Uhlmann Fidelity and Fidelity Susceptibility for Integrable Spin Chains at Finite Temperature: Exact Results

TL;DR

This paper derives exact formulas for Uhlmann fidelity in the quantum XY model, analyzes fidelity susceptibility in the Ising model, and reveals how including all parity sectors affects thermal state behavior across temperatures.

Contribution

It provides the exact expression for Uhlmann fidelity in finite quantum XY models and clarifies the impact of parity sectors on fidelity susceptibility in the Ising model.

Findings

Maximal fidelity susceptibility is enhanced when odd parity states are included.

The enhancement persists in the thermodynamic limit and scales quadratically with system size.

Low-temperature behavior is accurately described by the two lowest energy states.

Abstract

We derive the exact expression for the Uhlmann fidelity between arbitrary thermal Gibbs states of the quantum XY model in a transverse field with finite system size. Using it, we conduct a thorough analysis of the fidelity susceptibility of thermal states for the Ising model in a transverse field. We compare the exact results with a common approximation that considers only the positive-parity subspace, which is shown to be valid only at high temperatures. The proper inclusion of the odd parity subspace leads to the enhancement of maximal fidelity susceptibility in the intermediate range of temperatures. We show that this enhancement persists in the thermodynamic limit and scales quadratically with the system size. The correct low-temperature behavior is captured by an approximation involving the two lowest many-body energy eigenstates, from which simple expressions are obtained for the thermal susceptibility and specific heat.