Symmetry-resolved R\'enyi fidelities and quantum phase transitions

TL;DR

This paper introduces symmetry-resolved quantum Rnyi fidelities, providing exact formulas for Gaussian states and demonstrating their effectiveness in detecting quantum phase transitions in the XX spin chain.

Contribution

It develops a framework for symmetry-resolved fidelities, deriving exact formulas for Gaussian states and applying them to analyze phase transitions in the XX spin chain.

Findings

Symmetry-resolved fidelities can detect quantum phase transitions.

Exact formulas for Gaussian states are derived.

Fidelities reveal the reorganization of charge sectors at critical points.

Abstract

We introduce a family of quantum R\'enyi fidelities and discuss their symmetry resolution. We express the symmetry-resolved fidelities as Fourier transforms of charged fidelities, for which we derive exact formulas for Gaussian states. These results also yield a formula for the total fidelities of Gaussian states, which we expect to have applications beyond the scope of this paper. We investigate the total and symmetry-resolved fidelities in the XX spin chain, and focus on (i) fidelities between thermal states, and (ii) fidelities between reduced density matrices at zero temperature. Both thermal and reduced fidelities can detect the quantum phase transition of the XX spin chain. Moreover, we argue that symmetry-resolved fidelities are sensitive to the inner structure of the states. In particular, they can detect the phase transition through the reorganisation of the charge sectors at the critical point. This a main feature of symmetry-resolved fidelities which we expect to be general. We also highlight that reduced fidelities can detect quantum phase transitions in the thermodynamic limit.