Computing observables without eigenstates: applications to Bloch Hamiltonians

Abstract

Calculating the observables of a Hamiltonian requires taking matrix elements of operators in the eigenstate basis. Since eigenstates are only defined up to arbitrary phases that depend on Hamiltonian parameters, analytical expressions for observables are often difficult to simplify. In this work, we show how for small Hilbert space dimension N all observables can be expressed in terms of the Hamiltonian and its eigenvalues using the properties of the SU(N) algebra, and we derive explicit expressions for N=2,3,4. Then we present multiple applications specializing to the case of Bloch electrons in crystals, including the computation of Berry curvature, quantum metric and orbital moment, as well as a more complex observable in non-linear response, the linear photogalvanic effect (LPGE). As a physical example we consider multiband Hamiltonians with nodal degeneracies to show first how constraints between these observables are relaxed when going from two to three-band models, and second how quadratic dispersion can lead to constant LPGE at small frequencies.

8 pages, 2 figures and appendix

References in corpus (31)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Weyl and Dirac Semimetals in Three Dimensional Solids
• Topological Crystalline Insulators
• Higher-Order Topological Insulators
• Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals
• Nearly-flat bands with nontrivial topology
• Model Hamiltonian for Topological Insulators
• Quantized circular photogalvanic effect in Weyl semimetals
• Non Fermi-Liquid States and Pairing of a general Model of Copper-Oxide Metals
• Multiple types of topological fermions in transition metal silicides
• Experimental realization and characterization of an electronic Lieb lattice
• Colossal Bulk Photovoltaic Effect in a Weyl Semimetal
• Large Fermi Arcs in Unconventional Weyl Semimetal RhSi
• Discovery of topological chiral crystals with helicoid arc states
• New classes of chiral topological nodes with non-contractible surface Fermi arcs in CoSi
• Observation of Chiral Fermions with a Large Topological Charge and and Associated Fermi-Arc Surface States in CoSi
• Nearly flat band with Chern number C=2 on the dice lattice
• Chiral Topological Semimetal with Multifold Band Crossings and Long Fermi arcs
• Robust zero-energy modes in an electronic higher-order topological insulator: the dimerized Kagome lattice
• Topological Phases for Fermionic Cold Atoms on the Lieb Lattice
• Chiral Optical Response of Multifold Fermions
• Dichroic f-sum rule and the orbital magnetization of crystals
• Observation and control of maximal Chern numbers in a chiral topological semimetal
• Invariant tensors for simple groups
• Difference frequency generation in topological semimetals
• Revealing tensor monopoles through quantum-metric measurements
• SU(3) Spin-Orbit Coupling in Systems of Ultracold Atoms
• Berry phase and Anomalous Hall Effect in a Three-orbital Tight-binding Hamiltonian
• Divergent bulk photovoltaic effect in Weyl semimetals
• Artificial $SU(3)$ Spin-Orbit Coupling and Exotic Mott Insulators
• Spiral spin textures of bosonic Mott insulator with SU(3) spin-orbit coupling

Cited by in corpus (14)

• Relations between topology and the quantum metric for Chern insulators
• Berry Curvature and Quantum Metric in $N$-band systems -- an Eigenprojector Approach
• Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals
• Revealing Chern number from quantum metric
• Quantum Fluctuation of the Quantum Geometric Tensor and its Manifestation as Intrinsic Hall Signatures in Time-Reversal Invariant Systems
• Interplay of Band Geometry and Topology in Ideal Chern Insulators in Presence of External Electromagnetic Fields
• The multi-state geometry of shift current and polarization
• Quantum Geometric Origin of the Intrinsic Nonlinear Hall Effect
• Electronic Crystal Phases in the Presence of Non-Uniform Berry Curvature and Tunable Berry Flux: The ${\lambda }_N$-Jellium model
• Geometric semimetals and their simulation in synthetic matter
• Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric
• Precise quantum-geometric electronic properties from first principles
• Analytic Formulas for the Anomalous Hall Effect in Itinerant Magnets
• Suppression of second-harmonic generation from linear bands in the topological multifold semimetal RhSi