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 T^2. 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 T^2{₊₋}. Here ₊₋0 is the dimension of the (irrelevant or marginal) charge density operator J^t (, k₋) in the locally critical theory, at the lattice momentum k₋. 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 (log1T) ^-1.