Monotone Riemannian metrics and dynamic structure factor in condensed matter physics

TL;DR

This paper develops an analytical method to compute monotone Riemannian metrics on quantum states, linking them to thermodynamic quantities and dynamic structure factors, with applications to condensed matter models.

Contribution

It introduces a new analytical expansion for monotone Riemannian metrics based on operator commutators and sum rules, connecting them to thermodynamic susceptibilities.

Findings

Derived new expressions for monotone Riemannian metrics

Linked metrics to thermodynamic susceptibilities and variances

Validated approach on condensed matter models

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.