Essay: Where Can Quantum Geometry Lead Us?

Abstract

Quantum geometry defines the phase and amplitude distances between quantum states. The phase distance is characterized by the Berry curvature and thus relates to topological phenomena. The significance of the full quantum geometry, including the amplitude distance characterized by the quantum metric, has started to receive attention in the last few years. Various quantum transport and interaction phenomena have been found to be critically influenced by quantum geometry. For example, quantum geometry allows counterintuitive flow of supercurrent in a flat band where single electrons are immobile. In this Essay, I will discuss my view of the important open problems and future applications of this research topic and will try to inspire the reader to come up with further ideas. At its best, quantum geometry can open a new chapter in band theory and lead to breakthroughs as transformative as room-temperature superconductivity. However, first, more experiments directly showing the effect of quantum geometry are needed. We also have to integrate quantum geometry analysis in our most advanced numerical methods. Further, the ramifications of quantum geometry should be studied in a wider range, including electric and electromagnetic responses and interaction phenomena in free- and correlated-electron materials, bosonic systems, optics, and other fields.

Part of a series of Essays in Physical Review Letters which concisely present author visions for the future of their field. Not a review; due to the format, extremely small number of references

References in corpus (3)

• Signatures of Fractional Quantum Anomalous Hall States in Twisted MoTe2 Bilayer
• Observation of Fractionally Quantized Anomalous Hall Effect
• Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure

Cited by in corpus (113)

• Roadmap for Photonics with 2D Materials
• Quantum geometry in condensed matter
• Revealing Quantum Geometry in Nonlinear Quantum Materials
• Quantized Integrated Shift Effect in Multigap Topological Phases
• The quantum metric of electrons with spin-momentum locking
• Cavity engineering of solid-state materials without external driving
• Non-Abelian Hopf-Euler insulators
• Theory of Generalized Landau Levels and Implication for non-Abelian States
• Extrinsic Orbital Hall Effect: Orbital Skew Scattering and Crossover Between Diffusive and Intrinsic Orbital Transport
• Thermodynamic Response and Neutral Excitations in Integer and Fractional Quantum Anomalous Hall States Emerging from Correlated Flat Bands
• Equivalence of semiclassical and response theories for second-order nonlinear ac Hall effects
• Instantaneous response and quantum geometry of insulators
• Floquet engineering of topological phase transitions in quantum spin Hall $\alpha$-$T_3$ system
• Identifying non-Hermitian critical points with quantum metric
• Cooper pairing, flat-band superconductivity and quantum geometry in the pyrochlore-Hubbard model
• Quantum Metric in Step Response
• Quantum entanglement and quantum geometry measured with inelastic X-ray scattering
• Quantum Geometry Probed by Chiral Excitonic Optical Response of Chern Insulators
• Altermagnetic Instabilities from Quantum Geometry
• Excitonic topology and quantum geometry in organic semiconductors
• Giant Third-Order Nonlinearity Induced by the Quantum Metric Quadrupole in Few-Layer WTe2
• Corner Charge Fluctuation as an Observable for Quantum Geometry and Entanglement in Two-dimensional Insulators
• Quantum Metric Nonlinear Spin-Orbit Torque Enhanced by Topological Bands
• Scaling of the Integrated Quantum Metric in Disordered Topological Phases
• Intrinsic nonlinear Hall effect in two-dimensional honeycomb topological antiferromagnets
• Disorder-Induced Delocalization in Magic-Angle Twisted Bilayer Graphene
• Exact projected entangled pair ground states with topological Euler invariant
• Quantum geometric superfluid weight in multiband superconductors: A microscopic interpretation
• Disorder-Induced Topological Transitions in a Multilayer Topological Insulator
• Coherence length and quantum geometry in a dilute flat-band superconductor
• Electric field induced second-order anomalous Hall transport in unconventional Rashba systems
• A Clarification on Quantum-Metric-Induced Nonlinear Transport
• Geometric and conventional contributions of superconducting diode effect: Application to flat-band systems
• Flat-band Fulde-Ferrell-Larkin-Ovchinnikov State from Quantum Geometric Discrepancy
• Intrinsic nonlinear Nernst and Seebeck effect
• Universal Wilson Loop Bound of Quantum Geometry
• Nonlinear optical responses in multi-orbital topological superconductors
• Quantum geometry and $X$-wave magnets with $X=p,d,f,g,i$
• Quantum response theory and momentum-space gravity
• Optical signatures of Euler superconductors
• Nonlinear Hall effects with an exceptional ring
• Friedel oscillations in two-dimensional materials with inverted bands and Mexican-hat dispersion
• Low-energy optical absorption in correlated insulators: Projected sum rules and the role of quantum geometry
• Pair size and quantum geometry in a multiband Hubbard model
• Density matrix renormalization group study of quantum-geometry-facilitated pair density wave order
• Quantum Geometric Origin of the Intrinsic Nonlinear Hall Effect
• Trapping Hard-Core Bosons in Flatband Lattices
• Flat bands in condensed-matter systems -- perspective for magnetism and superconductivity
• Manifestation of the Berry connection in chiral lattice systems
• Structure factors and quantum geometry in multiband BCS superconductors
• Magnetic Field Induced Quantum Metric Dipole in Dirac Semimetal Cd3As2
• Quantum geometric bounds in spinful systems with trivial band topology
• Functional approach to superfluid stiffness: Role of quantum geometry in unconventional superconductivity
• Many-Body Quantum Geometric Dipole
• Thermal and thermoelectric transport in flat bands with non-trivial quantum geometry
• Quantum geometrical effects in non-Hermitian systems
• From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory
• Quantum-geometric dipole: a topological boost to flavor ferromagnetism in flat bands
• Nonlinear planar magnetotransport as a probe of the topology of surface states
• Shift photocurrent vortices from topological polarization textures
• Ferromagnetism vs. Antiferromagnetism in Narrow-Band Systems: Competition Between Quantum Geometry and Band Dispersion
• Coexisting Kagome and Heavy Fermion Flat Bands in YbCr${}_6$Ge${}_6$
• Family of Aperiodic Tilings with Tunable Quantum Geometric Tensor
• Quantum Metric-induced Oscillations in Flat Bands
• Superfluid stiffness bounds in time-reversal symmetric superconductors
• Topological invariants and topological charges in photonic systems
• Quantum geometry and geometric entanglement entropy of one-dimensional Floquet topological matter
• Fast entangling quantum gates with almost-resonant modulated driving
• Ideal quantum geometry of the surface states of rhombohedral graphite and its effects on the surface superconductivity
• Quantum geometric tensors from sub-bundle geometry
• Generalized Peierls substitution for Wannier obstructions: response to disorder and interactions
• Phonon spectra, quantum geometry, and the Goldstone theorem
• On the geometry of topological defects in glasses
• Direct measurement of the quantum geometric tensor in pseudo-Hermitian systems
• Universal intrinsic orbital dynamics from Berry curvature in electronic two-band systems
• Charged impurity scattering in two-dimensional topological insulators with Mexican-hat dispersion
• Quantum geometry embedded in unitarity of evolution: revealing its impacts as geometric oscillation and dephasing in spin resonance and crystal bands
• Giant Room-Temperature Third-Order Electrical Transport in a Thin-Film Altermagnet Candidate
• Quantum Geometric Origin of Orbital Magnetization
• Superfluid stiffness of superconductors with delicate topology
• Band Geometry Induced Third-Harmonic Generation
• Normal state quantum geometry, non-locality and superconductivity
• Optimal spectral transport of non-Hermitian systems
• Spin-polarized scanning tunneling microscopy measurement scheme for determining the quantum geometric tensor
• Diamagnetic Meissner response of odd-frequency superconducting pairing from quantum geometry
• Frustration-free free fermions
• Probing quantum geometric nonlinear magnetization via second-harmonic magneto-optical Kerr effect
• Quantum geometry, localization, and topological bounds of spin fluctuations
• Quantum geometry in correlated electron phases: from flat band to dispersive band
• Quantum geometry and low-frequency optical conductivity of nodal planes
• Third-harmonic generation in superconductors: Role of quantum geometry in the competition between Higgs mode and quasiparticles
• Superconductivity and geometric superfluid weight of a tunable flat band system
• Linear thermal noise induced by Berry curvature dipole in a four-terminal system
• Superconducting magnetoelectric effects in mesoscopic hybrid structures
• Gaplessness from disorder and quantum geometry in gapped superconductors
• Variational approach for the two-body problem in a multiband extended-Hubbard model
• Analysis of the singular band structure occurring in one-dimensional topological normal and superfluid fermionic systems: A pedagogical description
• From generating functions to the geometric Binder cumulant
• Exploring Many-Body Quantum Geometry Beyond the Quantum Metric with Correlation Functions: A Time-Dependent Perspective
• Probing the Quantum Geometry of Correlated Metals using Optical Conductivity
• Deformation of Matrix Geometry via Landau Level Evolution
• Quantum geometric map of magnetotransport
• Lattice-induced wavefunction effects on trapped superfluids
• Identifying geometric third-order nonlinear transport in disordered materials
• Quantum geometric tensor in systems with fractional band dispersion
• Distinct topological excitonic insulators characterized by quantum geometry
• Electric field controlled second-order anomalous Hall effect in altermagnets
• From Divergent Series to Geometry: Resurgence of the Quantum Metric
• Intertwined geometries in collective modes of two dimensional Dirac fermions
• Quantum Christoffel Nonlinear Magnetization
• Dissipation-Shaped Quantum Geometry in Nonlinear Transport
• Scaling of Quantum Geometry Near the Non-Hermitian Topological Phase Transitions
• Orbital magnetization reveals multiband topology