Theory of the Uhlmann Phase in Quasi-Hermitian Quantum Systems

TL;DR

This paper develops a comprehensive theory of the Uhlmann phase for quasi-Hermitian quantum systems with parameter-dependent metrics, revealing new geometric features and topological phase behaviors at finite temperatures.

Contribution

It introduces a generalized purification and parallel transport framework for quasi-Hermitian systems, extending geometric phase concepts to non-Hermitian contexts with temperature-dependent topological phases.

Findings

Finite-temperature topological phase diagrams with multiple transitions

Quasi-Hermitian parameters affect phase stability against temperature

Measurable interferometric protocols for geometric phases in experiments

Abstract

Geometric phases play a fundamental role in understanding quantum topology, yet extending the Uhlmann phase to non-Hermitian systems poses significant challenges due to parameter-dependent inner product structures. In this work, we develop a comprehensive theory of the Uhlmann phase for quasi-Hermitian systems, where the physical Hilbert space metric varies with external parameters. By constructing a generalized purification that respects the quasi-Hermitian inner product, we derive the corresponding parallel transport condition and Uhlmann connection. Our analysis reveals that the dynamic metric induces emergent geometric features absent in the standard Hermitian theory. Applying this formalism to solvable two-level models, we uncover rich finite-temperature topological phase diagrams, including multiple transitions between trivial and nontrivial phases driven by thermal fluctuations. Crucially, the quasi-Hermitian parameters are shown to profoundly influence the stability of topological regimes against temperature, enabling nontrivial phases to persist within finite-temperature windows. Furthermore, by extending established interferometric protocols originally developed for Hermitian systems, the geometric amplitude can be recast as a measurable Loschmidt fidelity between purified states, providing a practical and experimentally accessible pathway to investigate quasi-Hermitian mixed-state geometric phases and their finite-temperature transitions. This work establishes a unified framework for understanding mixed-state geometric phases in non-Hermitian quantum systems and opens a practical avenue for their experimental investigation.