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Good metals are characterised by diffusive transport of coherentquasi-particle states and the resistivity is much less than theMott-Ioffe-Regel (MIR) limit, $\frac{ha}{e^{2}}$, where $a$ is the latticeconstant. In bad metals, such as many strongly correlated electron materials,the resistivity exceeds the Mott-Ioffe-Regel limit and the transport isincoherent in nature. Hartnoll, loosely motivated by holographic duality(AdS/CFT correspondence) in string theory, recently proposed a lower bound tothe charge diffusion constant, $D \gtrsim \hbar v_{F}^{2}/(k_{B}T)$, in theincoherent regime of transport, where $v_F$ is the Fermi velocity and $T$ thetemperature. Using dynamical mean field theory (DMFT) we calculate the chargediffusion constant in a single band Hubbard model at half filling. We show thatin the strongly correlated regime the Hartnoll's bound is violated in thecrossover region between the coherent Fermi liquid region and the incoherent(bad metal) local moment region. The violation occurs even when the bare Fermivelocity $v_F$ is replaced by its low temperature renormalised value,$v_F^*$.The bound is satisfied at all temperatures in the weakly and moderatelycorrelated systems as well as in strongly correlated systems in the hightemperature region where the resistivity is close to linear in temperature. Ourcalculated charge diffusion constant, in the incoherent regime of transport,also strongly violates a proposed quantum limit of spin diffusion, $D_{s} \sim1.3 \hbar/m$, where $m$ is the fermion mass, experimentally observed andtheoretically calculated in a cold degenerate Fermi gas in the unitary limit ofscattering.

Absence of a quantum limit to charge diffusion in bad metals