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 $\mathit{H}=H(sin\theta cos\varphi ,sin\theta sin\varphi ,cos\theta )$ 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 $\alpha <1$. We report a divergence of the Berry curvature at ${\alpha }_c=1$ for magnetic fields aligned along the equator $\theta =\pi /2$. 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 Schrödinger approach we find that, for a fixed field sweep velocity $\theta \left(t\right)=vt$, the bath induces a crossover from (quasi)adiabatic to nonadiabatic dynamical behavior when the spin bath coupling $\alpha$ increases. We also investigate the particular regime $H/{\omega }_c\ll v/H\ll 1$ with large bath cutoff frequency ${\omega }_c$, where the dynamical Chern number vanishes already at $\alpha =1/2$. 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 $\alpha =1/2$. 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.

Figure 1

(Left) Adiabatic sweep protocol of the magnetic field $\mathit{H}\left[\theta \right(t\left)\right]$ with linearly varying polar angle $\theta =v(t-t_0)$. Grey arrow denotes external magnetic field, magenta arrow the spin direction in the ground state. Red path is for a free spin (or weak spin-bath coupling), blue bath is for stronger spin-bath coupling which leads to renormalization of transvere field $\mathrm{\Delta }=Hsin\theta \to {\mathrm{\Delta }}_r$. (Three right panels) The red circles are parametrized by $(Hsin\theta ,H_0+Hcos\theta )$ with $H\ne 0$. We have $H_0/H=0$ (left), $0<H_0/H<1$ (middle), and $H_0/H>1$ (right) and the black dot shows the position of the origin in each case. The arrows show the orientation of the Bloch vector for each value of $\theta$. The vertical component is given by ${\langle {\sigma }^z\rangle }_{\mathrm{eq}}$ while the horizontal component is given by ${\langle {\sigma }^x\rangle }_{\mathrm{eq}}$.

Figure 2

(Top) Evolution of the Berry curvature at the equator ${\mathcal{F}}_{\varphi \theta =\pi /2}$ from Bethe ansatz results, Eq. (12) for $H/{\omega }_c=0.2$, allowing to access the vicinity of the quantum phase transition at ${\alpha }_c=1$ at equilibrium. (Bottom) Evolution of the Berry curvature ${\mathcal{F}}_{\varphi \theta }$ in the single-polaron picture obtained with a variational approach valid at low coupling, with respect to $\theta$ for $\alpha =0$ (black), $\alpha =0.2$ (yellow), $\alpha =0.4$ (green), and $\alpha =0.6$ (red). We have chosen the particular value $H/{\omega }_c=0.2$. The inset shows the evolution of the Chern number with $\alpha$. The dots are obtained from the integration of the Berry curvature obtained from the single-polaron ansatz.

Figure 3

Spin variables $\langle {\sigma }^z\rangle$ (black), $\langle {\sigma }^x\rangle$ (yellow) and $\langle {\sigma }^y\rangle$ (red) and their time-evolution during the sweep, with $v/H=0.08$. From left to right and top to bottom, we take $\alpha =0.05, \alpha =0.15, \alpha =0.26, \alpha =0.3, \alpha =0.34$ and $\alpha =0.36$. We have $H/{\omega }_c=0.01$. In the low-coupling regime $\alpha \ll 1$, we recover the two different frequencies for the dynamics $v$ and $\sqrt{H^2+v^2}$, as shown by the description in the rotating frame [Eq. (14) and discussion below].

Figure 4

Evolution of the dynamical Chern number $C_{\text{dyn}}$ with respect to $v/{\mathrm{\Delta }}_r(\theta =\pi /2)$ for $\alpha =0.3$ (red points), $\alpha =0.36$ (blue points), $\alpha =0.38$ (yellow up pointing triangles), $\alpha =0.4$ (magenta squares) and $\alpha =0.42$ (green up-pointing triangles). $C_{\text{dyn}}$ is a good estimate for $C$ as long as the quasiadiabaticity criterion (20) is fulfilled.

Figure 5

(Top) The main panel shows the evolution of the measured Chern number $C_{\text{dyn}}$ with respect to $\alpha$ for $v/H=0.08$ (black), 0.06 (red), 0.04 (blue), and 0.02 (green). The inset zooms on the region around $\alpha =1/2$, and the dashed lines represent predictions based on the mapping with an interacting resonance level valid around $\alpha =1/2$ (see Sec. 3c). We have $v/H<1$ and $v{\omega }_c/H^2>1$. (Bottom) Evolution of $C_{\text{dyn}}$ with respect to $\alpha$ and $H_0/H$ for $v/H=0.08$. The red line shows the asymptotic jump line of $C_{\text{dyn}}$ from 1 to 0 expected when decreasing the velocity while keeping $v{\omega }_c/H^2$ constant.

Figure 6

Effective field $|\langle h\rangle |/H$ felt by the spin at the end of the dynamical protocol as a function of $\lambda$, for $v/H=0.01$ (magenta), 0.02 (black), 0.03 (yellow), 0.04 (red), 0.05 (cyan), 0.06 (green), 0.07 (blue), and 0.08 (magenta). (Inset) Fit of the value of the coupling ${\lambda }^0$, for which $|\langle h\rangle |/H$=1, giving the scaling ${\lambda }^0/H={(v/H)}^{1/2}$.

Figure 7

(Top) Evolution of the final entanglement entropy (left), and effective temperature as a function of $\alpha$ for fixed $v/H=0.08$ and $H/{\omega }_c=0.01$. (Bottom) Effective temperature $T^{*}/H$ as a function of $\alpha$. The temperature $T^{*}$ becomes negative when $\langle {\sigma }^z\left(t_f\right)\rangle =0$ and the entanglement entropy $\mathcal{E}$ is maximal.

Figure 8

Evolution of $\langle {\sigma }^y\rangle$ with respect to time at $\alpha =0.1$ for $v/H=0.06, H/{\omega }_c=0.01$, and $p=5$ (red), 6 (black), 7 (yellow), 8 (blue), and 10 (green). For greater values of $p$, the curve remains unchanged.

Figure 9

Results from the dynamics using Keldysh formalism in the scaling regime. (Left) Evolution of $\langle d^{†}d\rangle (t_f=t_0+\pi /v)$ with respect to $u$ for a fixed value of $v/H=0.3$. (Right) Evolution of $\langle d^{†}d\rangle (t_f=t_0+\pi /v)$ with respect to $H/v$ for a fixed value of $u=0.008$. Each point corresponds to a given value of ${\omega }_c/H$, regularly spaced from ${\omega }_c/H=500$ to ${\omega }_c/H=5000$. The black lines correspond to linear fits.