Viyuela, O.Rivas, A.Martín-Delgado Alcántara, Miguel Ángel2023-06-182023-06-182015-092053-158310.1088/2053-1583/2/3/034006https://hdl.handle.net/20.500.14352/24334© 2015 IOP Publishing Ltd. We thank the Spanish MINECO grant FIS2012-33152, FIS2009-10061, CAM research consortium QUITEMAD+S2013/ICE-2801, European Commission PICC: FP7 2007-2013 grant No. 249958, UCM-BS grant GICC-910758, FPU MEC grant and Residencia de Estudiantes.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.engjournal articlehttp://dx.doi.org/10.1088/2053-1583/2/3/034006http://www.arxiv.orghttp://iopscience.iop.org/open access535Topological insulatorsTopological superconductorsThermal propertiesÓptica (Física)2209.19 Óptica Física