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2D Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers

Oscar Viyuela, ́Angel Rivas, Miguel Angel Martin-Delgado

Published May 23, 2014• 0 citations
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Abstract

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 ${\rm n}_{\rm U}$. With it, we study thermal topological phases in several 2D models of topological insulators and superconductors, computing phase diagrams where the temperature $T$ 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.

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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.
We have applied the recently developed theory of topological Uhlmann numbers to a representative model of a topological insulator in two dimensions, the Qi–Wu–Zhang model. We have found a stable symmetry-protected topological phase under external thermal fluctuations in two dimensions. A complete phase diagram for this model is computed as a function of temperature and coupling constants in the original Hamiltonian. It shows the appearance of large stable phases of matter with topological properties compatible with thermal fluctuations or external noise and the existence of critical lines separating abruptly trivial phases from topological phases. These novel critical temperatures represent thermal topological phase transitions. The initial part of the paper comprises a self-contained explanation of the Uhlmann geometric phase needed to understand the topological properties that it may acquire when applied to topological insulators and superconductors.
We have applied the recently developed theory of topological Uhlmann numbers to a representative model of a topological insulator in two dimensions, the Qi-Wu-Zhang model. We have found a stable symmetry-protected topological phase under external thermal fluctuations in two dimensions. A complete phase diagram for this model is computed as a function of temperature and coupling constants in the original Hamiltonian. It shows the appearance of large stable phases of matter with topological properties compatible with thermal fluctuations or external noise and the existence of critical lines separating abruptly trivial phases from topological phases. These novel critical temperatures represent thermal topological phase transitions. The initial part of the paper comprises a self-contained explanation of the Uhlmann geometric phase needed to understand the topological properties that it may acquire when applied to topological insulators and superconductors.
We introduce the Uhlmann geometric phase as a tool to characterize symmetry-protected topological phases in one-dimensional fermion systems, such as topological insulators and superconductors. Since this phase is formulated for general mixed quantum states, it provides a way to extend topological properties to finite temperature situations. We illustrate these ideas with some paradigmatic models and find that there exists a critical temperature Tc at which the Uhlmann phase goes discontinuously and abruptly to zero. This stands as a borderline between two different topological phases as a function of the temperature. Furthermore, at small temperatures we recover the usual notion of topological phase in fermion systems.
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.