Finite-temperature topological phase and phase transitions of spin-j systems in Uhlmann processes

Topological properties of noninteracting systems in their ground states have been extensively studied. However, finite-temperature topological properties still await more research. The Uhlmann phase and Loschmidt amplitude have been shown to offer promising ways to characterize topological phase and phase transitions of mixed quantum states at finite-temperatures. We present general expressions of the Uhlmann phase and Loschmidt amplitude of a quantum spin-j system under a magnetic field in a Uhlmann process. Our analyses of the spin-1/2 and spin-1 systems show finite-temperature topological phase transitions (TQPTs) corresponding to the zeros of the Loschmidt amplitude. The Uhlmann phase jumps at the TQPT, and the number of TQPTs is determined by the winding number in the parameter space of the Uhlmann process. The Uhlmann processes of spin-j systems may be realized and studied in cold-atom quantum simulators.