We determine the shear viscosity and the dc electrical conductivity of interacting three-dimensional Luttinger semimetals, which have a quadratic band touching point in the energy spectrum, in the hydrodynamic regime. It is well known that when the chemical potential is right at the band touching point, the long-range Coulomb interaction induces the Luttinger-Abrikosov-Beneslavskii (LAB) phase at $T=0$, which is an interacting, scale-invariant, non-Fermi-liquid state of electrons. Upon combining the renormalization-group (RG) analysis near the upper critical spatial dimension of 4 with the Boltzmann kinetic equation, we determine the universal ratio of viscosity over entropy and the electrical dc conductivity of the system at the interacting LAB fixed point of the RG flow. The projection of the Coulomb interaction on the eigenstates of the system is found to play an important quantitative role for the scattering amplitude in the collision integral, and the so-called Auger processes make a large numerical contribution to the inverse scattering time in the transport quantities. The obtained leading-order result suggests that the universal ratio of the viscosity over entropy, when extrapolated to the physical three spatial dimensions, is above, but could be rather close to, the Kovtun-Son-Starinets lower bound.

• Received 18 November 2019
• Revised 3 February 2020
• Accepted 3 March 2020

DOI:https://doi.org/10.1103/PhysRevB.101.125128