Locally critical umklapp scattering and holography

Efficient momentum relaxation through umklapp scattering, leading to a power law in temperature d.c. 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 Gamma ~ 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 Gamma ~ T^(2\Delta(k_L)). Here \Delta(k_L) \geq 0 is the dimension of the (irrelevant or marginal) charge density operator J^t(w,k_L) in the locally critical theory, at the lattice momentum k_L. 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 Gamma ~ 1/log(1/T)