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Motivated by recent experimental efforts on three-dimensional semimetals, weinvestigate the static and dynamic density response of the nodal line semimetalby computing the polarizability for both undoped and doped cases. The nodalline semimetal in the absence of doping is characterized by a ring-shape zeroenergy contour in momentum space, which may be considered as a collection ofDirac points. In the doped case, the Fermi surface has a torus shape and twoindependent processes of the momentum transfer contribute to the singularfeatures of the polarizability even though we only have a single Fermi surface.In the static limit, there exist two independent singularities in the secondderivative of the static polarizability. This results in the highly anisotropicFriedel oscillations which show the angle-dependent algebraic power law and thebeat phenomena in the oscillatory electron density near a charged impurity.Furthermore, the dynamical polarizability has two singular lines along$\hbar\omega = \gamma p$ and $\hbar\omega = \gamma p \sin\eta$, where $\eta$ isthe angle between the external momentum $\vec{p}$ and the plane where the nodalring lies. From the dynamical polarizability, we obtain the plasmon modes inthe doped case, which show anisotropic dispersions and angle-dependent plasmafrequencies. Qualitative differences between the low and high doping regimesare discussed in light of future experiments.

Anisotropic density fluctuations, plasmons, and Friedel oscillations in nodal line semimetal