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

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.