As pointed out in Ref., the Berry phase has a profound geometrical origin because an adiabatic and cyclic process of a quantum state is mathematically equivalent to parallel transporting it along a loop, which connects to the concept of holonomy in geometry. Hence, the Berry phase bridges physics and geometry, making it extremely important in the understanding of topological phenomena, such as integer quantum Hall effect, topological insulators and superconductors, and others. The description of the Berry phase relies on the properties of a pure state of a quantum systems at zero temperature. Meanwhile, mixed quantum states, including thermal state at finite temperatures, are more common. Therefore, mixed-state generalizations of the Berry phase have been an important task. Uhlmann made a breakthrough by constructing the Uhlmann connection for exploring the topology of finite-temperature systems. As the Berry holonomy arises from parallel transport of a state-vector along a closed path, the Uhlmann holonomy is generated by parallel-transporting the amplitude of a density matrix. defined by W = √ρU. Here the amplitude W is the mixed-state counterpart of the wavefunction, and U is a phase factor. A geometrical phase is deduced from the initial and final amplitudes. However, Uhlmann’s definition of parallel transport is rather abstract and may involve non-unitary processes, complicating a direct and clear physical interpretation. Moreover, the fiber bundle built upon Uhlmann’s formalism is trivial, which severely restricts its applications in physical systems.Purification of a mixed state leads to purified state, round flower buckets a state-vector equivalent to the amplitude of a density matrix. The lack of a one-to-one correspondence between the density matrix and its purified states gives rise to a phase factor, similar to the phase of a wave function.
In a branch of quantum field theory called thermal field theory, there is a similar structure for describing the thermal-equilibrium state of a system by constructing the corresponding thermal vacuum by duplicating the system state as an ancilla and forming a composite state. It plays a crucial role in the formalism of traversable wormholes induced by the holographic correspondence between a quantum field theory and a gravitational theory of one higher dimensions. Importantly, purified states of a two level system has been demonstrated on the IBM quantum computer while the thermal vacuum of a transverse field Ising model in its approximate form has been realized on a trapped-ion quantum computer. Despite the superficial similarity, a major difference between a thermal vacuum and a purified state is a partial transposition of the ancilla to ensure the Hilbert-Schmidt product is well defined. In quantum information theory, a partial transposition is closely related to entanglement of mixed states. Importantly, partial transpositions of composite systems have been approximately realized in experiments by utilizing structural physical approximations in suitable quantum computing platforms. Although ordinary observables cannot discern the partial transposition between the purified state and thermal vacuum, here we will show that at least two types of generalizations of the Berry phase to mixed states are capable of differentiating the two representations of finite temperature systems. Among many attempts to generalize the Berry phase or related geometric concepts to mixed states , a frequently mentioned approach was proposed in Ref.. Instead of decomposing the density matrix to obtain a matrix-valued phase factor, a geometrical phase is di-rectly assigned to a mixed state after parallel transport by an analogue of the optical process of the MachZehnder interferometer. Hence, the geometrical phase generated in this way is often referred to as the interferometric phase. The interferometric phase has been generalized to nonunitary processes, but the transformations are still on the system only. Moreover, it is essentially different from Uhlmann’s theory since the conceptual structure of holonomy is not incorporated.
We will first derive a mixed-state generalization of the parallel-transport condition for generalizing the Berry phase without invoking holonomy. This approach unifies the necessary condition for both the interferometric phase and Uhlmann phase . Two ways to implement the parallel-transport condition based on how the system of interest undergoes adiabatic evolution will be introduced, and they lead to different generalizations of the Berry phase. We will name one thermal Berry phase and the other generalized Berry phase. Importantly, the partial transposition of the ancilla between a purified state and thermal vacuum will be shown to produces observable geometrical effects in both thermal Berry phase and generalized Berry phase. Through explicit examples, the two generalized phases are shown to differentiate the two finite-temperature representations, a task beyond the capability of the conventional interferometric phase or Uhlmann phase. The rest of the paper is organized as follows. Sec. II summarizes the Berry phase in a geometrical framework with an introduction of the parallel-transport condition for pure quantum states. In Sec. III, we review the representations of mixed states via purified states and thermal vacua and then explain the difference of the partial transposition of the ancilla. In Sec. IV, we introduce the thermal Berry phase via generalized adiabatic processes. While the thermal Berry phase can differentiate a purified state from a thermal vacuum, it may contain non-geometrical contributions. In Sec. V, we generalize the parallel-transport condition to involve the system and ancilla and derive the general Berry phase according to the generalized condition. While the generalized Berry phase only carries geometrical information, its ability of differentiating a purified state from a thermal vacuum depends on the setup and protocol. We present examples of the thermal and generalized Berry phases. Sec. VI concludes our study. Some details and derivations are given in the Appendix.While purified states of a two-level system incorporating environmental effects have been simulated on the IBM quantum platform, thermal vacua of the transverse Ising model has been experimentally realized on an ion-trap quantum computer by the quantum approximate optimization algorithm. Moreover, partial transposition of a composite system has been approximately realized on quantum computers with various numbers of qubits. Therefore, a comparison of the geometric effects reflected by the generalizations of the Berry phase of purified states or thermal vacua is expected to be achievable in future experiments on quantum computers or quantum simulators. For example, one may consider two identical composite quantum systems of Example V.1 of the generalized Berry phase and then apply a partial transposition to one of the composite systems. As a consequence, the composite system with a partial transposition corresponds to a purified state while the one without partial transposition may be viewed as a thermal vacuum. By applying parallel transport that involves the ancilla to both composite systems and extract their generalized Berry phase after a cycle, a π-phase difference is expected between the two composite systems. Given the large phase difference between them after a cycle, the result is robust against small perturbations or noise from the hardware and offers another demonstration of geometrical protection of information. We have presented two generalizations of the Berry phase, the thermal Berry phase and generalized Berry phase, for distinguishing the two state-vector representations of mixed states via the purified state and thermal vacuum. From the geometrical and physical points of view, the generalized Berry phase has more desirable properties since the thermal Berry phase is generated by a temperature-dependent thermal Hamiltonian and may carry non-geometrical information. We caution that while the transformations can be on the system, ancilla, or both in the construction of the generalized Berry phase, an operation on the ancilla is necessary if we want to differentiate the purified state and thermal vacuum.The earliest classification of the forms of matter we see around us, typically presented to us in our early school days, consists of solids, liquids and gases.
High school physics textbooks and experience later teach us that solids can be further classified based on their electronic properties as conductors and insulators. Solid state physics courses in college add semiconductors, semimetals and superconductors to that list, plastic flower buckets wholesale and explain the basic physics that governs the electronic properties of these phases. More precisely, as long as the electrons in a solid are non-interacting, solids with partially filled bands are shown to be metals or conductors while those with no partially filled bands and a gap between the valence and the conduction bands are insulators or semiconductors. If the gap is extremely small or vanishing or if there is a very small overlap between the valence and the conduction bands, the material is semimetallic. Superconductors are argued to be the fate of a metal cooled to extremely low temperatures. However, vast quantitative as well as qualitative differences between the properties of materials within a single category are often observed. For instance, some insulators have conducting surfaces while others do not. Moreover, the surface conduction is stable against perturbations as well as deformations in the band structure as long as the underlying symmetries of the system are preserved and the system remains insulating after the deformations. Similarly, interactions and certain perturbations can gap out some semimetals and turn them insulating but not others . These observations call for further refinement in the classification of solids, especially one that explains why the properties of some phases are robust to certain deformations, interactions and perturbations. In other words, an understanding for why some phases are topological while other are not, is required. The study of the topology of the bands provides a powerful unifying framework for accomplishing this task. Topology shows up in two distinct ways in the band structures of non-interacting Hamiltonians depending on whether the spectrum is gapped or gapless. In systems with gapped band structures, which include insulators and most superconductors, the wavefunctions wind non-trivially across the Brillouin zone in the topological phases, as described in Sec. 1.1. What constitutes a non-trivial winding depends strongly on the symmetries and the dimensionality of the system under consideration, thus, revealing a rich substrucutre within the insulating and superconducting phases. On the other hand, gapless band structures contain topological objects in momentum space which can be characterized by the winding of theGreen’s function around the object, as exemplified in Sec. 1.2. A key feature of topological media is that they typically have unconventional surface states which, in many cases, cannot exist independently of the bulk phase in one lower dimension. For instance, the two dimensional surface of a three dimensional topological insulator hosts pseudorelativistic electrons and cannot exist as an independent two dimensional system. Because of this intimate surface-bulk connection and because the surface is usually more accessible than the bulk, experiments usually probe the surface states in order to identify the bulk topological phase. The surface states are also extremely valuable from a practical point of view, since the unconventional properties bestowed upon them by the non-trivial bulk topology may be exploited to design novel electronic devices.The role topology plays in the band structures of gapped non-interacting Hamiltonians is analogous to its manifestation in an early example of topology in physics – Gauss’s law in electrostatics. Gauss’s law states that the total electric flux piercing a closed surface is determined only by the charge enclosed by it and is independent of its shape or the precise charge distribution. Thus, the total flux through each surface in Figure 1.1.1 is four units. For gapped noninteracting lattice Hamiltonians, the analog of the Gaussian surface is the Brillouin zone, and different Gaussian surfaces correspond to different Brillouin zones in the extended zone scheme or to different bands in the reduced zone scheme. The electric charge, then, maps to an appropriate topological invariant whose exact form depends on the symmetries of the system under consideration. This topological invariant can be written as an integral over the Brillouin zone of an appropriate field derived from the Bloch functions of the occupied bands, analogous to how the electric charge enclosed by a Gaussian surface is equal to an integral of the electric field over it. The analogy is generalizable to continuum Hamiltonians, defined over all of momentum space, as well, provided the points at infinity are identified. This identification compactifies momentum space to the topological equivalent of a sphere, which, being a closed surface, permits the application of Gauss’s law. Next, assuming every charge that exists in the universe is inside a Gaussian surface, the only way to change the amount of charge enclosed by a single surface is by fusing it with another Gaussian surface, moving charges across the junction and pinching the junction ofto get the original surfaces back, as illustrated in Figure 1.1.2. Similarly, for band structures, topological invariants can only be defined for Brillouin zones wave functions in the extended zone scheme or equivalently, for bands in the more common reduced zone scheme.