Two-Dimensional Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers

Abstract

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 nU. 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. Although these studies provide a successful picture for quantum systems in pure states, typically the ground state, very little is known about the fate of those topological phases of matter when the system is in a mixed quantum state represented by a density matrix. In fact, the correct understanding of this situation becomes particularly relevant in order to address unavoidable thermal effects on topological phases and nonequilibrium dynamics under dissipation [19].Over the last few years, there have been some results establishing the absence of stable topological phases subject to thermal effects, like for TOs with spins [20][21][22], for SP-TOs with spins [23], or SP-TOs with fermions under certain conditions [24]. However, in a recent work [25] we have shown that it is possible to characterize a thermal topological phase for topological insulators and superconductors in one-dimensional systems. This thermal topological phase, classified by Uhlmann holonomies [26,27], is separated from a trivial phase by a critical finite temperature, at which the topological phase abruptly disappears. This result paves the way towards the characterization of SP-TOs with fermions in thermal states, or more general density matrices.In this paper, we have achieved this goal for twodimensional fermion systems, either insulating or superconducting. This extension from one to two-dimensional systems is nontrivial in the sense that a direct generalization of the topological invariants for pure states (Chern numbers) to density matrices via the Uhlmann approach leads to trivial results. However, we have succeeded in circumventing this problem and introducing a suitable notion of topological Uhlmann numbers. These are gauge invariant and observable quantities which allows for a classification of topological phases of density matrices in two-dimensional (2D) quantum systems. Specifically, we

Topology of density matrices

“…In Section III, topological invariants for density matrices with various spectral assumptions are defined. We compare the present approach to the construction of Uhlmann phase winding numbers that have recently been proposed [24,25] to classify thermal Chern insulators in Section IV. Concluding remarks are presented in Section V.…”

“…As both k x in the BZ and the phase φ kx U are defined on a circle, the mapping k x → φ kx U is characterized by an integer quantized winding number which is exactly measured by Eq. (25).…”

Majorana Fermions and Topology in Superconductors

“…Recently, there have been some advances in this direction. [243][244][245][246][247][248][249][250][251][252][253] The effects of superconducting fluctuations on Majorana fermion states have also been studied recently. [254][255][256][257] It was revealed that the topological degeneracy associated with Majorana fermions is maintained even for the case of quasi-long-range order with power-law decay of the superconducting correlation, for which the total electron charge is conserved, and the phase of the superconducting gap fluctuates.…”

Cellular dynamical mean-field theory study of an interacting topological honeycomb lattice model at finite temperature

“…Recent theoretical work also shows that there is a precise sense in which finite temperature topological phases retain a quantized invariant. 64,65 With the caveats mentioned earlier regarding the magnetic order in the strong coupling regime, one may discuss a finite temperature phase diagram for our model that includes both topological phases and magnetic phases.…”

“…Thus, it is highly desirable to investigate the effect of interactions on topological systems, particularly at finite temperature, which is important and relevant to real materials. Recent work has shown that a finite-temperature topological invariant can be defined 64,65 , which effectively extends the notation of topological phases to finite temperatures.…”