  The generalization of the geometric phase to the realm of mixed states is
known as Uhlmann phase. Recently, applications of this concept to the field of
topological insulators have been made and an experimental observation of a
characteristic critical temperature at which the topological Uhlmann phase
disappears has also been reported. Surprisingly, to our knowledge, the Uhlmann
phase of such a paradigmatic system as the spin-$j$ particle in presence of a
slowly rotating magnetic field has not been reported to date. Here we study the
case of such a system in a thermal ensemble. We find that the Uhlmann phase is
given by the argument of a complex valued second kind Chebyshev polynomial of
order $2j$. Correspondingly, the Uhlmann phase displays $2j$ singularities,
occurying at the roots of such polynomials which define critical temperatures
at which the system undergoes topological order transitions. Appealing to the
argument principle of complex analysis each topological order is characterized
by a winding number, which happen to be $2j$ for the ground state and decrease
by unity each time increasing temperature passes through a critical value. We
hope this study encourages experimental verification of this phenomenon of
thermal control of topological properties, as has already been done for the
spin-$1/2$ particle.
