Locally Critical Resistivities from Umklapp Scattering

Efficient momentum relaxation through umklapp scattering, leading to a power law in temperature dc resistivity, requires a significant low energy spectral weight at finite momentum. One way to achieve this is via a Fermi surface structure, leading to the well-known relaxation rate Γ~T². We observe that local criticality, in which energies scale but momenta do not, provides a distinct route to efficient umklapp scattering. We show that umklapp scattering by an ionic lattice in a locally critical theory leads to Γ~T^2Δ_kL. Here Δ_kL≥0 is the dimension of the (irrelevant or marginal) charge density operator Jᵗ(ω,kL) in the locally critical theory, at the lattice momentum kL. We illustrate this result with an explicit computation in locally critical theories described holographically via Einstein-Maxwell theory in Anti--de Sitter spacetime. We furthermore show that scattering by random impurities in these locally critical theories gives a universal Γ~(log1/T)^-1.