  We argue that the a.c. conductivity $\sigma(\omega)$ in the many-body
localized phase is a power law of frequency $\omega$ at low frequency:
specifically, $\sigma(\omega) \sim \omega^\alpha$ with the exponent $\alpha$
approaching 1 at the phase transition to the thermal phase, and asymptoting to
2 deep in the localized phase. We identify two separate mechanisms giving rise
to this power law: deep in the localized phase, the conductivity is dominated
by rare resonant pairs of configurations; close to the transition, the dominant
contributions are rare regions that are locally critical or in the thermal
phase. We present numerical evidence supporting these claims, and discuss how
these power laws can also be seen through polarization-decay measurements in
ultracold atomic systems.
