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Running Shear Viscosities in Anisotropic Holographic Superfluids

We have examined holographic renormalization group($RG$) flows of the shear viscosities in anisotropic holographic superfluids via their gravity duals, Einstein-SU(2) Yang-Mills system. In anisotropic phase, below the critical temperature $T_c$, the SO(3) isometry(spatial rotation) in the dual gravity system is broken down to the residual SO(2). The shear viscosities in the symmetry broken directions of the conformal fluids defined on $AdS$ boundary present non-universal values which depend on the chemical potential $μ$ and temperature $T$ of the system and also satisfy non-trivial holographic $RG$-flow equations. The shear viscosities flow down to the specific values in $IR$ region, in fact which are given by the ratios of the metric components in the symmetry unbroken direction to those in the broken directions, evaluated at the black brane horizon in the dual gravity system.