Notes on hyperscaling violating Lifshitz and shear diffusion

TL;DR

This paper extends the membrane paradigm analysis of shear diffusion to hyperscaling violating Lifshitz theories, deriving diffusion constants and exploring universal behavior and temperature scaling, including special cases with logarithmic behavior.

Contribution

It generalizes shear diffusion analysis to theories with gauge fields and hyperscaling violation, providing new expressions for diffusion constants and insights into universal viscosity bounds.

Findings

Shear diffusion constant scales with temperature in these theories.

Universal behavior observed for certain exponents $z, heta$.

Logarithmic behavior appears at the critical case $z=4- heta$.

Abstract

We explore in greater detail our investigations of shear diffusion in hyperscaling violating Lifshitz theories in arXiv:1604.05092 [hep-th]. 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 $z, \theta$ satisfying $z<4-\theta$ 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 behaviour in relation to the viscosity bound. For $z=4-\theta$, we find logarithmic behaviour.