Dynamic quantum phase transitions (DQPT) following quantum quenches exhibit singular behavior of the overlap between the initial and evolved states. Here we present a formalism to incorporate a geometric phase into quench dynamics of mixed quantum states, a process named the Uhlmann quench, based on the Uhlmann parallel transport. To overcome the incompatibility between the Uhlmann parallel-transport condition and Hamiltonian dynamics, we formulate the evolution of purification of the density matrix in a form which not only respects the dynamics according to the density matrix but also follows the Uhlmann parallel-transport condition to generate a geometric phase after a quantum quench. For cyclic processes exemplified by a spin-1/2 system, geometric DQPTs (GDQPTs) can emerge with both singular behavior in the dynamic analog of the free energy and jumps of the geometric phase. Moreover, the Uhlmann phase reflecting the holonomy is generated at the end of each cycle. The Uhlmann quench thus paves the way for investigating the interplay between quantum dynamics and geometric processes in mixed states.
Dynamic quantum phase transitions (DQPT) following quantum quenches exhibit singular behavior of the overlap between the initial and evolved states. Here we present a formalism to incorporate a geometric phase into quench dynamics of mixed quantum states, a process named the Uhlmann quench, based on the Uhlmann parallel transport. To overcome the incompatibility between the Uhlmann parallel-transport condition and Hamiltonian dynamics, we formulate the evolution of purification of the density matrix in a form which not only respects the dynamics according to the density matrix but also follows the Uhlmann parallel-transport condition to generate a geometric phase after a quantum quench. For cyclic processes exemplified by a spin-1/2 system, geometric DQPTs (GDQPTs) can emerge with both singular behavior in the dynamic analog of the free energy and jumps of the geometric phase. Moreover, the Uhlmann phase reflecting the holonomy is generated at the end of each cycle. The Uhlmann quench thus paves the way for investigating the interplay between quantum dynamics and geometric processes in mixed states.
Preparing an integrable system in a mixed state described by a thermal density matrix, we subject it to a sudden quench and explore the subsequent unitary dynamics. To address the question of whether the nonanalyticities, namely, the dynamical quantum phase transitions (DQPTs), persist when the initial state is mixed, we consider two versions of the generalized Loschmidt overlap amplitude (GLOA). Our study shows that the GLOA constructed using the Uhlmann approach does not show any signature of DQPTs at any nonzero initial temperature. On the other hand, a GLOA defined in the interferometric phase approach through the purifications of the time-evolved density matrix, indeed shows that nonanalyiticies in the corresponding ``dynamical free-energy density'' persist, thereby establishing the existence of mixed state dynamical quantum phase transitions (MSDQPTs). Our work provides a framework that perfectly reproduces both the nonanalyticities and also the emergent topological structure in the pure state limit. These claims are corroborated by analyzing the nonequilibrium dynamics of a transverse Ising chain initially prepared in a thermal state and subjected to a sudden quench of the transverse field.
While a pure quantum state may accumulate both the Berry phase and dynamic phase as it undergoes a cyclic path in the parameter space, the situation is more complicated when mixed quantum states are considered. From the Ulhmann bundle, a mixed quantum state can accumulate the Ulhmann phase if the parallel-transport condition is satisfied. However, we show that the Ulhmann process is in general not compatible with the evolution equation of the density matrix governed by the Hamiltonian. Thus a mixed quantum state usually accumulates a dynamic phase during its time evolution. We present the expression of the dynamic phase for mixed quantum states. In examples of one-dimensional two-band models and simple harmonic oscillator, the dynamic phase can take multiple discrete values in cyclic, quasistatic processes at infinitely high temperature due to the resonant points. However, the behavior differs if the energy spectrum is continuous without a band gap. Moreover, there is no natural analog of the dynamic phase in classical systems.
The Loschmidt amplitude of the purified states of mixed-state density matrices is shown to have zeros when the system undergoes a quasistatic, quench, or Uhlmann process. While the Loschmidt-amplitude zero of a quench process corresponds to a dynamical quantum phase transition (DQPT) accompanied by the diverging dynamical free energy, the Loschmidt-amplitude zero of the Uhlmann process corresponds to a topological phase transition (TQPT) accompanied by a jump of the Uhlmann phase. Although the density matrix remains intact in a quasistatic process, the Loschmidt amplitude can have zeros not associated with a phase transition. We present examples of two-level and three-level systems exhibiting finite- or infinite-temperature DQPTs and finite-temperature TQPTs associated with the Loschmidt-amplitude zeros. Moreover, the dynamical phase or geometrical phase of mixed states can be extracted from the Loschmidt amplitude. Those phases may become quantized or exhibit discontinuity at the Loschmidt-amplitude zeros. A spinor representation of the purified states of a general two-level system is presented to offer more insights into the change of purification in different processes. The quasistatic process, for example, is shown to cause a rotation of the spinor.
The Uhlmann phase extends geometric phases to mixed quantum states via a parallel-transport condition on purification amplitudes, yet its direct implementation under standard Hamiltonian dynamics is obstructed by the non-Hermitian nature of the purification. We establish that for any smooth one-dimensional closed loop of full-rank qubit density matrices, there exists a four-level Hermitian parent Hamiltonian whose doubly degenerate ground-state subspace carries a Wilczek--Zee connection exactly equal to the Uhlmann connection. Consequently, the Uhlmann holonomy is faithfully reproduced by adiabatic evolution in the enlarged system. We further prove that this auxiliary-field construction is obstructed in generic two-dimensional parameter spaces by a Frobenius integrability condition, which we derive explicitly. The one-dimensional Uhlmann phase is thus placed on the same footing as the non-Abelian Berry phase, offering a purely Hermitian, Hamiltonian-based route to simulating mixed-state geometric phases. Numerical integration of the adiabatic dynamics confirms the exact correspondence and validates the convergence to the Uhlmann holonomy in the large-gap limit.
Two geometric phases of mixed quantum states, known as the interferometric phase and Uhlmann phase, are generalizations of the Berry phase of pure states. After reviewing the two geometric phases and examining their parallel-transport conditions, we specify a class of cyclic processes that are compatible with both conditions and therefore accumulate both phases through their definitions, respectively. Those processes then facilitate a fair comparison between the two phases. We present exact solutions of two-level and three-level systems to contrast the two phases. While the interferometric phase exhibits finite-temperature transitions only in the three-level system but not the two-level system, the Uhlmann phase shows finite-temperature transitions in both cases. Thus, using the two geometric phases as finite-temperature topological indicators demonstrates the rich physics of topology of mixed states.
Geometric phases play a fundamental role in understanding the geometric structure of quantum states, 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 parameter-dependent metric modifies the Uhlmann connection and leads to a finite-temperature distribution of Uhlmann-phase regions that differs from the standard Hermitian case. Applying this formalism to solvable two-level models, we uncover tunable finite-temperature Uhlmann-phase diagrams, where the parameter-dependent metric shifts and deforms the zeros of the Uhlmann amplitude, thereby reshaping the temperature intervals in which nontrivial Uhlmann phases occur. Furthermore, by extending established interferometric protocols originally developed for Hermitian systems, the geometric amplitude can be recast as a measurable Loschmidt amplitude 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 quasi-Hermitian quantum systems and, through its natural relation to the mixed-state quantum geometric tensor, opens new avenues for exploring local geometry in quasi-Hermitian thermal states.