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 σ(ω) 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≤28 and the frequency resolution. We identify differences between the general interacting case Δ>0 and the anisotropy Δ=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-ω dependence of the conductivity. Results for the dc value σ₀ 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 α exhibits a small departure from α~1 only.