Viyuela García, ÓscarRivas Vargas, ÁngelMartin-Delgado Alcántara, Miguel Ángel2023-06-192023-06-192014-08-130031-900710.1103/PhysRevLett.113.076408https://hdl.handle.net/20.500.14352/35592© 2014 American Physical Society. We are thankful for the following: the Spanish MINECO Grants No. FIS2012-33152 and No. FIS2009-10061, the CAM research consortium QUITEMAD S2009-ESP-1594, the European Commission PICC: FP7 2007-2013, Grant No. 249958, the UCM-BS Grant No. GICC-910758, and the FPU MEC Grant and Residencia de Estudiantes.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.engjournal articlehttp://dx.doi.org/10.1103/PhysRevLett.113.076408https://journals.aps.orgopen access53SuperconductorsStatesInsulatorscomputationDegeneracyStatisticsVorticesParitySpaceModel.Física-Modelos matemáticos