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arXiv:1405.4567 · Physical Review Letters

Topological Indices for Open and Thermal Systems Via Uhlmann’s Phase

Zhoushen Huang, Daniel P. Arovas

Published Aug 13, 2014• 102 citations
arXivHTMLPDF

Abstract

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 $T$ or dephasing Lindbladian. This scheme reduces to the familiar Thouless-Kohmoto-Nightingale--den Nijs classification for $T<{T}_{\mathrm{c},1}$, becomes trivial for $T>{T}_{\mathrm{c},2}$, 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.

DOI
10.1103/physrevlett.113.076407

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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.
Topological properties of quantum systems at finite temperatures, described by mixed states, pose significant challenges due to the triviality of the Uhlmann bundle. By inserting the density matrix into the Chern character, we introduce the thermal Uhlmann-Chern number, a generalization of the Chern number that reduces to the pure-state value in the zero-temperature limit and vanishes at infinite temperature, providing a framework to study the temperature-dependent evolution of topological features in mixed states. We provide, for the first time, a rigorous mathematical proof that the first- and higher-order Uhlmann-Chern numbers converge to the corresponding Chern numbers in the zero-temperature limit, differing only by a factor of 1/D1/D1/D for DDD-fold degenerate ground states. We demonstrate the utility of this framework through applications to a two-level system, the coherent state model, the 2D Haldane model, and a four-band model, highlighting the temperature-dependent behavior of topological invariants. Our results establish a robust bridge between the topological properties of pure and mixed states, offering new insights into finite-temperature topological phases.
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 a 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 nontopological 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.
We construct a topological invariant that classifies density matrices of symmetry-protected topological orders in 2D fermionic systems. As it is constructed out of the previously introduced Uhlmann phase, we refer to it as the topological Uhlmann number nU{\rm n}_{\rm U}nU​. With it, we study thermal topological phases in several 2D models of topological insulators and superconductors, computing phase diagrams where the temperature TTT is on equal footing with the coupling constants in the Hamiltonian. Moreover, we find novel thermal-topological transitions between two non-trivial phases in a model with high Chern numbers. At small temperature we recover the standard topological phases as the Uhlmann number approaches to the Chern number.
We construct a topological invariant that classifies density matrices of symmetry-protected topological orders in two-dimensional fermionic systems. As it is constructed out of the previously introduced Uhlmann phase, we refer to it as the topological Uhlmann number n_{U}. With it, we study thermal topological phases in several two-dimensional models of topological insulators and superconductors, computing phase diagrams where the temperature T is on an equal footing with the coupling constants in the Hamiltonian. Moreover, we find novel thermal-topological transitions between two nontrivial phases in a model with high Chern numbers. At small temperatures we recover the standard topological phases as the Uhlmann number approaches to the Chern number.
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.
The discovery of integer quantum Hall effect and its subsequent theoretical formulation heralded a new paradigm of thinking in condensed matter physics, which has by now blossomed into the rapidly growing field of topological phases. In this work we investigate several mutually related topics in the framework of topological band theory. In Chapter 2, we study solutions to boundary states on a lattice and see how they are related to the bulk topology. To elicit a real space manifestation of the non-trivial topology, the presence of a physical edge is not strictly necessary. We study two other possibilities, namely the entanglement spectrum associated with an imaginary spatial boundary, and the localization centers of Wannier functions, in Chapters 3,4, and 5. Topological classification through discrete indices is so far possible only for systems described by pure quantum states - in the existing scheme, quantization is lost for systems in mixed states. In Chapter 6, we present a program through which discrete topological indices can be defined for topological band systems at finite temperature, based on Uhlmann's parallel transport of density matrices. The potential of topologocal insulators in realistic applications lies in the existence of Dirac nodes on its surface spectrum. Dirac physics, however, is not exclusive to TI surfaces. In a recently discovered class of materials known as Weyl semimetals, energy nodes which emit linear dispersions also occur in the bulk material. In Chapter 7, we study the possibility of resonance states induced by localized impurities near the nodal energy in Weyl semimetals, which will help us in understanding the stability of density-of-state suppression at the energy nodes. Finally, in Chapter 8, we apply the topological characterization developed for noninteracting particles to a class of interacting spin models in 3D, which are generalizations of Kitaev's honeycomb model, and identify several exotic quantum phases such as spin metals and spin semimetals