chern insulators with a particular focus on the connection to the fractional quantum Hall effect. While the latter conventionally arises in semiconductor heterostructures at low temperatures and in high magnetic fields, interacting Chern insulators at fractional band filling may host phases with the same topological properties, but stabilized at the lattice scale, potentially leading to high-temperature topological order. We discuss the construction of topological flat band models, provide a survey of numerical results, and establish the connection between the Chern band and the continuum Landau problem. We then briefly summarize various aspects of Chern band physics that have no natural continuum analogs, before turning to a discussion of possible experimental realizations. We close with a survey of future directions and open problems, as well as a discussion of extensions of these ideas to higher dimensions and to other topological phases.

30 pages, 4 figures; review submitted to topical issue of Comptes Rendus Physique devoted to topological insulators and Dirac matter. Pre-publication version; comments are invited

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Cited by in corpus (305)

• Atomic quantum gases in periodically driven optical lattices
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• Fractional Chern insulators in magic-angle twisted bilayer graphene
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• Nearly Flat Chern Bands in Moiré Superlattices
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• Detangling Flat Bands into Fano Lattices
• Ferromagnetism in narrow bands of moiré superlattices
• Essay: Where Can Quantum Geometry Lead Us?
• Spontaneous fractional Chern insulators in transition metal dichalcogenides Moire superlattices
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• Fractional Chern Insulator in Twisted Bilayer MoTe${}_2$
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• Twisted bilayer graphene in a parallel magnetic field
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• Quantum geometry and stability of the fractional quantum Hall effect in the Hofstadter model
• Origin of flat-band superfluidity on the Mielke checkerboard lattice
• Two-body problem in a multiband lattice and the role of quantum geometry
• Detecting Topological Phases of Microwave Photons in a Circuit Quantum Electrodynamics Lattice
• Characterization of quasiholes in fractional Chern insulators
• Floquet-engineering of nodal rings and nodal spheres and their characterization using the quantum metric
• Probing photon correlations in the dark sites of geometrically frustrated cavity lattices
• Imaging topology of Hofstadter ribbons
• Flat Topological Bands and Eigenstate Criticality in a Quasiperiodic Insulator
• Interaction induced doublons and embedded topological subspace in a complete flat-band system
• Flat bands and long range Coulomb interactions: conducting or insulating?
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• Charge and statistics of lattice quasiholes from density measurements: a Tree Tensor Network study
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• Fractional Excitonic Insulator
• Hidden mechanism for embedding the flat bands of Lieb, kagome, and checkerboard lattices in other structures
• Coupled Atomic Wires in a Synthetic Magnetic Field
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• Detecting Fractional Chern Insulators in Optical Lattices through Quantized Displacement
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• Higher vortexability: zero field realization of higher Landau levels
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• Suppression of non-equilibrium quasiparticle transport in flat band superconductors
• Fractional Chern Insulators in Singular Geometries
• Ultraviolet-Infrared Mixing in Marginal Fermi Liquids
• Superconducting circuit simulator of Bose-Hubbard model with a flat band
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• Designing topology and fractionalization in narrow gap semiconductor films via electrostatic engineering
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• Flat-band ferromagnetism in a topological Hubbard model
• Wave Functions for Fractional Chern Insulators on Disk Geometry
• Enhanced Kohn-Luttinger topological superconductivity in bands with nontrivial geometry
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• Diagnosis of pairing symmetry by vortex and edge spectra in kagome superconductors
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• Real-space probe for lattice quasiholes
• Tuning the topological insulator states of artificial graphene
• Correlations and entanglement in flat band models with variable Chern numbers
• Flat Band Induced Metal-Insulator Transitions for Weak Magnetic Flux and Spin-Orbit Disorder
• Magnetization process and low-temperature thermodynamics of a spin-1/2 Heisenberg octahedral chain
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• Displacement field-controlled fractional Chern insulators and charge density waves in a graphene/hBN moiré superlattice
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• Fractional Chern insulator edges and layer-resolved lattice contacts
• Higher angular momentum band inversions in two dimensions
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• Fermionic non-Abelian fractional Chern insulators from dipolar interactions
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• Bosonic fractional quantum Hall conductance in shaken honeycomb optical lattices without flat bands
• Quantum anomalous Hall crystals in moiré bands with higher Chern number
• Interaction-induced conductance from zero modes in a clean magnetic graphene waveguide
• Dynamics of the Wigner Crystal of Composite Particles
• Two-dimensional topological order of kinetically constrained quantum particles
• Broken Symmetry in Ideal Chern Bands
• Area-law entanglement from quantum geometry
• One-dimensional flat bands and Dirac cones in narrow zigzag dice lattice ribbons
• Spectra of Magnetoroton and Chiral Graviton Modes of Fractional Chern Insulator
• Ferromagnetism in armchair graphene nanoribbon heterostructures
• A Recipe for Flat Bands in Pyrochlore Materials: A Chemist's Perspective
• Response of fermions in Chern bands to spatially local quenches
• Particle-hole asymmetric ferromagnetism and spin textures in the triangular Hubbard-Hofstadter model
• Information geometry of quantum critical submanifolds: relevant, marginal and irrelevant operators
• Light driven magnetic transitions in transition metal dichalcogenide heterobilayers
• Anyon superconductivity and plateau transitions in doped fractional quantum anomalous Hall insulators
• Characterization of fractional Chern insulator quasiparticles in twisted homobilayer MoTe${}_2$
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• Low-energy optical absorption in correlated insulators: Projected sum rules and the role of quantum geometry
• On the robustness of strongly correlated multi-photon states in frustrated driven-dissipative cavity lattices
• Fermionic tensor networks for higher order topological insulators from charge pumping
• Finite-wavevector Electromagnetic Response in Lattice Quantum Hall Systems
• Evolution between quantum Hall and conducting phases: simple models and some results
• Singular flat bands in the modified Haldane-Dice model
• Three-body problem in a multiband Hubbard model
• Spin separation due to an inherent spontaneous symmetry breaking of the fractional topological insulator
• Parafermions in Moiré Minibands
• A Novel Perspective on Ideal Chern Bands with Strong Short-Range Repulsion: Applications to Correlated Metals, Superconductivity, and Topological Order
• A tale of two localizations: coexistence of flat bands and Anderson localization in a photonics-inspired amorphous system
• Isolated nearly flat higher Chern band in monolayer transition metal trihalides
• Noncommutative Geometry and Deformation Quantization in the Quantum Hall Fluids with Inhomogeneous Magnetic Fields
• Fractional quantum anomalous Hall phase for Raman superarray of Rydberg atoms
• Floquet Fractional Chern Insulators and Competing Phases in Twisted Bilayer Graphene
• Electronic States and Anomalous Hall Effect in Strongly Correlated Topological Systems
• Twistronics and moiré superlattice physics in 2D transition metal dichalcogenides
• Interplay of many-body interactions and quasiperiodic disorder in the all-band-flat diamond chain
• Topological states in strongly correlated systems
• Anomalous fractional quantum Hall effect and multi-valued Hamiltonians
• Stability of $(N+1)$-body fermion clusters in multiband Hubbard model
• Fractional quantum Hall states with gapped boundaries in an extreme lattice limit
• Multi-Band Exact Diagonalization and an Iteration Approach to Hunt For Fractional Chern Insulators in Rhombohedral Multilayer Graphene
• Zero-energy excitation in the classical kagome antiferromagnet NaBa${}_2$Mn${}_3$F${}_{11}$
• Constructing vortex functions and basis states of Chern insulators: ideal condition, inequality from index theorem, and coherent-like states on von Neumann lattice
• Simulating chiral spin liquids with projected entangled-pair states
• Quantum mechanics of composite fermions
• Stability of quasi-particle creation and multiband geometry in fractional Chern insulators under magnetic fields
• Interband excitations in the 1D limit of two-band fractional Chern insulators
• Symmetry-based classification of exact flat bands in single and bilayer moiré systems
• Structure of the nearly-degenerate manifold of lattice quasiholes on a torus
• Properties of Laughlin states on fractal lattices
• Quantum-geometric dipole: a topological boost to flavor ferromagnetism in flat bands
• Coexisting Kagome and Heavy Fermion Flat Bands in YbCr${}_6$Ge${}_6$
• Electronic Crystal Phases in the Presence of Non-Uniform Berry Curvature and Tunable Berry Flux: The ${\lambda }_N$-Jellium model
• Adiabatic preparation of fractional Chern insulators from an effective thin-torus limit
• Reordering Fractional Chern Insulators into Stripes of Fractional Charges with Long-Range Interactions
• Crystalline invariants of fractional Chern insulators
• Family of Aperiodic Tilings with Tunable Quantum Geometric Tensor
• Geometric transport signatures of strained multi-Weyl semimetals
• Anomalous magnetic transport and extra quantum oscillation in semi-metallic photon-like fermion gas
• Non-Abelian geometric potentials and spin-orbit coupling for periodically driven systems
• Interaction-driven plateau transition between integer and fractional Chern Insulators
• Boundary Flat Bands with Topological Spin Textures Protected by Sub-chiral Symmetry
• Symmetry-protected flatband condition for Hamiltonians with local symmetry
• Topological Triviality of Flat Hamiltonians
• Higher bracket structure of density operators in Weyl fermion systems and topological insulators
• Impurities and Landau level mixing in a fractional quantum Hall state in a flatband lattice model
• Exciton fractional Chern insulators in moiré heterostructures
• Ideal quantum geometry of the surface states of rhombohedral graphite and its effects on the surface superconductivity
• Unconventional Fractional Phases in Multi-Band Vortexable Systems
• Optical signatures of -1/3 fractional quantum anomalous Hall state in twisted MoTe2
• Universal Performance Gap of Neural Quantum States Applied to the Hofstadter-Bose-Hubbard Model
• Identifying Anyonic Topological Order in Fractional Quantum Anomalous Hall Systems
• Elastic Response and Instabilities of Anomalous Hall Crystals
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• Quantum Magnetism in Wannier-Obstructed Mott Insulators
• Chiral Graviton Modes in Fermionic Fractional Chern Insulators
• Family of multilayer graphene superconductors with tunable chirality: Momentum-space vortices nucleated by a ring of Berry curvature
• Disorder-driven phase transitions in bosonic fractional quantum Hall liquids
• Fractional Wannier Orbitals and Tight-Binding Gauge Fields for Kitaev Honeycomb Superlattices with Flat Majorana Bands
• Reduced Wannier representation for topological bands
• Flat bands on spherical surface: from Landau levels to giant-quantum-number orbitals
• Phase transitions of the dimerized Kane-Mele model with/without the strong interaction
• Topological two-body interaction obstructing trivial ground states: an indicator of fractional Chern insulators
• Flux attachment theory of fractional excitonic insulators
• Gaplessness from disorder and quantum geometry in gapped superconductors
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• Anyon Dispersion in Aharonov-Casher Bands and Implications for Twisted MoTe${}_2$
• On flat bands in the $J_1$-$J_2$-$J_3$ XXZ sawtooth chain
• Topological Order Without Band Topology in Moiré Graphene
• Intervalley Band Crossing and Transition of Fractional Chern Insulators in Floquet Twisted Bilayer MoTe${}_2$
• Generalizing the composite fermion theory for fractional Chern insulators
• Singular flat bands in three dimensions: Landau level spreading, quantum geometry, and Weyl reconstruction
• Abelian and non-Abelian fractionalized states in twisted MoTe${}_2$: A generalized Landau-level theory
• Exceptional flat bands in bipartite non-Hermitian lattices
• Evolution of the optimal trial wave function with interactions in fractional Chern insulators
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• Exciton-Anyon Binding in Fractional Chern Insulators: Spectral Fingerprints
• Higher Chern bands in helical homotrilayer transition metal dichalcogenides
• Theory of Hofstadter Superconductors