Monotone Riemannian metrics and dynamic structure factor in condensed matter physics

Abstract

An analytical approach is developed to the problem of computation of monotone Riemannian metrics (e.g. Bogoliubov-Kubo-Mori, Bures, Chernoff, etc.) on the set of quantum states. The obtained expressions originate from the Morozova, Chencov and Petz correspondence of monotone metrics to operator monotone functions. The used mathematical technique provides analytical expansions in terms of the thermodynamic mean values of iterated (nested) commutators of a model Hamiltonian T with the operator S involved through the control parameter $h$. Due to the sum rules for the frequency moments of the dynamic structure factor new presentations for the monotone Riemannian metrics are obtained. Particularly, relations between any monotone Riemannian metric and the usual thermodynamic susceptibility or the variance of the operator $S$ are discussed. If the symmetry properties of the Hamiltonian are given in terms of generators of some Lie algebra, the obtained expansions may be evaluated in a closed form. These issues are tested on a class of model systems studied in condensed matter physics.

typos corrected, ref.49 added, minor change in Section II to mach the published version

References in corpus (17)

• Fidelity, dynamic structure factor, and susceptibility in critical phenomena
• The Quantum Chernoff Bound
• Mixed-state fidelity and quantum criticality at finite temperature
• The quantum Chernoff bound as a measure of distinguishability between density matrices: application to qubit and Gaussian states
• Bures metric over thermal state manifolds and quantum criticality
• Quantum correlations and distinguishability of quantum states
• Quantum Fidelity and Thermal Phase Transitions
• Quantum state estimation with informationally overcomplete measurements
• Thermodynamic length for far from equilibrium quantum systems
• Finite temperature fidelity susceptibility for one-dimensional quantum systems
• Generalized fidelity susceptibility at phase transitions
• Solving two-mode squeezed harmonic oscillator and $k$th-order harmonic generation in Bargmann-Hilbert spaces
• Quantum Chernoff Bound metric for the XY model at finite temperature
• Finite-Temperature Fidelity-Metric Approach to the Lipkin-Meshkov-Glick Model
• Generalized inequalities for the Bogoliubov-Duhamel inner product with applications in the Approximating Hamiltonian Method
• Mixed-state fidelity susceptibility through iterated commutator series expansion
• Frequency moments of the Coulomb dynamic structure factor and related integrals

Cited by in corpus (1)

• Quasirandom estimations of two-qubit operator-monotone-based separability probabilities