Viyuela, O.Rivas, A.Martín-Delgado Alcántara, Miguel Ángel2023-06-192023-06-192014-04-020031-900710.1103/PhysRevLett.112.130401https://hdl.handle.net/20.500.14352/35600© 2014 American Physical Society. We are grateful to the anonymous referee and to Z. Huang for pointing out a correction in a previous version of the manuscript. We thank the Spanish MINECO Grants No. FIS2012-33152 and No. FIS2009-10061, CAM research consortium QUITEMAD S2009-ESP-1594, European Commission PICC: FP7 2007-2013, Grant No. 249958, UCM-BS Grant No. GICC-910758, FPU MEC Grant and Residencia de Estudiantes.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.engjournal articlehttp://dx.doi.org/10.1103/PhysRevLett.112.130401https://journals.aps.orgopen access53Parallel transportMixed statesInsulatorsSupercondunctorsBandsFísica-Modelos matemáticos