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Absence of a quantum limit to charge diffusion in bad metals

Good metals are characterised by diffusive transport of coherent quasi-particle states and the resistivity is much less than the Mott-Ioffe-Regel (MIR) limit, $\frac{ha}{e^{2}}$, where $a$ is the lattice constant. In bad metals, such as many strongly correlated electron materials, the resistivity exceeds the Mott-Ioffe-Regel limit and the transport is incoherent in nature. Hartnoll, loosely motivated by holographic duality (AdS/CFT correspondence) in string theory, recently proposed a lower bound to the charge diffusion constant, $D \gtrsim \hbar v_{F}^{2}/(k_{B}T)$, in the incoherent regime of transport, where $v_F$ is the Fermi velocity and $T$ the temperature. Using dynamical mean field theory (DMFT) we calculate the charge diffusion constant in a single band Hubbard model at half filling. We show that in the strongly correlated regime the Hartnoll's bound is violated in the crossover region between the coherent Fermi liquid region and the incoherent (bad metal) local moment region. The violation occurs even when the bare Fermi velocity $v_F$ is replaced by its low temperature renormalised value, $v_F^*$.The bound is satisfied at all temperatures in the weakly and moderately correlated systems as well as in strongly correlated systems in the high temperature region where the resistivity is close to linear in temperature. Our calculated charge diffusion constant, in the incoherent regime of transport, also strongly violates a proposed quantum limit of spin diffusion, $D_{s} \sim 1.3 \hbar/m$, where $m$ is the fermion mass, experimentally observed and theoretically calculated in a cold degenerate Fermi gas in the unitary limit of scattering.