Density Matrix Topological Insulators
In the first part of this talk, I will present recent results regarding the stability of fermionic topological orders in the presence of dissipation. Particularly, we address the behavior of topological insulators in the presence of thermal baths. These systems present non-vanishing topological conductivity at zero temperature, as their conduction and valence bands are connected by the so-called (topologically protected) edge states. I shall explain that, in general, these edge states are no longer protected when the system is in contact with a thermal bath. However, for some kind of environments, it is possible to obtain and characterize topologically ordered phases even in the presence of thermal dissipation. We will illustrate both results with examples for models with gauge symmetries: the Creutz Ladder in 1D and the Haldane model in 2D. In the second part of this talk, I will show how to 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 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.