Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals
arXiv:2412.03637 · doi:10.1103/qscv-qxqt
Abstract
The importance of simple geometrical invariants, such as the Berry curvature and quantum metric, constructed from the Bloch states of a crystal has become well-established over four decades of research. More complex aspects of geometry emerge in properties linking multiple bands, such as optical responses. In the companion work [arXiv:2409.16358], we identified novel multi-state geometrical invariants using an explicitly gauge-invariant formalism based on projection operators, which we used to clarify the relation between the shift current and the theory of electronic polarization among other advancements for second-order non-linear optics. Here, we provide considerably more detail on the projector formalism and the geometrical invariants arising in the vicinity of a specific value of crystal momentum. We combine the introduction to multi-state quantum geometry with broadly relevant algebraic relationships and detailed example calculations, enabling extensions toward future applications to topological and geometrical properties of insulators and metals.
13 + 5 pages, 1 table. Joint submission with arXiv:2409.16358
References in corpus (32)
- Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials
- Superfluidity in topologically nontrivial flat bands
- Band geometry of fractional topological insulators
- Quantum metric-induced nonlinear transport in a topological antiferromagnet
- Low-frequency divergence and quantum geometry of the bulk photovoltaic effect in topological semimetals
- Consequences of Time-reversal-symmetry Breaking in the Light-Matter Interaction: Berry Curvature, Quantum Metric and Diabatic Motion
- Vortexability: A Unifying Criterion for Ideal Fractional Chern Insulators
- Resonance-enhanced optical nonlinearity in the Weyl semimetal TaAs
- Kähler geometry and Chern insulators: Relations between topology and the quantum metric
- Topological Floquet Engineering of Twisted Bilayer Graphene
- Fundamental bound on topological gap
- Berry Curvature and Quantum Metric in -band systems -- an Eigenprojector Approach
- Light-matter coupling and quantum geometry in moiré materials
- Fluctuations, uncertainty relations, and the geometry of quantum state manifolds
- Geometric photon-drag effect and nonlinear shift current in centrosymmetric crystals
- Revealing Quantum Geometry in Nonlinear Quantum Materials
- Nontrivial quantum geometry of degenerate flat bands
- Intrinsic Fermi Surface Contribution to the Bulk Photovoltaic Effect
- Unifying semiclassics and quantum perturbation theory at nonlinear order
- Quantized Integrated Shift Effect in Multigap Topological Phases
- Generalized Wilson Loop Method for Nonlinear Light-Matter Interaction
- Longitudinal conductivity and Hall coefficient in two-dimensional metals with spiral magnetic order
- Quantum weight: A fundamental property of quantum many-body systems
- Universal relation between energy gap and dielectric constant
- Non-Abelian Hopf-Euler insulators
- Beyond the Berry Phase: Extrinsic Geometry of Quantum States
- Anomalous shift and optical vorticity in the steady photovoltaic current
- Computing observables without eigenstates: applications to Bloch Hamiltonians
- Quantum metric on the Brillouin Zone in correlated electron systems and its relation to topology for Chern insulators
- Spectral Density and Sum Rules for Second-Order Response Functions
- Drude weight of an interacting flat-band metal
- Geometric bound on structure factor
Cited by in corpus (15)
- The multi-state geometry of shift current and polarization
- Quantum geometric bounds for observables: Linear responses, Drude weight, and orbital magnetization
- Quantum response theory and momentum-space gravity
- Quantum Geometric Origin of the Intrinsic Nonlinear Hall Effect
- Quantization of spin circular photogalvanic effect in altermagnetic Weyl semimetals
- Functional approach to superfluid stiffness: Role of quantum geometry in unconventional superconductivity
- Electronic Crystal Phases in the Presence of Non-Uniform Berry Curvature and Tunable Berry Flux: The -Jellium model
- From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory
- Real-Space Switching of Local Moments Driven by Quantum Geometry in Correlated Graphene Heterostructures
- Orbital Magnetization and Streda Formula of Interacting Electrons in the Mean-field Approximation
- Quantum geometric tensors from sub-bundle geometry
- Gradient flow and Bogomolny bounds for quantum metric actions
- Quantum geometry and low-frequency optical conductivity of nodal planes
- Exploring Many-Body Quantum Geometry Beyond the Quantum Metric with Correlation Functions: A Time-Dependent Perspective
- Two dimensional sub-wavelength topological dark state lattices