  We analyze the topological deformations of a spin-1/2 in an effective
magnetic field induced by an ohmic quantum dissipative environment at zero
temperature. From Bethe Ansatz results and a variational approach, we confirm
that the Chern number is preserved in the delocalized phase for $\alpha<1$. We
report a divergence of the Berry curvature at the equator when $\alpha_c=1$
that appears at the localization Kosterlitz-Thouless quantum phase transition
in this model. Recent experiments in quantum circuits have engineered
non-equilibrium protocols in time to access topological properties at
equilibrium from the measure of the (quasi-)adiabatic out-of-equilibrium spin
expectation values. Applying a numerically exact stochastic Schr\"{o}dinger
equation we find that, for a fixed sweep velocity, the bath induces a crossover
from (quasi-)adiabatic to non-adiabatic dynamical behavior when the spin bath
coupling increases. We also investigate the particular regime $H/\omega_c \ll
v/H \ll 1$, where the dynamical Chern number observable built from
out-of-equilibrium spin expectation values vanishes at $\alpha=1/2$. In this
regime, the mapping to an interacting resonance level model enables us to
characterize the evolution of the dynamical Chern number in the vicinity of
$\alpha=1/2$. Then, we provide an intuitive physical explanation of the
breakdown of adiabaticity in analogy to the Faraday effect in electromagnetism.
We demonstrate that the driving of the spin leads to the production of a large
number of bosonic excitations in the bath, which in return strongly affect the
spin dynamics. Finally, we quantify the spin-bath entanglement and build an
analogy with an effective model at thermal equilibrium.
