Topological Flat Band Models and Fractional Chern Insulators

Abstract

Topological insulators and their intriguing edge states can be understood in a single-particle picture and can as such be exhaustively classified. Interactions significantly complicate this picture and can lead to entirely new insulating phases, with an altogether much richer and less explored phenomenology. Most saliently, lattice generalizations of fractional quantum Hall states, dubbed fractional Chern insulators, have recently been predicted to be stabilized by interactions within nearly dispersionless bands with non-zero Chern number, $C$. Contrary to their continuum analogues, these states do not require an external magnetic field and may potentially persist even at room temperature, which make these systems very attractive for possible applications such as topological quantum computation. This review recapitulates the basics of tight-binding models hosting nearly flat bands with non-trivial topology, $C\ne 0$, and summarizes the present understanding of interactions and strongly correlated phases within these bands. Emphasis is made on microscopic models, highlighting the analogy with continuum Landau level physics, as well as qualitatively new, lattice specific, aspects including Berry curvature fluctuations, competing instabilities as well as novel collective states of matter emerging in bands with $|C|>1$. Possible experimental realizations, including oxide interfaces and cold atom implementations as well as generalizations to flat bands characterized by other topological invariants are also discussed.

Invited review. 46 pages, many illustrations and references. V2: final version with minor improvements and added references

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Cited by in corpus (343)

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• Chiral groundstate currents of interacting photons in a synthetic magnetic field
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• Interacting topological insulators: a review
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• Dynamic optical lattices of sub-wavelength spacing for ultracold atoms
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• Non-Hermitian Mott Skin Effect
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• Non-Abelian Fractional Chern Insulators from Long-Range Interactions
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• Exceptional band touching for strongly correlated systems in equilibrium
• Bosonic integer quantum Hall effect in optical flux lattices
• Fractional Chern insulators with strong interactions far exceeding bandgaps
• Creating anomalous Floquet Chern insulators with magnetic quantum walks
• Transfer learning relaxation, electronic structure and continuum model for twisted bilayer MoTe${}_2$
• Geometric quench and nonequilibrium dynamics of fractional quantum Hall states
• Phase transitions out of quantum Hall states in moiré materials
• ${\mathbb{Z}}_2$ fractional topological insulators in two dimensions
• Dissipation-induced topological insulators: A no-go theorem and a recipe
• Haldane Statistics for Fractional Chern Insulators with an Arbitrary Chern number
• Model Fractional Chern Insulators
• Anomalous Hall Crystals in Rhombohedral Multilayer Graphene II: General Mechanism and a Minimal Model
• Parent Berry curvature and the ideal anomalous Hall crystal
• Multiple Chern bands in twisted MoTe${}_2$ and possible non-Abelian states
• The single-mode approximation for fractional Chern insulators and the fractional quantum Hall effect on the torus
• Interaction-induced lattices for bound states: Designing flat bands, quantized pumps and higher-order topological insulators for doublons
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• Anyons and Fractional Quantum Hall Effect in Fractal Dimensions
• Exact Solutions of Fractional Chern Insulators: Interacting Particles in the Hofstadter Model at Finite Size
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• Three-level Haldane-like model on dice optical lattice
• Circuit Quantum Electrodynamics Simulator of Flat Band Physics in Lieb lattice
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• Flatband Line States in Photonic Super-Honeycomb Lattices
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• Compact Discrete Breathers on Flat Band Networks
• Flat-band-based multifractality in the all-band-flat diamond chain
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• Fractional Topological Phases in Generalized Hofstadter Bands with Arbitrary Chern Numbers
• Fractional Excitonic Insulator
• Fractional Chern Insulator phase at the transition between checkerboard and Lieb lattice
• Evolution of Floquet Topological Quantum States in Driven Semiconductors
• Hidden mechanism for embedding the flat bands of Lieb, kagome, and checkerboard lattices in other structures
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• Two-component quantum Hall effects in topological flat bands
• Topology and localization of a periodically driven Kitaev model
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• Coupled Atomic Wires in a Synthetic Magnetic Field
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• Interacting bosons in topological optical flux lattices
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• Localized modes in nonlinear fractional systems with deep lattices
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• Correlations and entanglement in flat band models with variable Chern numbers
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• Strongly interacting photons in 2D waveguide QED
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• Truncation of lattice fractional quantum Hall Hamiltonians derived from conformal field theory
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• Non-Abelian Fractional Chern Insulators and Competing States in Flat Moiré Bands
• Microscopic Diagnosis of Universal Geometric Responses in Fractional Quantum Hall Liquids
• Topological insulators with arbitrarily tunable entanglement
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• Exact Topological Flat Bands from Continuum Landau Levels
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• Chains of nanoparticles for flat-band emission and lasing
• Disorder effects in two-dimensional flat-band system with next-nearest-neighbor hopping