Topology of a dissipative spin: Dynamical Chern number, bath-induced nonadiabaticity, and a quantum dynamo effect

We analyze the topological deformations of the ground state manifold of a quantum spin-1/2 in a magnetic field $\mathbit{H}=H(sin\ensuremath{\theta}cos\ensuremath{\phi},sin\ensuremath{\theta}sin\ensuremath{\phi},cos\ensuremath{\theta})$ induced by a coupling to an ohmic quantum dissipative environment at zero temperature. From Bethe ansatz results and a variational approach, we confirm that the Chern number associated with the geometry of the reduced spin ground state manifold is preserved in the delocalized phase for $\ensuremath{\alpha}<1$. We report a divergence of the Berry curvature at ${\ensuremath{\alpha}}_c=1$ for magnetic fields aligned along the equator $\ensuremath{\theta}=\ensuremath{\pi}/2$. This divergence is caused by the complete quenching of the transverse magnetic field by the bath associated with a gap closing that occurs at the localization Kosterlitz-Thouless quantum phase transition in this model. Recent experiments in quantum circuits have engineered nonequilibrium protocols to access topological properties from a measurement of a dynamical Chern number defined via the out-of-equilibrium spin expectation values. Applying a numerically exact stochastic Schr\"odinger approach we find that, for a fixed field sweep velocity $\ensuremath{\theta}(t)=vt$, the bath induces a crossover from (quasi)adiabatic to nonadiabatic dynamical behavior when the spin bath coupling $\ensuremath{\alpha}$ increases. We also investigate the particular regime $H/{\ensuremath{\omega}}_c\ensuremath{\ll}v/H\ensuremath{\ll}1$ with large bath cutoff frequency ${\ensuremath{\omega}}_c$, where the dynamical Chern number vanishes already at $\ensuremath{\alpha}=1/2$. In this regime, the mapping to an interacting resonance level model enables us to analytically describe the behavior of the dynamical Chern number in the vicinity of $\ensuremath{\alpha}=1/2$. We further provide an intuitive physical explanation of the bath-induced 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 strongly affect the spin dynamics. Finally, we quantify the spin-bath entanglement and formulate an analogy with an effective model at thermal equilibrium.