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

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, $η$, 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, $η^\star = -\lim_{ω\to 0}(1/ω) \text{Im} G^{R}_{T^{xy}T^{xy}}(ω,k=0)$, even at the leading order in disorder strength. In a simple holographic model with broken translational symmetry, we show that both $η/s$ and $η^\star/s$ violate the bound of viscosity-entropy ratio for arbitrary disorder strength.