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arXiv:2112.15567 · Physical review. B./Physical review. B

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

Aixin Pi, Ye Zhang, Yan He, Chih-Chun Chien

Published Feb 22, 2022• 5 citations
arXivHTMLPDF

Abstract

The ensemble geometric phase (EGP) has been proposed as a topological indicator for finite-temperature quantum 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.

DOI
10.1103/physrevb.105.085418

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The ensemble geometric phase (EGP) has been proposed as a topological indicator for finite-temperature quantum 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.
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 ground states of noninteracting fermions in one-dimension with chiral symmetry form a class of topological band insulators, described by a topological invariant that can be related to the Zak phase. Recently, a generalization of this quantity to open systems---known as the ensemble geometric phase (EGP)---has emerged as a robust way to describe topology at nonzero temperature. By using this quantity, we explore the nature of topology allowed for dissipation beyond a Lindblad description, to allow for coupling to external baths at finite temperatures. We introduce two main aspects to the theory of open-system topology. First, we discover topological phase transitions as a function of the temperature TTT, manifesting as changes in differences of the EGP accumulated over a closed loop in parameter space. We characterize the nature of these transitions and reveal that the corresponding nonequilibrium steady state can exhibit a nontrivial structure---contrary to previous studies where it was found to be in a fully mixed state. Second, we demonstrate that the EGP itself becomes quantized when key symmetries are present, allowing it to be viewed as a topological marker which can undergo equilibrium topological transitions at nonzero temperatures.
The topological classification of fermion systems in mixed states is a long-standing quest. For Gaussian states, reminiscent of noninteracting unitary fermions, some progress has been made. While the topological quantization of certain observables such as the Hall conductivity is lost for mixed states, directly observable many-body correlators exist which preserve the quantized nature and naturally connect to known topological invariants in the ground state. For systems that break time-reversal (TR) symmetry, the ensemble geometric phase was identified as such an observable which can be used to define a Chern number in (1+1)(1+1)(1+1) and two dimensions. Here we propose a corresponding Z2{\mathbb{Z}}_{2}Z2​ topological invariant for systems with TR symmetry. We show that this mixed-state invariant is identical to well-known Z2{\mathbb{Z}}_{2}Z2​ invariants for the ground state of the so-called fictitious Hamiltonian, which for thermal states is just the ground state of the system Hamiltonian itself. We illustrate our findings for finite-temperature states of a paradigmatic Z2{\mathbb{Z}}_{2}Z2​ topological insulator, the Kane-Mele model.
Two-dimensional topological phases are characterized by Thouless-Kohmoto-Nightingale--den Nijs integers, which classify Bloch energy bands or groups of Bloch bands. However, quantization does not survive thermal averaging or dephasing to mixed states. We show that using Uhlmann's parallel transport for density matrices [Rep. Math. Phys. 24, 229 (1986)], an integer classification of topological phases can be defined for a finite generalized temperature TTT or dephasing Lindbladian. This scheme reduces to the familiar Thouless-Kohmoto-Nightingale--den Nijs classification for T<Tc,1T<{T}_{\mathrm{c},1}T<Tc,1​, becomes trivial for T>Tc,2T>{T}_{\mathrm{c},2}T>Tc,2​, and exhibits a ``gapless'' intermediate regime where topological indices are not well defined. We demonstrate these ideas in detail, applying them to Haldane's honeycomb lattice model and the Bernevig-Hughes-Zhang model, and we comment on their generalization to multiband Chern insulators.
For a long time, it was believed that the Ginzburg-Landau formalism was able to classify all different types of phase transitions. This view changed with the discovery of the quantum Hall effect and topological insulators. Topological insulators (superconductors) are materials that host symmetry-protected metallic edge states in an insulating (superconducting) bulk. Although topological insulators are well understood by now, a thermodynamic description of their behavior remained elusive, firstly because the edges are lost in the thermodynamic limit, when the system size goes to infinity, and secondly because topological quantum field theories involve non-local order parameters, and hence cannot be captured by the conventional Ginzburg-Landau formalism. Recently, this challenge has been overcome: by using Hill thermodynamics to describe the paradigmatic Bernevig-Hughes-Zhang model in two dimensions, it has been shown that at the topological phase change the thermodynamic potential does not scale extensively due to the boundary effects.\n\nHere, we extend this thermodynamic approach firstly to the Kitaev chain in one-dimension and then to different topological models in various dimensions (the 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, we find that all these models exhibit the same universal behavior in the order of the topological-phase transition, which depends on the dimensionality of the system. Next, we use this thermodynamic approach to describe topological phases at finite-temperatures. We calculate the entropy and heat capacity for the Kitaev chain and verify its behavior in comparison with conformal field theory. Furthermore, we extend the topological phase transition to finite-temperatures, by numerically deriving the topological phase diagram for the bulk of the system, and show that it displays a good agreement with the one calculated analytically from the Uhlmann phase. Finally, we calculate measurable quantities such as the density of states and the heat capacity for different topological insulators. Our work reveals unexpected universalities and opens the path to a thermodynamic description of systems with a non-local order parameter, provided that nonlinear terms are appropriately taken into account for the thermodynamic extensive variables.
We generalize the concept of topological invariants for mixed states based on the ensemble geometric phase (EGP) to two-dimensional band structures. In contrast to the geometric Uhlmann phase for density matrices, the EGP leads to a proper Chern number for Gaussian, finite-temperature, or nonequilibrium steady states. The Chern number can be expressed as an integral of the ground-state Berry curvature of a fictitious lattice Hamiltonian, constructed from single-particle correlations. For the Chern number to be nonzero the fictitious Hamiltonian has to break time-reversal symmetry.