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arXiv:2103.15340 · Physical Review A

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

Xu-Yang Hou, Hao Guo, Chih-Chun Chien

Published Aug 4, 2021• 22 citations
arXivHTMLPDF

Abstract

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 formulas 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 $j=\frac{1}{2}$ and $j=1$ systems show topological regimes that survive only 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.

DOI
10.1103/physreva.104.023303

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Bulletin of the American Physical Society•Mar 5, 2015•0 citations
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-jjj quantum paramagnet in a magnetic field undergoing an Uhlmann process and derive general formulas of the Uhlmann phase and Loschmidt amplitude for arbitrary jjj 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=12j=\frac{1}{2}j=21​ and j=1j=1j=1 systems show topological regimes that survive only 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.
The generalization of the geometric phase to the realm of mixed states is known as the 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. Here we study the case of the Uhlmann phase of a paradigmatic system such as the spin-jjj particle in the presence of a slowly rotating magnetic field at finite temperature in an exact analytical form. We find that the Uhlmann phase is given by the argument of a complex-valued second-kind Chebyshev polynomial of order 2j2j2j. Correspondingly, the Uhlmann phase displays 2j2j2j singularities, occurring at the roots of such polynomials which define the 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 happens to be 2j2j2j for the ground state and decreases by unity each time the 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-1/21/21/2 particle.
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.
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 Z2{Z}_{2}Z2​ index. While the phases of the eigenvalues of the Wilson loop reflect the Z2{Z}_{2}Z2​ 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.
We investigate the Uhlmann phase, a generalization of the celebrated Berry phase, for Bose-Einstein condensates (BECs) at finite temperature. The Uhlmann phase characterizes topological properties of mixed states, in contrast to the Berry phase, which is defined for pure states at zero temperature. Using the SU(1,1)\text{SU}(1,1)SU(1,1) symmetry of the Bogoliubov Hamiltonian, we derive a general formula for the Uhlmann phase of BECs. Numerical calculations reveal that the Uhlmann phase can differ from the Berry phase in the zero-temperature limit, contrary to previous studies. As the temperature increases, the Uhlmann phase exhibits a winding behavior, and we relate the total winding degree to the Berry phase. This winding indicates that the Uhlmann phase takes values on a Riemann surface. Furthermore, we propose an experimental scheme to measure the Uhlmann phase of BECs by purifying the density matrix using an atomic interferometer.
We study the geometric Uhlmann phase of mixed states at finite temperature in a system of two coupled spin-12\frac{1}{2}21​ particles driven by a magnetic field applied to one of the spins. In the parameter space of temperature and coupling, we show the emergence of two topological Uhlmann phase transitions when the magnetic field evolves around the equator, where a winding number can characterize each temperature range. For small couplings, the width of the temperature gap of the nontrivial phase is roughly the critical temperature Tc{T}_{c}Tc​ of one-dimensional fermion systems with two-band Hamiltonians. The first phase transition in the low-temperature regime and small values of the coupling corresponds to the peak of the Schottky anomaly of the heat capacity, typical of a two-level system in solid-state physics involving the ground and first excited states. The second phase transition occurs at temperatures very close to the second maximum of the heat capacity associated with a multilevel system. We also derive analytical expressions for the thermal Uhlmann phase for both subsystems, showing that they exhibit phase transitions. In the driven subsystem, for minimal ggg, a topological phase transition appears at Tc{T}_{c}Tc​ again. However, for larger values of ggg, the transitions occur at lower-temperature values, and they disappear when the coupling reaches a critical value gc{g}_{c}gc​. The latter is not the case for the undriven subsystem, where at low temperatures, a single phase transition occurs at gc{g}_{c}gc​. Nevertheless, as the temperature rises, we demonstrate the emergence of two phase transitions defining a coupling gap, where the phase is nontrivial and vanishes as the temperature reaches a critical value.
I will introduce the Uhlmann geometric phase as a tool to characterise density matrices of 1D and 2D topological insulators and superconductors. We achieve this goal by constructing new topological invariants called Topological Uhlmann numbers. Since this phase is formulated for general mixed quantum states, it provides a way to extend topological properties to finite temperature situations. New effects appear such as the existence of critical temperatures, novel thermal-topological transitions in models with high Chern numbers, breakdown of the usual bulk-edge correspondence, … Moreover, as the Uhlmann phase is an observable itself, we analyse potential measurement schemes that could be applicable to current experimental setups like cold atoms in optical lattices.