Drag viscosity of metals and its connection to Coulomb drag

Shear viscosity is a key parameter in the hydrodynamic description of fluids, including electronic matter - an interdisciplinary subject that has attracted an increasing amount of interest recently. Related to the rate of momentum transport, viscosity is given by the retarded correlation function of the momentum current, i.e., the stress tensor, in the linear response theory. In this work, we find that there exists a previously overlooked contribution to viscosity, which originate from the interacting part of the stress tensor. This new contribution, which we name drag viscosity, is connected to the frictional drag forces induced by the long-range interactions. A related phenomenon is the Coulomb drag of a double-layers electronic system, which measures the rate of charge transport originating from scattering between quasiparticles in different layers in the presence of Coulomb interactions. Using the diagrammatic approach in the Keldysh formalism, we derive the drag viscosity of the 2D and 3D electronic systems as well as the drag resistivity of the 2D double-layers systems. At low enough temperatures, both the drag viscosity and resistivity exhibit quadratic temperature dependence which becomes linear as temperature increases.