Spin bath dynamics and dynamical renormalization group

We discuss the quantum dynamics of the central spin model in a regime where the central spin and bath are slaved to each other. The exact solution is found when the bath is static and is compared with the effect of an external field, finding that they are inequivalent due to the quantum nature of the environment. When the bath has dynamics, we analyze the differences between the numerical simulation using time-dependent perturbation theory and the equation of motion technique, which shows better accuracy. We demonstrate that the use of dynamical renormalization group (dRG), simultaneously with the equation of motion technique, provides a suitable analytical tool to understand the physics, to capture the main physical processes, and a powerful method to eliminate secular terms. In addition, this approach allows to separate classical nonlinear behavior from corrections due to quantum correlations.

Figure 1

Comparison between the free dynamics and the case with a bath of spins for different spectral functions $J\left(\alpha \right)$ characterized by $\sigma$. We have chosen $B_{\perp }/B_z=0.5$ and an initial spin up state.

Figure 2

Comparison between the exact dynamics for ${\mathrm{\Delta }}_{\perp }/B_{\perp }=0$ (black) and 0.03 (red), for the case of homogeneous couplings $A/B_{\perp }=0.05, N=100$, and $B_z={\mathrm{\Delta }}_z=0$. The dynamics of the bath spins, produced by ${\mathrm{\Delta }}_{\perp }\ne 0$, leads to instantonlike transitions in the central spin at long timescales, when $N\gg 1$ (this transition gets more abrupt as $N$ increases). (Blue) Numerical solution of the mean-field equations. When ${\mathrm{\Delta }}_{\perp }/B_{\perp }\ll 1$ the short time dynamics is identical to the case ${\mathrm{\Delta }}_{\perp }=0$, with a static bath. The initial condition is a product state with central spin up and all bath spins up.

Figure 3

Numerical solution of the mean-field equations (black) and the solution for the slow component $M_z\left(\tau \right)$ (red) using the flow equation. Parameters are $B_z={\mathrm{\Delta }}_z=0, {\mathrm{\Delta }}_{\perp }/B_{\perp }=0.03, A/B_{\perp }=0.05$, and $N=100$, with central spin initially up and a fully polarized bath. The slow component perfectly describes the instanton transition.

Figure 4

Comparison between the mean-field solution (red), the solution including spin-bath correlations (black) and the exact simulation (green) for the same parameters as Fig. 2.

Figure 5

Comparison between the exact dynamics (dashed, green) and the lowest-order solution using dRG (blue) for the same parameters as Fig. 2. The red dot-dashed line shows $M_z\left(\tau \right)$, which plays the role of the envelope function for the faster oscillations. The lowest-order solution correctly captures the suppression of oscillations, but the frequency is shifted because backreaction between the mean-field solution and the correlations has been neglected at this order of dRG.

Figure 6

Comparison between the central spin dynamics between the dRG result to first order (blue) and the exact diagonalization one (yellow). The red dashed line shows the function $M_z\left(t\right)$, which modulates the fast coherent oscillations of the central spin. Parameters are ${\mathrm{\Delta }}_{\perp }/B_{\perp }=0.03, A/B_{\perp }=0.05$, and $N=100$ for the initial condition $\langle S^z\left(0\right)\rangle =\frac{1}{2}$ and $\langle P_z\left(0\right)\rangle =N/2$.

Figure 7

Comparison between different decoupling schemes.

Figure 8

Comparison between the exact Hierarchy of correlations (black), its lowest-order approximation using dRG (red) and the exact dynamics for the central spin (green), at long times. Parameters: $\mathrm{\Delta }/B=0.03, J/B=0.05$ and $N=100$ for the initial condition $\langle S^z\left(0\right)\rangle =\frac{1}{2}$ and $\langle P_z\left(0\right)\rangle =N/2$.

Figure 9

Numerical solution of the flow equations for the running of the different initial conditions. (Red, solid line) $M_z\left(t\right)$ still displays instantonlike behavior, but the profile is modified due to correlations with respect to the uncorrelated case (see Fig. 6, red-dashed line). (Blue, dashed line) $M_x\left(t\right)$ also shows large corrections but it is always negative, which is expected due to the positive value chosen for the interaction parameter. (Green, dot-dashed line) $a_{zy}\left(t\right)$ displays complicated oscillations, correlated with the bath and the central spin magnetization, but importantly, acquires large values that can largely modify the mean-field dynamics.