Uhlmann phase as a topological measure for one-dimensional fermion systems.

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.

Two-dimensional density-matrix topological fermionic phases: topological Uhlmann numbers.

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.

Topological phases in nodeless tetragonal superconductors.

We compute the topological phase diagram of 2D tetragonal superconductors for the only possible nodeless pairing channels compatible with that crystal symmetry. Subject to a Zeeman field and spin-orbit coupling, we demonstrate that these superconductors show surprising topological features: non-trivial high Chern numbers, massive edge states, and zero-energy modes out of high symmetry points, even though the edge states remain topologically protected. Interestingly, one of these pairing symmetries, [Formula: see text], has been proposed to describe materials such as water-intercalated sodium cobaltates, bilayer silicene or highly doped monolayer graphene, which opens the way for further applications of our results.