Flow of shear response functions in hyperscaling violating Lifshitz theories

TL;DR

This paper investigates the flow of shear response functions in hyperscaling violating Lifshitz theories, revealing conditions under which shear viscosity saturates bounds and diverges, with implications for holographic dualities and UV behavior.

Contribution

It adapts the membrane paradigm to analyze shear response functions in hvLif theories, deriving new insights into shear viscosity bounds and boundary divergences.

Findings

Shear viscosity saturates the KSS bound at leading order.

Logarithmic scaling of shear viscosity correction when z=d_i-θ.

Divergence of gauge field response near the boundary for z>d_i+2-θ.

Abstract

We study the flow equations of the shear response functions for hyperscaling violating Lifshitz (hvLif) theories, with Lifshitz and hyperscaling violating exponents $z$ and $\theta$. Adapting the membrane paradigm approach of analysing response functions as developed by Iqbal and Liu, we focus specifically on the shear gravitational modes which now are coupled to the perturbations of the background gauge field. Restricting to the zero momenta sector, we make further simplistic assumptions regarding the hydrodynamic expansion of the perturbations. Analysing the flow equations shows that the shear viscosity at leading order saturates the Kovtun-Son-Starinets (KSS) bound of $\frac{1}{4\pi}$. When $z=d_i-\theta$, ($d_i$ being the number of spatial dimension in the dual field theory) the first-order correction to shear viscosity exhibits logarithmic scaling, signalling the emergence of a scale in the UV regime for this class of hvLif theories. We further show that the response function associated to the gauge field perturbations diverge near the boundary when $z>d_i+2-\theta$. This provides a holographic understanding of the origin of such a constraint and further vindicates results obtained in previous works that were obtained through near horizon and quasinormal mode analysis.