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arXiv:2205.05658 · Physical review. B./Physical review. B

Uhlmann holonomy against Lindblad dynamics of topological systems at finite temperatures

Yan He, Chih-Chun Chien

Published Jul 25, 2022• 6 citations
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Abstract

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.

DOI
10.1103/physrevb.106.024310

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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.
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.
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.
Topological insulators (superconductors) are materials that host symmetry-protected metallic edge states in an insulating (superconducting) bulk. Although they are well understood, a thermodynamic description of these materials remained elusive, firstly because the edges yield a non-extensive contribution to the thermodynamic potential, and secondly because topological field theories involve non-local order parameters, and cannot be captured by the Ginzburg-Landau formalism. Recently, this challenge has been overcome: by using Hill thermodynamics to describe the Bernevig-Hughes-Zhang model in two dimensions, it was shown that at the topological phase transition the thermodynamic potential does not scale extensively due to boundary effects. Here, we extend this approach to different topological models in various dimensions (the Kitaev chain and Su-Schrieffer-Heeger model in one dimension, the Kane-Mele model in two dimensions and the Bernevig-Hughes-Zhang model in three dimensions) at zero temperature. Surprisingly, all models exhibit the same universal behavior in the order of the topological-phase transition, depending on the dimension. Moreover, we derive the topological phase diagram at finite temperature using this thermodynamic description, and show that it displays a good agreement with the one calculated from the Uhlmann phase. Our work reveals unexpected universalities and opens the path to a thermodynamic description of systems with a non-local order parameter.
The Wilczek-Zee (WZ) holonomy arises in degenerate states while the Uhlmann holonomy characterizes finite-temperature topology. We investigate possible relationships between the Uhlmann phase and the scalar WZ phase, which reflects the Uhlmann and WZ holonomy respectively, in an exemplary four-level model with two doubly degenerate subspaces. Through some solvable exact solutions, we contrast the behavior of the Uhlmann and WZ connections and their associated phases. In the zero-temperature limit, the Uhlmann phase may or may not agree with the scalar WZ phase of the degenerate ground states due to obstructions from the Hamiltonian manifested as Dirac points. This is in stark contrast to nondegenerate systems where the correspondence between the Uhlmann and Berry phases in general holds. Our analyses further show that for the example studied here, the Uhlmann phase catches the singular behavior at the Dirac points while the WZ connection and scalar WZ phase vanish along a zero-field axis. We also briefly discuss possible experimental implications.
We study the time evolution of geometric phases of one-dimensional topological models under the quench dynamics. Taking the Creutz ladder model as an example, it is found that the Berry phase is fixed as the parameter is suddenly tuned across the topological phase boundary, given that the inversion symmetry of the model is preserved. At finite temperature, the Uhlmann phase displays abrupt jumps between the two quantized values, which indicates the topological transition at certain times after the quench. Both the Berry and Uhlmann phase will deviate from quantized values if the inversion symmetry if the model is broken.
Ultracold atomic quantum systems offer an unprecedentedly powerful platform for exploring interacting symmetry-protected topological phases. Here, we show that their characteristic control over temperature provides a resource for engineering effective topological regimes with no zero-temperature counterpart. Employing finite-temperature matrix-product-state calculations, we investigate experimentally accessible Hamiltonians - the interacting bosonic Su-Schrieffer-Heeger and spin-1 XXZ chains - that host a topological phase at zero temperature, as well as a fermionic dipolar ladder model whose ground state exhibits no topological order. By tracking complementary topological markers, we identify an intriguing mechanism rooted in the structure of the zero-temperature phase diagrams: in finite-size systems, excitation gaps with different magnitudes define an intermediate-temperature window where the correlations associated with the lower gap are suppressed, while those protected by the larger gap persist. This enables the emergence of effective finite-temperature regimes exhibiting topological features that are either absent or qualitatively different from those at zero temperature. Our results establish a novel route toward exploring effective topological phases in ultracold atomic quantum systems.