  The Uhlmann process is built on the density matrix of a mixed quantum state
and offers a way to characterize topological properties at finite temperatures.
We analyze an ideal spin-j quantum paramagnet in a magnetic field undergoing an
Uhlmann process and derive general formulae of the Uhlmann phase and Loschmidt
amplitude for arbitrary $j$ as the system traverses a great circle in the
parameter space. A quantized jump of the Uhlmann phase signifies a topological
quantum phase transition (TQPT) of the underlying process, which is accompanied
by a zero of the Loschmidt amplitude. The exact results of j=1/2 and j=1
systems show topological regimes that only survive at finite temperatures but
not at zero temperature, and the number of TQPTs is associated with the winding
number in the parameter space. Our results pave the way for future studies on
finite-temperature topological properties, and possible experimental protocols
and implications are discussed.
