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

We study the frequency dependence of the optical conductivity $$\text{Re}$ , \sigma(\omega)$oftheHeisenbergspin-$1/2$ chain in the thermal and near the transition to the many-body localized phase induced by the strength of a random $z$-directed magnetic field. Using the method of dynamical quantum typicality, we calculate the real-time dynamics of the spin-current autocorrelation function and obtain the Fourier transform $\text{Re}\sigma (\omega )$ for system sizes much larger than accessible to standard exact-diagonalization approaches. We find that the low-frequency behavior of $$\text{Re}$ , \sigma(\omega)$iswelldescribedby$$\text{Re}$ , \sigma(\omega) \approx \sigma_$\text{dc}$ + a , |\omega|^\alpha$,with$\alpha \approx 1$ in a wide range within the thermal phase and close to the transition. We particularly detail the decrease of ${\sigma }_{\text{dc}}$ in the thermal phase as a function of increasing disorder for strong exchange anisotropies. We further find that the temperature dependence of ${\sigma }_{\text{dc}}$ is consistent with the existence of a mobility edge.