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Friedel oscillation near a van Hove singularity in two-dimensional Dirac materials

We consider Friedel oscillation in the two-dimensional Dirac materials when Fermi level is near the van Hove singularity. Twisted graphene bilayer and the surface state of topological crystalline insulator are the representative materials which show low-energy saddle points that are feasible to probe by gating. We approximate the Fermi surface near saddle point with a hyperbola and calculate the static Lindhard response function. Employing a theorem of Lighthill, the induced charge density $δn$ due to an impurity is obtained and the algebraic decay of $δn$ is determined by the singularity of the static response function. Although a hyperbolic Fermi surface is rather different from a circular one, the static Lindhard response function in the present case shows a singularity similar with the response function associated with circular Fermi surface, which leads to the $δn\propto R^{-2}$ at large distance $R$. The dependences of charge density on the Fermi energy are different. Consequently, it is possible to observe in twisted graphene bilayer the evolution that $δn\propto R^{-3}$ near Dirac point changes to $δn\propto R^{-2}$ above the saddle point. Measurements using scanning tunnelling microscopy around the impurity sites could verify the prediction.