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

Abstract

We study the frequency dependence of the optical conductivity Re sigma(omega) of the Heisenberg spin-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 Re sigma(omega) for system sizes much larger than accessible to standard exact-diagonalization approaches. We find that the low-frequency behavior of Re sigma(omega) is well described by Re sigma(omega) approximate to sigma(dc) a|omega|(alpha), with alpha approximate to 1 in a wide range within the thermal phase and close to the transition. We particularly detail the decrease of sigma(dc) in the thermal phase as a function of increasing disorder for strong exchange anisotropies. We further find that the temperature dependence of alpha(dc) is consistent with the existence of a mobility edge.