  Geometric phase of an open quantum system that is interacting with a thermal
environment (bath) is studied through some simple examples. The system is
considered to be a simple spin-half particle which is weakly coupled to the
bath. It is seen that even in this regime the geometric phase can vary with
temperature. In addition, we also consider the system under an adiabatically
time-varying magnetic field which is weakly coupled to the bath. An important
feature of this model is that it reveals existence of a temperature-scale in
which adiabaticity condition is preserved and beyond which the geometric phase
is varying quite rapidly with temperature. This temperature is exactly the one
in which the geometric phase vanishes. This analysis has some implications in
realistic implementations of geometric quantum computation.
