Macroscopic Distinguishability Between Quantum States Defining Different Phases of Matter: Fidelity and the Uhlmann Geometric Phase

TL;DR

This paper explores how the fidelity approach and the Uhlmann geometric phase can detect and characterize quantum and thermal phase transitions in various models, linking non-analyticity to physical response functions.

Contribution

It extends the fidelity approach to thermal phase transitions and analyzes the role of the Uhlmann geometric phase in non-commuting Hamiltonian systems near criticality.

Findings

Fidelity drops sharply at phase transition lines.

Non-analyticity of fidelity relates to susceptibility and heat capacity.

Uhlmann geometric phase signals eigenvector changes near transitions.

Abstract

We study the fidelity approach to quantum phase transitions (QPTs) and apply it to general thermal phase transitions (PTs). We analyze two particular cases: the Stoner-Hubbard itinerant electron model of magnetism and the BCS theory of superconductivity. In both cases we show that the sudden drop of the mixed state fidelity marks the line of the phase transition. We conduct a detailed analysis of the general case of systems given by mutually commuting Hamiltonians, where the non-analyticity of the fidelity is directly related to the non-analyticity of the relevant response functions (susceptibility and heat capacity), for the case of symmetry-breaking transitions. Further, on the case of BCS theory of superconductivity, given by mutually non-commuting Hamiltonians, we analyze the structure of the system's eigenvectors in the vicinity of the line of the phase transition showing that their sudden change is quantified by the emergence of a generically non-trivial Uhlmann mixed state geometric phase.