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

Abstract

We study the frequency dependence of the optical conductivity $\text{Re}\phantom{\rule{0.16em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega})$ of the Heisenberg spin-$\frac{1}{2}$ 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 $\text{Re}\phantom{\rule{0.16em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega})$ for system sizes much larger than accessible to standard exact-diagonalization approaches. We find that the low-frequency behavior of $\text{Re}\phantom{\rule{0.16em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega})$ is well described by $\text{Re}\phantom{\rule{0.16em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\approx}{\ensuremath{\sigma}}_{\text{dc}}+a{|\ensuremath{\omega}|}^{\ensuremath{\alpha}}$, with $\ensuremath{\alpha}\ensuremath{\approx}1$ in a wide range within the thermal phase and close to the transition. We particularly detail the decrease of ${\ensuremath{\sigma}}_{\text{dc}}$ in the thermal phase as a function of increasing disorder for strong exchange anisotropies. We further find that the temperature dependence of ${\ensuremath{\sigma}}_{\text{dc}}$ is consistent with the existence of a mobility edge.