  We compute the geometric phase for a spin-1/2 particle under the presence of
a composite environment, composed of an external bath (modeled by an infinite
set of harmonic oscillators) and another spin-1/2 particle. We consider both
cases: an initial entanglement between the spin-1/2 particles and an initial
product state in order to see if the initial entanglement has an enhancement
effect on the geometric phase of one of the spins. We follow the nonunitary
evolution of the reduced density matrix and evaluate the geometric phase for a
single two-level system. We also show that the initial entanglement enhances
the sturdiness of the geometric phase under the presence of an external
composite environment.
