Resistivity bound for hydrodynamic bad metals

We obtain a rigorous upper bound on the resistivity ρ 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 ρ≲AΓ. The coefficient A is independent of temperature and inhomogeneity lengthscale, and Γ is a microscopic momentum-preserving scattering rate. In this way, we obtain a unified mechanism—without umklapp—for ρ∼T2 in a Fermi liquid and the crossover to ρ∼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.