Shear viscosity in holography and effective theory of transport without translational symmetry

TL;DR

This paper investigates shear viscosity in holographic and effective hydrodynamic models with broken translational symmetry, revealing deviations from traditional calculations and violations of the viscosity-entropy ratio bound.

Contribution

It introduces a new constitutive relation for shear viscosity in systems with broken translational symmetry, derived from fluid/gravity correspondence and holography, highlighting deviations from standard formulas.

Findings

Shear viscosity deviates from the usual Green's function-based expression.

Both η/s and η*/s ratios violate the viscosity bound at arbitrary disorder levels.

The new constitutive relation captures effects of translational symmetry breaking.

Abstract

We study the shear viscosity in an effective hydrodynamic theory and holographic model where the translational symmetry is broken by massless scalar fields. We identify the shear viscosity, $\eta$, from the coefficient of the shear tensor in the modified constitutive relation, constructed from thermodynamic quantities, fluid velocity and the scalar fields, which break the translational symmetry explicitly. Our construction of constitutive relation is inspired by those derived from the fluid/gravity correspondence in the weakly disordered limit $m/T \ll 1$. We show that the shear viscosity from the constitutive relation deviates from the one obtained from the usual expression, $\eta^\star = -\lim_{\omega\to 0}(1/\omega) \text{Im} G^{R}_{T^{xy}T^{xy}}(\omega,k=0)$, even at the leading order in disorder strength. In a simple holographic model with broken translational symmetry, we show that both $\eta/s$ and ${\eta}^\star/s$ violate the bound of viscosity-entropy ratio for arbitrary disorder strength.