Drude weight and Mazur-Suzuki bounds in holography

We investigate the Drude weight and the related Mazur-Suzuki (MS) bound in a broad variety of strongly coupled field theories with a gravity dual at finite temperature and chemical potential. We revisit the derivation of the recently proposed universal expression for the Drude weight for Einstein-Maxwell-dilaton (EMd) theories and extend it to the case of theories with multiple massless gauge fields. We show that the MS bound, which in the context of condensed matter provides information on the integrability of the theory, is saturated in these holographic theories including R-charged backgrounds. We then explore the limits of this universality by studying EMd theories with $U(1)$ spontaneous symmetry breaking and gravity duals of non-relativistic field theories including an asymptotically Lifshitz EMd model with two massless gauge fields and the Einstein-Proca model. In all these cases, the Drude weight, computed analytically, deviates from the universal result and the MS bound is not saturated. In general it is not possible to deduce the low temperature dependence of the Drude weight by simple dimensional analysis. Finally we study the effect of a weak breaking of translational symmetry by coupling the EMd action, with and without $U(1)$ spontaneous symmetry breaking, to an axion field. We show the coherent part of the conductivity in this limit is simply the product of the MS bound and the scattering time obtained from the leading quasinormal mode. For asymptotically $AdS$ theories it seems that the MS bound sets a lower bound on the DC conductivity for a given scattering time.