Typicality approach to the optical conductivity in thermal and many-body localized phases

We study the frequency dependence of the optical conductivity Reσ(ω) of the Heisenberg spin-12 chain in the thermal and near the transition to the many-body localized phase induced by the strength of a random z-directed magnetic field. Using the method of dynamical quantum typicality, we calculate the real-time dynamics of the spin-current autocorrelation function and obtain the Fourier transform Reσ(ω) for system sizes much larger than accessible to standard exact-diagonalization approaches. We find that the low-frequency behavior of Reσ(ω) is well described by Reσ(ω)≈σdc+a|ω|α, with α≈1 in a wide range within the thermal phase and close to the transition. We particularly detail the decrease of σdc in the thermal phase as a function of increasing disorder for strong exchange anisotropies. We further find that the temperature dependence of σdc is consistent with the existence of a mobility edge.