Scaling behaviour and superconducting instability in anisotropic non-Fermi liquids

We study the scaling behaviour of the optical conductivity (σ), free energy density (F) and shear viscosity of the quantum critical point associated with spin density wave phase transition for a two-dimensional metallic system with C2 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, σ 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.