Spin transport of weakly disordered Heisenberg chain at infinite temperature

We study the disordered Heisenberg spin chain, which exhibits many body localization at strong disorder, in the weak to moderate disorder regime. A continued fraction calculation of dynamical correlations is devised, using a variational extrapolation of recurrents. Good convergence for the infinite chain limit is shown. We find that the local spin correlations decay at long times as $C \sim t^{-\beta}$, while the conductivity exhibits a low frequency power law $\sigma \sim \omega^{\alpha}$. The exponents depict sub-diffusive behavior $\beta < 1/2, \alpha> 0$ at all finite disorders, and convergence to the scaling result, $\alpha+2\beta = 1$, at large disorders.