Figure 1

The time dependence of the expectation value $⟨{\psi }_0|S^z\left(t\right)-\frac{1}{2}|{\psi }_0⟩$ is shown for $g=-1/2$.Reuse & Permissions

Figure 2

We depict the ratio $r\left(g\right)=\frac{\overline{⟨{\psi }_0|S^z\left(t\right)-1/2|{\psi }_0⟩}}{{⟨S^z-1/2⟩}_{\text{eq}}}$ as a function of the coupling strength $g$. In the weak-coupling limit one finds the universal ratio $r\left(0\right)=2$, see text.Reuse & Permissions

Figure 3

The forward-backward transformation scheme induces a nonperturbative solution of the Heisenberg equations of motion for an operator. $U$ denotes the full unitary transformation that relates the $B=0$ to the $B=\infty$ basis (Ref. 16).Reuse & Permissions

Figure 4

(Color online) We show $⟨S^z\left(t\right)⟩$ for isotropic coupling $j$ and short times. For very short times $t⪡D^{-1}$, the perturbative result from Eq. (28) (dashed line) is asymptotically coinciding with a fully numerical solution of the flow Eq. (21) (full lines). The renormalization of $J^{\perp }$ by $J^{\parallel }$ sets in at energy scales $E<D$, leading to a reduction in spin-flip scattering. Beyond the short-time regime $t⪡D^{-1}$, the magnetization relaxes therefore slower than predicted by the perturbative result.Reuse & Permissions

Figure 5

(Color online) We show $⟨S^z\left(t\right)⟩$ for different isotropic couplings $j$. In panel (a) the full numerical solution of the flow equations (full lines in both panels) is compared to a fit to the asymptotic analytical behavior of Eq. (43) (dashed lines, fit parameters given in Table ). Beyond the perturbative short-time regime, described by unrenormalized spin-flip scattering according to Eq. (28), the logarithmic renormalization of the spin-flip scattering coupling $J^{\perp }$ sets in. This leads to a logarithmically slow relaxation of the magnetization and the asymptotic value $⟨S^z\left(t\to \infty \right)⟩=0.5+\frac{j}{2}$ [dashed lines in panel (b)] is reached extremely slowly.Reuse & Permissions

Figure 6

(Color online) Short-time behavior of the magnetization for anisotropic couplings from a fully numerical solution of the flow equations in comparison with the perturbative result from Eq. (28) (dashed line). The perpendicular coupling is fixed to $J^{\perp }=-0.04$, the curve with $J^{\parallel }=-0.04$ corresponds to the isotropic case. For increasing $\left|J^{\parallel }\right|$, the spin-flip scattering coupling $J^{\perp }$ is stronger renormalized, and the magnetization tends to decay slower than predicted by the unrenormalized perturbative result.Reuse & Permissions

Figure 7

(Color online) Long-time result of the magnetization relaxation for anisotropic couplings and comparison with a long-time fit. All parameters are equivalent to those from Fig. 6. In panel (a), the full numerical solution of the flow equations (full lines in both panels) is fitted against the analytical power-law behavior stated in Eq. (49) (dashed lines). At time scales where renormalization of $J^{\perp }$ by $J^{\parallel }$ becomes significant and leads to a deviation from the perturbative result, again the asymptotic fit describes the relaxation process very well. In panel (b), a comparison with the saturation value $⟨S^z\left(t\to \infty \right)⟩=1/2+{\alpha }^2{\rho }_F^2/\left(4{\tilde{j}}^{\parallel }\right)$ (dashed lines) is given. Clearly visible, anisotropic couplings $J^{\perp }<J^{\parallel }$ lead to much faster saturation of the magnetization than isotropic couplings. Fitting parameters are given in Table .Reuse & Permissions