Dynamical conductivity and its fluctuations along the crossover to many-body localization

We present a numerical study of the many-body localization (MBL) phenomenon in the high-temperature limit within an anisotropic Heisenberg model with random local fields. Taking the dynamical spin conductivity $\sigma(\omega)$ as the test quantity, we investigate the full frequency dependence of sample-to-sample fluctuations and their scaling properties as a function of the system size $L\leq 28$ and the frequency resolution. We identify differences between the general interacting case $\Delta>0$ and the anisotropy $\Delta=0$, the latter corresponding to the standard Anderson localization. Except for the extreme MBL case when the relative sample-to-sample fluctuations became large, numerical results allow for the extraction of the low-$\omega$ dependence of the conductivity. Results for the d.c. value $\sigma_0$ indicate a crossover into the MBL regime, i.e. an exponential-like variation with the disorder strength $W$. For the same regime, our numerical analysis indicates that the low-frequency exponent $\alpha$ exhibits a small departure from $\alpha\sim 1$ only.