Quantum geometry in condensed matter

Abstract

One of the most celebrated accomplishments of modern physics is the description of fundamental principles of nature in the language of geometry. As the motion of celestial bodies is governed by the geometry of spacetime, the motion of electrons in condensed matter can be characterized by the geometry of the Hilbert space of their wave functions. Such quantum geometry, comprising of Berry curvature and quantum metric, can thus exert profound influences on various properties of materials. The dipoles of both Berry curvature and quantum metric produce nonlinear transport. The quantum metric plays an important role in flat-band superconductors by enhancing the transition temperature. The uniformly distributed momentum-space quantum geometry stabilizes the fractional Chern insulators and results in the fractional quantum anomalous Hall effect. We here review in detail quantum geometry in condensed matter, paying close attention to its effects on nonlinear transport, superconductivity, and topological properties. Possible future research directions in this field are also envisaged.

Review article accepted by National Science Review, unproofread; 17 pages, 8 figures, 1 table

References in corpus (78)

• Magic-angle graphene superlattices: a new platform for unconventional superconductivity
• Berry Phase Effects on Electronic Properties
• Moire bands in twisted double-layer graphene
• Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene
• Intrinsic quantized anomalous Hall effect in a moiré heterostructure
• High temperature fractional quantum Hall states
• Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials
• Fractional quantum Hall states at zero magnetic field
• Nearly-flat bands with nontrivial topology
• Tunable Phase Boundaries and Ultra-Strong Coupling Superconductivity in Mirror Symmetric Magic-Angle Trilayer Graphene
• Origin of Magic Angles in Twisted Bilayer Graphene
• Observation of the nonlinear Hall effect under time reversal symmetric conditions
• Signatures of Fractional Quantum Anomalous Hall States in Twisted MoTe2 Bilayer
• Observation of Fractionally Quantized Anomalous Hall Effect
• Superfluidity in topologically nontrivial flat bands
• Fractional Chern insulators in magic-angle twisted bilayer graphene
• Maximally-localized Wannier orbitals and the extended Hubbard model for the twisted bilayer graphene
• Integer and fractional Chern insulators in twisted bilayer MoTe2
• Topological Flat Band Models and Fractional Chern Insulators
• Strongly Correlated Chern Insulators in Magic-Angle Twisted Bilayer Graphene
• Field Induced Positional Shift of Bloch Electrons and its Dynamical Implications
• Nonlinear Hall Effects
• Geometric origin of superfluidity in the Lieb lattice flat band
• Superfluidity and Quantum Geometry in Twisted Multilayer Systems
• Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure
• The Insulating State of Matter: A Geometrical Theory
• Band geometry of fractional topological insulators
• Observation of fractional Chern insulators in a van der Waals heterostructure
• Quantum metric-induced nonlinear transport in a topological antiferromagnet
• Band geometry, Berry curvature and superfluid weight
• Topology-Bounded Superfluid Weight In Twisted Bilayer Graphene
• Fractional Chern Insulator States in Twisted Bilayer Graphene: An Analytical Approach
• Disorder-induced nonlinear Hall effect with time-reversal symmetry
• Unconventional Superconductivity
• Intrinsic nonlinear Hall effect in antiferromagnetic tetragonal CuMnAs
• Room temperature nonlinear Hall effect and wireless RF rectification in Weyl semimetal TaIrTe4
• Band signatures for strong nonlinear Hall effect in bilayer WTe${}_2$
• Flavour Hund's Coupling, Correlated Chern Gaps, and Diffusivity in Moiré Flat Bands
• Exact Landau Level Description of Geometry and Interaction in a Flatband
• Intrinsic Second-Order Anomalous Hall Effect and Its Application in Compensated Antiferromagnets
• Superfluid weight and Berezinskii-Kosterlitz-Thouless transition temperature of twisted bilayer graphene
• Geometric and conventional contribution to superfluid weight in twisted bilayer graphene
• Topology and geometry under the nonlinear electromagnetic spotlight
• Particle-Hole Duality, Emergent Fermi Liquids and Fractional Chern Insulators in Moiré Flatbands
• Essay: Where Can Quantum Geometry Lead Us?
• Spontaneous fractional Chern insulators in transition metal dichalcogenides Moire superlattices
• High-temperature topological superconductivity in twisted double layer copper oxides
• Anomalous Hall metal and fractional Chern insulator in twisted transition metal dichalcogenides
• Fractional Chern Insulators and the W-Infinity Algebra
• Unification of Nonlinear Anomalous Hall Effect and Nonreciprocal Magnetoresistance in Metals by the Quantum Geometry
• Programming Correlated Magnetic States via Gate Controlled Moiré Geometry
• Quantum metric and effective mass of a two-body bound state in a flat band
• Quantum geometry induced nonlinear transport in altermagnets
• Giant spontaneous Hall effect in a nonmagnetic Weyl-Kondo semimetal
• Composite Fermi Liquid at Zero Magnetic Field in Twisted MoTe${}_2$
• Interplay of Fractional Chern Insulator and Charge-Density-Wave Phases in Twisted Bilayer Graphene
• Giant $c$-axis nonlinear anomalous Hall effect in T${}_d$-MoTe${}_2$ and WTe${}_2$
• Geometric stability of topological lattice phases
• Giant nonlinear Hall effect in twisted WSe${}_2$
• Zero-field composite Fermi liquid in twisted semiconductor bilayers
• Berry connection polarizability tensor and third-order Hall effect
• Pressure--enhanced fractional Chern insulators in moiré transition metal dichalcogenides along a magic line
• Non-linear Hall effect in three-dimensional Weyl and Dirac semimetals
• Berry Curvature and Quantum Metric in $N$-band systems -- an Eigenprojector Approach
• Orbital Magneto-Nonlinear Anomalous Hall Effect in Kagome Magnet Fe${}_3$Sn${}_2$
• Superconductivity in Twisted Double Bilayer Graphene Stabilized by WSe${}_2$
• Orbital Origin of Intrinsic Planar Hall Effect
• Fractional Chern insulator on a triangular lattice of strongly correlated $t_{2g}$ electrons
• Semiclassical equations of motion for disordered conductors: extrinsic interband velocity, corrected collision integral and spin-orbit torques
• Recent Developments in Fractional Chern Insulators
• Third-order intrinsic anomalous Hall effect with generalized semiclassical theory
• Disorder in the non-linear anomalous Hall effect of $\mathcal{P}\mathcal{T}$-symmetric Dirac fermions
• Non-linear ballistic response of quantum spin-Hall edge states
• Quantum kinetic theory of quadratic responses
• Topological nature of the proper spin current and the spin-Hall torque
• Quantum geometry beyond projective single bands
• Quantum-geometry-induced anomalous Hall effect in nonunitary superconductors and application to Sr${}_2$RuO${}_4$
• Fractional quantum Hall states with gapped boundaries in an extreme lattice limit

Cited by in corpus (41)

• Revealing Quantum Geometry in Nonlinear Quantum Materials
• Quantum kinetic theory of quadratic responses
• Planar Hall Plateau in Magnetic Weyl Semimetals
• Scaling of the Integrated Quantum Metric in Disordered Topological Phases
• TensorSymmetry: a package to get symmetry-adapted tensors disentangling spin-orbit coupling effect and establishing analytical relationship with magnetic order
• Quantum geometric superfluid weight in multiband superconductors: A microscopic interpretation
• A Clarification on Quantum-Metric-Induced Nonlinear Transport
• Intrinsic nonlinear Nernst and Seebeck effect
• Quantum geometry and $X$-wave magnets with $X=p,d,f,g,i$
• Quantum response theory and momentum-space gravity
• Nonlinear Hall effects with an exceptional ring
• Quantum geometry and the electric magnetochiral anisotropy in noncentrosymmetric polar media
• Family of Aperiodic Tilings with Tunable Quantum Geometric Tensor
• Electronic Crystal Phases in the Presence of Non-Uniform Berry Curvature and Tunable Berry Flux: The ${\lambda }_N$-Jellium model
• Quantum-geometric dipole: a topological boost to flavor ferromagnetism in flat bands
• Ideal quantum geometry of the surface states of rhombohedral graphite and its effects on the surface superconductivity
• Quantum Geometric Origin of Orbital Magnetization
• Giant Room-Temperature Third-Order Electrical Transport in a Thin-Film Altermagnet Candidate
• Superfluid stiffness of superconductors with delicate topology
• Quantum geometric tensors from sub-bundle geometry
• Probing quantum geometric nonlinear magnetization via second-harmonic magneto-optical Kerr effect
• Normal state quantum geometry, non-locality and superconductivity
• Quantum geometry in correlated electron phases: from flat band to dispersive band
• Non-Hermitian wave-packet dynamics and its realization within a non-Hermitian chiral cavity
• Quantum geometry, localization, and topological bounds of spin fluctuations
• Band Geometry Induced Third-Harmonic Generation
• Intrinsic nonlinear Hall effect beyond Bloch geometry
• Linear thermal noise induced by Berry curvature dipole in a four-terminal system
• Dissipation-Shaped Quantum Geometry in Nonlinear Transport
• Quantum Christoffel Nonlinear Magnetization
• Quantum geometry and low-frequency optical conductivity of nodal planes
• Quantum geometric contribution to the diffusion constant
• Quantum geometric tensor in systems with fractional band dispersion
• Correlation lengths of flat-band superconductivity from quantum geometry
• Superconductivity and geometric superfluid weight of a tunable flat band system
• Quantum geometric map of magnetotransport
• Chern insulators in two and three dimensions: A global perspective
• Identifying geometric third-order nonlinear transport in disordered materials
• Quantum geometric magnetic monopole and two-phase superconductivity in CeRh${}_2$As${}_2$
• Quantum theory of nonlinear electromagnetic response
• Dissipation-Driven Transition of Particles from Dispersive to Flat Bands