Relations between topology and the quantum metric for Chern insulators

Abstract

We investigate relations between topology and the quantum metric of two-dimensional Chern insulators. The quantum metric is the Riemannian metric defined on a parameter space induced from quantum states. Similar to the Berry curvature, the quantum metric provides a geometrical structure associated to quantum states. We consider the volume of the parameter space measured with the quantum metric, which we call the quantum volume of the parameter space. We establish an inequality between the quantum volume of the Brillouin zone and that of the twist-angle space. Exploiting this inequality and the inequality between the Chern number and the quantum volume, we investigate how the quantum volume can be used as a good measure to infer the Chern number. The inequalities are found to be saturated for fermions filling Landau levels. Through various concrete models, we elucidate conditions when the quantum volume gives a good estimate of the topology of the system.

14 pages, 5 figures

References in corpus (13)

• Topological Acoustics
• Nearly-flat bands with nontrivial topology
• Band geometry, Berry curvature and superfluid weight
• Topological flat band models with arbitrary Chern numbers
• Experimental Measurement of the Quantum Metric Tensor and Related Topological Phase Transition with a Superconducting Qubit
• Position-Momentum Duality and Fractional Quantum Hall Effect in Chern Insulators
• Kähler geometry and Chern insulators: Relations between topology and the quantum metric
• Orbital magnetism of coupled bands models
• Perturbative Approach to Flat Chern Bands in the Hofstadter Model
• Steady-state Hall response and quantum geometry of driven-dissipative lattices
• Computing observables without eigenstates: applications to Bloch Hamiltonians
• Chirality-driven edge flow and non-Hermitian topology in active nematic cells
• Euler characteristic number of the energy band and the reason for its non-integer values

Cited by in corpus (101)

• Exact Landau Level Description of Geometry and Interaction in a Flatband
• Vortexability: A Unifying Criterion for Ideal Fractional Chern Insulators
• 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
• Kähler geometry and Chern insulators: Relations between topology and the quantum metric
• Anomalous Hall Crystals in Rhombohedral Multilayer Graphene I: Interaction-Driven Chern Bands and Fractional Quantum Hall States at Zero Magnetic Field
• Hierarchy of Ideal Flatbands in Chiral Twisted Multilayer Graphene Models
• Pressure--enhanced fractional Chern insulators in moiré transition metal dichalcogenides along a magic line
• Origin of Model Fractional Chern Insulators in All Topological Ideal Flatbands: Explicit Color-entangled Wavefunction and Exact Density Algebra
• Bound on resistivity in flat-band materials due to the quantum metric
• Engineering geometrically flat Chern bands with Fubini-Study Kähler structure
• The quantum geometric origin of capacitance in insulators
• Non-Abelian fractionalization in topological minibands
• Exact Many-Body Ground States from Decomposition of Ideal Higher Chern Bands: Applications to Chirally Twisted Graphene Multilayers
• Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers
• Non-Abelian Fractional Quantum Anomalous Hall States and First Landau Level Physics in Second Moiré Band of Twisted Bilayer MoTe2
• Nontrivial quantum geometry of degenerate flat bands
• Extracting the Quantum Geometric Tensor of an Optical Raman Lattice by Bloch State Tomography
• Higher Landau-Level Analogs and Signatures of Non-Abelian States in Twisted Bilayer MoTe${}_2$
• Adiabatic Approximation and Aharonov-Casher Bands in Twisted Homobilayer TMDs
• Recent Developments in Fractional Chern Insulators
• Quantum geometric effect on Fulde-Ferrell-Larkin-Ovchinnikov superconductivity
• Quantum weight: A fundamental property of quantum many-body systems
• 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${}_2$
• Beyond the Berry Phase: Extrinsic Geometry of Quantum States
• Higher vortexability: zero field realization of higher Landau levels
• From Classical to Quantum Information Geometry: A Guide for Physicists
• Self-consistent theory for the fractional quantum anomalous Hall effect in rhombohedral pentalayer graphene
• Experimental demonstration of topological bounds in quantum metrology
• Nested-sphere description of the N-level Chern number and the generalized Bloch hypersphere
• Interplay of Band Geometry and Topology in Ideal Chern Insulators in Presence of External Electromagnetic Fields
• Topological properties of a non-Hermitian quasi-1D chain with a flat band
• Geometric superfluid weight of composite bands in multiorbital superconductors
• Analytic approach to quantum metric and optical conductivity in Dirac models with parabolic mass in arbitrary dimensions
• Unsupervised learning of topological phase diagram using topological data analysis
• Flat-band ratio and quantum metric in the superconductivity of modified Lieb lattices
• Quantum metric on the Brillouin Zone in correlated electron systems and its relation to topology for Chern insulators
• Quantum optics measurement scheme for quantum geometry and topological invariants
• Drude weight and the many-body quantum metric in one-dimensional Bose systems
• Pseudospin-orbit coupling and non-Hermitian effects in the Quantum Geometric Tensor of a plasmonic lattice
• Non-Abelian quantum geometric tensor in degenerate topological semimetals
• Nonlocal Moments in the Chern Bands of Twisted Bilayer Graphene
• Emergent Haldane Model and Photon-Valley Locking in Chiral Cavities
• Quantum Geometry and the Hidden Scales in Materials
• Quantum geometrical properties of topological materials
• Quantum-metric-induced quantum Hall conductance inversion and reentrant transition in fractional Chern insulators
• Scaling of the Integrated Quantum Metric in Disordered Topological Phases
• Revisiting the magnetic responses of bilayer graphene from the perspective of the quantum distance
• Polaron spectra and edge singularities for correlated flat bands
• Quantum geometric bounds for observables: Linear responses, Drude weight, and orbital magnetization
• Quantum multiparameter estimation enhanced by a topological phase transition
• Uniqueness of Landau levels and their analogs with higher Chern numbers
• Sj$\ddot{\text{o}}$qvist quantum geometric tensor of finite-temperature mixed states
• Electric field induced second-order anomalous Hall transport in unconventional Rashba systems
• Broken Symmetry in Ideal Chern Bands
• Inducing topological flat bands in bilayer graphene with electric and magnetic superlattices
• Anyon superconductivity and plateau transitions in doped fractional quantum anomalous Hall insulators
• Flat Bands and High Chern Numbers in Twisted Multilayer Graphene
• Information geometry of quantum critical submanifolds: relevant, marginal and irrelevant operators
• Quantum geometry and $X$-wave magnets with $X=p,d,f,g,i$
• Fubini-Study metric and topological properties of flat band electronic states: the case of an atomic chain with $s-p$ orbitals
• Geometric bound on structure factor
• Corner Charge Fluctuations and Many-Body Quantum Geometry
• Floquet Fractional Chern Insulators and Competing Phases in Twisted Bilayer Graphene
• 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
• Functional approach to superfluid stiffness: Role of quantum geometry in unconventional superconductivity
• Adiabatic charge transport in extended SSH models
• Ideal topological flat bands in chiral symmetric moiré systems from non-holomorphic functions
• Symmetry-based classification of exact flat bands in single and bilayer moiré systems
• Geometric semimetals and their simulation in synthetic matter
• Stability of quasi-particle creation and multiband geometry in fractional Chern insulators under magnetic fields
• Identifying Instabilities with Quantum Geometry in Flat Band Systems
• Coexisting Kagome and Heavy Fermion Flat Bands in YbCr${}_6$Ge${}_6$
• From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory
• Characterizing the Many Body Localization Crossover as a Metal-Insulator Transition: Localization length from Polarization and Quantum Metric
• Ideal quantum geometry of the surface states of rhombohedral graphite and its effects on the surface superconductivity
• Superfluid stiffness of superconductors with delicate topology
• Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric
• Quantum phase diagram and non-abelian Moore-Read state in double twisted bilayer graphene
• Precise quantum-geometric electronic properties from first principles
• A Brillouin torus decomposition for two-dimensional topological insulators
• Gradient flow and Bogomolny bounds for quantum metric actions
• Electronic bounds in magnetic crystals
• Instability of Laughlin FQH liquids into gapless power-law correlated states with continuous exponents in ideal Chern bands: rigorous results from plasma mapping
• Topologically nontrivial flat bands and quantum Hall crossovers in square-octagon lattice materials
• Relaxation toward an Ideal Chern Band through Coupling to a Markovian Bath
• Refining Heuristic Predictors of Fractional Chern Insulators using Machine Learning
• Orbital Description of Landau Levels
• Distinct topological excitonic insulators characterized by quantum geometry
• Quantum-geometric thermal conductivity of superconductors
• Quantum geometric tensor in systems with fractional band dispersion
• Quantum geometry and low-frequency optical conductivity of nodal planes
• Abelian and non-Abelian fractionalized states in twisted MoTe${}_2$: A generalized Landau-level theory
• Pressure-Tunable Generalized Wigner Crystal and Fractional Chern Insulator in twisted MoTe${}_2$
• Evolution of quantum geometric tensor of 1D periodic systems after a quench
• Chern insulators in two and three dimensions: A global perspective
• Exploring Many-Body Quantum Geometry Beyond the Quantum Metric with Correlation Functions: A Time-Dependent Perspective
• Exact Parent Hamiltonians for All Landau Level States in a Half-flux Lattice