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We investigate the Drude weight and the related Mazur-Suzuki (MS) bound in abroad variety of strongly coupled field theories with a gravity dual at finitetemperature and chemical potential. We revisit the derivation of the recentlyproposed universal expression for the Drude weight for Einstein-Maxwell-dilaton(EMd) theories and extend it to the case of theories with multiple masslessgauge fields. We show that the MS bound, which in the context of condensedmatter provides information on the integrability of the theory, is saturated inthese holographic theories including R-charged backgrounds. We then explore thelimits of this universality by studying EMd theories with $U(1)$ spontaneoussymmetry breaking and gravity duals of non-relativistic field theoriesincluding an asymptotically Lifshitz EMd model with two massless gauge fieldsand the Einstein-Proca model. In all these cases, the Drude weight, computedanalytically, deviates from the universal result and the MS bound is notsaturated. In general it is not possible to deduce the low temperaturedependence of the Drude weight by simple dimensional analysis. Finally we studythe effect of a weak breaking of translational symmetry by coupling the EMdaction, with and without $U(1)$ spontaneous symmetry breaking, to an axionfield. We show the coherent part of the conductivity in this limit is simplythe product of the MS bound and the scattering time obtained from the leadingquasinormal mode. For asymptotically $AdS$ theories it seems that the MS boundsets a lower bound on the DC conductivity for a given scattering time.

Drude weight and Mazur-Suzuki bounds in holography