Thermal Uhlmann phase in a locally driven two-spin system

Abstract: We study the geometric Uhlmann phase of mixed states at finite temperature in a system of two coupled spin-$\frac 1 2$ particles driven by a magnetic field applied to one of the spins. In the parameter space of temperature and coupling, we show the emergence of two topological Uhlmann phase transitions when the magnetic field evolves around the equator, where a winding number can characterize each temperature range. For small couplings, the width of the temperature gap of the non-trivial phase is roughly the critical temperature $T_c$ of one-dimensional fermion systems with two-band Hamiltonians. The first phase transition in the low-temperature regime and small values of the coupling corresponds to the peak of the \textit{Schottky anomaly} of the heat capacity, typical of a two-level system in solid-state physics involving the ground and first excited states. The second phase transition occurs at temperatures very close to the second maximum of the heat capacity associated with a multilevel system. We also derive analytical expressions for the thermal Uhlmann phase for both subsystems, showing that they exhibit phase transitions. In the driven subsystem, for minimal $g$, a topological phase transition phase appears at $T_c$ again. However, for larger values of $g$, the transitions occur at lower temperature values, and they disappear when the coupling reaches a critical value $g_c$. The latter is not the case for the undriven subsystem, where at low temperatures, a single phase transition occurs at $g_c$. Nevertheless, as the temperature rises, we demonstrate the emergence of two phase transitions defining a coupling gap, where the phase is non-trivial and vanishes as the temperature reaches a critical value.