Fidelity Between Partial States as Signature of Quantum Phase Transitions

TL;DR

This paper introduces a partial state fidelity method to detect quantum phase transitions in a superconducting lattice with a magnetic impurity, revealing abrupt changes in fidelity and correlations near the transition point.

Contribution

It presents a novel approach using partial state fidelity and geometric phase to identify quantum phase transitions and analyze correlations in a superconducting lattice with impurities.

Findings

Fidelity drops sharply at the quantum phase transition point.

Two-site fidelity detects the transition when one site is at the impurity.

Correlations show an abrupt change near the transition.

Abstract

We introduce a partial state fidelity approach to quantum phase transitions. We consider a superconducting lattice with a magnetic impurity inserted at its centre, and look at the fidelity between partial (either one-site or two-site) quantum states. In the vicinity of the point of the quantum phase transition, we observe a sudden drop of the fidelity between two one-site partial states corresponding to the impurity location and its close vicinity. In the case of two-site states, the fidelity reveals the transition point as long as one of the two electron sites is located at the impurity, while the other lies elsewhere in the lattice. We also determine the Uhlmann mixed state geometric phase, recently introduced in the study of the structural change of the system state eigenvectors in the vicinity of the lines of thermal phase transitions, and find it to be trivial, both for one- and two-site partial states, except when an electron site is at the impurity. This means that the system partial state eigenvectors do not contribute significantly to the enhanced state distinguishability around the point of this quantum phase transition. Finally, we use the fidelity to analyze the total amount of correlations contained within a composite system, showing that, even for the smallest two-site states, it features an abrupt quantitative change in the vicinity of the point of the quantum phase transition.