  Two indicators of finite-temperature topological properties based on the
Uhlmann connection, one generalizing the Wilson loop to the Uhlmann-Wilson loop
and the other generalizing the Berry phase to the Uhlmann phase, are
constructed explicitly for a time-reversal invariant topological insulators
with a $Z_2$ index. While the phases of the eigenvalues of the Wilson loop
reflect the $Z_2$ index of the model at zero temperature, it is found that the
signature from the Uhlmann-Wilson loop gradually fades away as temperature
increases. On the other hand, the Berry phase exhibits quantization due to the
underlying holonomy group. The Uhlmann phase retains the quantization at finite
temperatures and serves as an indicator of topological properties. A phase
diagram showing where jumps of the Uhlmann phase can be found is presented. By
modifying the model to allow higher winding numbers, finite-temperature
topological regimes sandwiched between trivial regimes at high and low
temperatures may emerge.
