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

Topology of a dissipative spin: dynamical Chern number, bath induced non-adiabaticity and a quantum dynamo effect