Resistivity bound for hydrodynamic bad metals

We obtain a rigorous upper bound on the resistivity rho of an electron fluid whose electronic mean free path is short compared with the scale of spatial inhomogeneities. When such a hydrodynamic electron fluid supports a nonthermal diffusion process-such as an imbalance mode between different bands-we show that the resistivity bound becomes rho less than or similar to A Gamma. The coefficient A is independent of temperature and inhomogeneity lengthscale, and Gamma is a microscopic momentum-preserving scattering rate. In this way, we obtain a unified mechanism-without umklapp-for rho similar to T-2 in a Fermi liquid and the crossover to rho similar to T in quantum critical regimes. This behavior is widely observed in transition metal oxides, organic metals, pnictides, and heavy fermion compounds and has presented a long-standing challenge to transport theory. Our hydrodynamic bound allows phonon contributions to diffusion constants, including thermal diffusion, to directly affect the electrical resistivity.