Notes on hyperscaling violating Lifshitz and shear diffusion

Abstract

We explore in greater detail our investigations of shear diffusion in hyperscaling violating Lifshitz theories in Phys. Lett. B 760, 86 (2016). This adapts and generalizes the membrane-paradigm-like analysis of Kovtun, Son, and Starinets for shear gravitational perturbations in the near horizon region given certain self-consistent approximations, leading to the shear diffusion constant on an appropriately defined stretched horizon. In theories containing a gauge field, some of the metric perturbations mix with some of the gauge field perturbations and the above analysis is somewhat more complicated. We find a similar near-horizon analysis can be obtained in terms of new field variables involving a linear combination of the metric and the gauge field perturbation resulting in a corresponding diffusion equation. Thereby as before, for theories with Lifshitz and hyperscaling violating exponents $z$, $\ensuremath{\theta}$ satisfying $z<4\ensuremath{-}\ensuremath{\theta}$ in four bulk dimensions, our analysis here results in a similar expression for the shear diffusion constant with power-law scaling with temperature suggesting universal behavior in relation to the viscosity bound. For $z=4\ensuremath{-}\ensuremath{\theta}$, we find logarithmic behavior.