Theory of the Uhlmann Phase in Quasi-Hermitian Quantum Systems

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.

Figures (4)

• Figure 1: The Uhlmann phase $\theta_U^{\eta}$ (inset) and the geometric factor $g = \cos^{-1}(\mathcal{G})$ as functions of temperature for $\Omega = 1$ (black solid), $\Omega = 2$ (blue dotted), and $\Omega = 3$ (red dashed). The diverging peaks indicate critical temperatures where the phase jumps by $\pi$.
• Figure 2: Zero-temperature Uhlmann fidelity $\mathcal{G}(\infty)$ for the quasi-Hermitian two-level model evolving along an equatorial closed loop, plotted as a function of the winding number $\Omega$. Parameters: $\varepsilon=0$, $a=5$, $b=4$, $\delta=0$.
• Figure 3: Uhlmann phase $\theta_U^{\eta_+}$ for the quasi-Hermitian two-level model along an equatorial closed loop as a function of temperature $T$ and the quasi-Hermitian parameter $b$. Left and right panels correspond to winding numbers $\Omega=1$ and $\Omega=2$, respectively. Yellow: $\theta_U^{\eta_+}=0$ (topologically trivial). Dark blue: $\theta_U^{\eta_+}=\pi$ (topologically nontrivial). Parameters: $\varepsilon=0$, $a=5$, $\delta=0$.
• Figure 4: Uhlmann phase $\theta_U^{\eta_+}$ for the quasi-Hermitian two-level model along an equatorial closed loop as a function of temperature $T$. Blue and red solid lines correspond to winding numbers $\Omega=1$ and $2$, respectively. Parameters: $\varepsilon=0$, $a=5$, $b=4$, $\delta=0$.