Uhlmann Phase Research Articles

Uhlmann phase winding in Bose-Einstein condensates at finite temperature

Uhlmann phase winding in Bose-Einstein condensates at finite temperature

Research on quantum correlations and Uhlmann phase in two-fermion mixed state system

Geometric phase, characterized by its unique properties has been an interesting subject since the discovery of Berry phase. A state inevitably becomes a mixed state in the environment. Uhlmann phase can be applied to mixed states and may be the generalization of Berry phase. In this paper, we use a pair of interacting spin-1/2 fermions, where one particle is influenced by an external magnetic field, while the other remains unaffected, to investigate quantum correlations and Uhlmann phase, we define g, p as the ratio of the particle coupling intensity to the magnetic field, and q as the depolarization parameter. Through calculation, we get the specific parameter range q of steering and Bell nonlocality under the depolarized noise channel, q∈0,1−4+g24+2g2 (steering) and q∈0,1−4+g24+3g2 (Bell nonlocality). Moreover, by using Uhlmann phase, we obtain a simple discrimination criterion of the steerability (Bell nonlocality) of the mixed system.

Uhlmann phase of a thermal spin-1 system with zero field splitting

We study the Uhlmann geometric phase of a spin-1 particle subjected to zero-field splitting (ZFS) interaction, modulated by a dimensionless parameter α, under the effect of an external magnetic field with a tilting angle θ. We show that the ZFS term induces a transition in the geometrical phase behavior, characterized by a critical parameter value, α=αc. For α<αc, this phase displays two critical temperatures at θ=π/2, similar to spin-1 systems without ZFS, but with a separation that varies with α. In contrast, for α>αc, the phase exhibits two singularities at a single critical temperature but at different field orientations θ≠π/2. The phase disappears for significantly large |α|, regardless of the values of the Hamiltonian parameters. This behavior clearly departs from the usual thermal Uhlmann phase observed in SU(2) systems. In addition, we analytically calculate the heat capacity, which, for θ=π/2 and nearby values, displays two different regimes according to the sign of α. For α<0, it develops two peaks associated to the multilevel nature of the system, while for α>0 only a single Schottky-anomaly like peak appears as in two level systems. Interestingly, when θ=π/2, the temperature centroids of the Uhlmann phase and the heat capacity coincide in the region between critical temperatures for a given value of α<αc. Furthermore, we demonstrate that when α=0, the Uhlmann phase, a global topological property of the system, can be expressed as a function of the thermal component of the Bures metric, a local geometric property related to the heat capacity.

Uhlmann phases in the Kitaev chain with NNN hopping interaction

We study the extended Kitaev chain with both nearest-neighbor and next-nearest-neighbor hopping terms and find the model system exhibiting nontrivial phases, which can be characterized by a nonzero Berry phase and winding number when the system is in a pure state. While in a mixed state, we investigate the robustness of the topological Uhlmann phases and show how it responses to the presence of next-nearest-neighbor hopping terms. Furthermore, we analyse the complicated behavior of the Uhlmann phase of the extended Kitaev chain at finite temperature as k moves along the Brillouin zone, and we think this may serve as a topological indicator for mixed states in condensed matter systems.

Thermal Uhlmann phase in a locally driven two-spin system

Thermal Uhlmann phase in a locally driven two-spin system

Mixed state behavior of Hermitian and non-Hermitian topological models with extended couplings

Geometric phase is an important tool to define the topology of the Hermitian and non-Hermitian systems. Besides, the range of coupling plays an important role in realizing higher topological indices and transition among them. With a motivation to understand the geometric phases for mixed states, we discuss finite temperature analysis of Hermitian and non-Hermitian topological models with extended range of couplings. To understand the geometric phases for the mixed states, we use Uhlmann phase and discuss the merit-limitation with respect extended range couplings. We extend the finite temperature analysis to non-Hermitian models and define topological invariant for different ranges of coupling. We include the non-Hermitian skin effect, and provide the derivation of topological invariant in the generalized Brillouin zone and their mixed state behavior also. We also adopt mixed geometric phases through interferometric approach, and discuss the geometric phases of extended-range (Hermitian and non-Hermitian) models at finite temperature.

Geometric phases of mixed quantum states: A comparative study of interferometric and Uhlmann phases

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.

Uhlmann phase of coherent states and the Uhlmann-Berry correspondence

We first compare the geometric frameworks behind the Uhlmann and Berry phases in a fiber-bundle language and then evaluate the Uhlmann phases of bosonic and fermionic coherent states. The Uhlmann phases of both coherent states are shown to carry geometric information and decrease smoothly with temperature. Importantly, the Uhlmann phases approach the corresponding Berry phases as temperature decreases. Together with previous examples in the literature, we propose a correspondence between the Uhlmann and Berry phases in the zero-temperature limit as a general property except some special cases and present a conditional proof of the correspondence.

Uhlmann holonomy against Lindblad dynamics of topological systems at finite temperatures

The Uhlmann phase, which reflects the holonomy as the purified state of a density matrix traverses a loop in the parameter space, has been used to characterize topological properties of several systems at finite temperatures. We test the Uhlmann holonomy against various system-environment couplings in quantum dynamics described by the Lindblad equations of three topological systems, including the Su-Schrieffer-Heeger (SSH) model, Kitaev chain, and Bernevig-Hughes-Zhang (BHZ) model. The Uhlmann phase is shown to remain quantized in all the examples if the initial state is topological and only certain types of the Lindblad jump operators are present. Topological protection at finite temperatures against environmental effects in quantum dynamics is therefore demonstrated albeit only for a restricted class of system-environment couplings.

Proxy ensemble geometric phase and proxy index of time-reversal invariant topological insulators at finite temperatures

The ensemble geometric phase (EGP) has been proposed as a topological indicator for finite-temperatures systems. The ensemble Wilson loop, or the transfer matrix, contains the crucial information in the EGP construction. We propose a proxy index and a proxy EGP directly from the transfer matrix and apply them to time-reversal invariant topological insulators exemplified by the Bernevig-Hughes-Zhang (BHZ) and Kane-Mele (KM) models. The quantized proxy index and proxy EGP smoothly generalize the ground-state topological index to finite temperatures. For the BHZ model, a comparison with another topological indicator, the Uhlmann phase, shows different transition behavior with temperature. For the KM model, the EGP have been generalized to the time-reversal EGP previously, but the proxy EGP does not require any splitting of the contributions. The proxy index and proxy EGP thus offer an efficient means for characterizing finite-temperature topological properties.

Islands and Uhlmann phase: explicit recovery of classical information from evaporating black holes

Recent work has established a route towards the semiclassical validity of the Page curve, and so provided evidence that information escapes an evaporating black hole. However, a protocol to explicitly recover and make practical use of that information in the classical limit has not yet been given. In this paper, we describe such a protocol, showing that an observer may reconstruct the phase space of the black hole interior by measuring the Uhlmann phase of the Hawking radiation. The process of black hole formation and evaporation provides an invertible map between this phase space and the space of initial matter configurations. Thus, all classical information is explicitly recovered. We assume in this paper that replica wormholes contribute to the gravitational path integral.

Comparison of finite-temperature topological indicators based on Uhlmann connection

Two indicators of finite-temperature topological properties based on the Uhlmann connection, one generalizing the Wilson loop to the Uhlmann-Wilson loop and the other generalizing the Berry phase to the Uhlmann phase, are constructed explicitly for a time-reversal invariant topological insulators with a $Z_2$ index. While the phases of the eigenvalues of the Wilson loop reflect the $Z_2$ index of the model at zero temperature, it is found that the signature from the Uhlmann-Wilson loop gradually fades away as temperature increases. On the other hand, the Berry phase exhibits quantization due to the underlying holonomy group. The Uhlmann phase retains the quantization at finite temperatures and serves as an indicator of topological properties. A phase diagram showing where jumps of the Uhlmann phase can be found is presented. By modifying the model to allow higher winding numbers, finite-temperature topological regimes sandwiched between trivial regimes at high and low temperatures may emerge.

Uhlmann phase in composite systems with entanglement

We study the geometric Uhlmann phase of entangled mixed states in a composite system made of two coupled spin-$\frac 1 2$ particles with a magnetic field acting on one of them. Within a depolarizing channel setup, an exact analytical expression for such a phase in each subsystem is derived. We find an explicit connection to the concurrence of the depolarizing channel density matrix, which allows to characterize the features of the Uhlmann phase in terms of the degree of entanglement in the system. In the space of field direction and coupling parameter, it exhibits a phase singularity revealing a topological transition between orders with different winding numbers. The transition occurs for fields lying in the equator of the sphere of directions and at critical values of the coupling which can be controlled by tuning the depolarization strength. Notably, under these conditions the concurrence of the composite system is bounded to the range $[0,1/2]$. We also compare the calculated Uhlmann phase to an interferometric phase, which has been formulated as an alternative for density matrices. The latter does not present a phase vortex, although they coincide in the weak entanglement regime, for vanishing depolarization (pure states). Otherwise they behave clearly different in the strong entanglement regime.

Finite-temperature topological phase transitions of spin- j systems in Uhlmann processes: General formalism and experimental protocols

The Uhlmann process is built on the density matrix of a mixed quantum state and offers a way to characterize topological properties at finite temperatures. We analyze an ideal spin-j quantum paramagnet in a magnetic field undergoing an Uhlmann process and derive general formulae of the Uhlmann phase and Loschmidt amplitude for arbitrary j as the system traverses a great circle in the parameter space. A quantized jump of the Uhlmann phase signifies a topological quantum phase transition (TQPT) of the underlying process, which is accompanied by a zero of the Loschmidt amplitude. The exact results of j=1/2 and j=1 systems show topological regimes that only survive at finite temperatures but not at zero temperature, and the number of TQPTs is associated with the winding number in the parameter space. Our results pave the way for future studies on finite-temperature topological properties, and possible experimental protocols and implications for atomic simulators and digital simulations are discussed.

Topological Uhlmann phase transitions for a spin-jparticle in a magnetic field

The generalization of the geometric phase to the realm of mixed states is known as Uhlmann phase. Recently, applications of this concept to the field of topological insulators have been made and an experimental observation of a characteristic critical temperature at which the topological Uhlmann phase disappears has also been reported. Surprisingly, to our knowledge, the Uhlmann phase of such a paradigmatic system as the spin-$j$ particle in presence of a slowly rotating magnetic field has not been reported to date. Here we study the case of such a system in a thermal ensemble. We find that the Uhlmann phase is given by the argument of a complex valued second kind Chebyshev polynomial of order $2j$. Correspondingly, the Uhlmann phase displays $2j$ singularities, occurying at the roots of such polynomials which define critical temperatures at which the system undergoes topological order transitions. Appealing to the argument principle of complex analysis each topological order is characterized by a winding number, which happen to be $2j$ for the ground state and decrease by unity each time increasing temperature passes through a critical value. We hope this study encourages experimental verification of this phenomenon of thermal control of topological properties, as has already been done for the spin-$1/2$ particle.

Ubiquity of zeros of the Loschmidt amplitude for mixed states in different physical processes and its implication

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.

Interferometric and Uhlmann phases of mixed polarization states

In our investigation into the effects of the degree of polarization in modulation of partially polarized light we assume general settings of the interferometry of partially polarized lightwaves and perform theoretical analysis of the Uhlmann and the interferometric phases. We introduce the relative Uhlmann phase determined by the Uhlmann holonomies of interfering beams and show that the interferometric phase generalized to the case of nonunitary evolution can, similar to the Uhlmann phase, be cast into the holonomy defined form. By using the technique based on a two-arm Mach-Zehnder interferometer, two different dynamical regimes of light modulation are experimentally studied: (a) modulation of the input light by the rotating quarter-wave plate (QWP); and (b) modulation of the testing beam by a birefringent plate with electrically controlled anisotropy represented by the deformed-helix ferroelectric liquid crystal (DHFLC) cell. In the setup with the rotating QWP, the interferometric phase is found to be equal to the relative Uhlmann phase. Experimental and theoretical results being in excellent agreement both show that this phase is an oscillating function of the QWP angle and increases with the degree of polarization. For modulation by the DHFLC cell, the data derived from our electro-optic measurements are fitted using the theory of the orientational Kerr effect in FLCs. This theory in combination with the results of fitting is used to evaluate electric field dependencies of the interferometric and the Uhlmann phases.

Thermal Uhlmann-Chern number from the Uhlmann connection for extracting topological properties of mixed states

The Berry phase is a geometric phase of a pure state when the system is adiabatically transported along a loop in its parameter space. The concept of geometric phase has been generalized to mixed states by the so called Uhlmann phase. However, the Uhlmann phase is constructed from the Uhlmann connection that possesses a well defined global section. This property implies that the Uhlmann connection is topologically trivial and as a consequence, the corresponding Chern character vanishes. We propose modified Chern character whose integral gives the thermal Uhlmann Chern number, which is related to the winding number of the mapping defined by the Hamiltonian. Therefore, the thermal Uhlmann Chern number reflects the topological properties of the underlying Hamiltonian of a mixed state. By including the temperature dependence in the volume integral, we also introduce the non-topological thermal Uhlmann Chern number which varies with temperature but is not quantized at finite temperatures. We illustrate the applications to a two-band model and a degenerate four-band model.

Topological footprints of the Kitaev chain with long-range superconducting pairings at a finite temperature

We study the one-dimensional Kitaev chain with long-range superconductive pairing terms at a finite temperature where the system is prepared in a mixed state in equilibrium with a heat reservoir maintained at a constant temperature $T$. In order to probe the footprint of the ground-state topological behavior of the model at finite temperature, we look at two global quantities extracted out of two geometrical constructions: the Uhlmann and the interferometric phase. Interestingly, when the long-range effect dominates, the Uhlmann phase approach fails to reproduce the topological aspects of the model in the pure-state limit; on the other hand, the interferometric phase which has a proper pure state reduction, shows a behavior independent of the ambient temperature.

Observation of topological Uhlmann phases with superconducting qubits

Topological insulators and superconductors at finite temperature can be characterized by the topological Uhlmann phase. However, a direct experimental measurement of this invariant has remained elusive in condensed matter systems. Here, we report a measurement of the topological Uhlmann phase for a topological insulator simulated by a system of entangled qubits in the IBM Quantum Experience platform. By making use of ancilla states, otherwise unobservable phases carrying topological information about the system become accessible, enabling the experimental determination of a complete phase diagram including environmental effects. We employ a state-independent measurement protocol which does not involve prior knowledge of the system state. The proposed measurement scheme is extensible to interacting particles and topological models with a large number of bands.