Scaling behaviour and superconducting instability in anisotropic non-Fermi liquids

We study the scaling behaviour of the optical conductivity $(\sigma )$, free energy density $(F)$, and shear viscosity of the quantum critical point associated with the spin density wave phase transition for a two-dimensional metallic system with $C_2$ symmetry. A non-Fermi liquid behaviour emerges at two pairs of isolated points on the Fermi surface, due to the coupling of a bosonic order parameter to fermionic excitations at those so-called ``hot-spots’'. We find that near the hot-spots, $\sigma$ and $F$ obey the scalings expected for such an anisotropic system, and the direction-dependent viscosity to entropy density ratio is not a universal number due to the anisotropy. Lastly, we also estimate the effect of the fermion-boson coupling at the hot-spots on superconducting instabilities.