plasmon excitations of surface states in a topological insulator: a second-order approximation

We studied the plasmon excitations in a topological insulator with a surface state hamiltonian containing a dirac term and a hexagonal warping term which yields a helical dirac cone energy dispersion. the existence of helical dirac fermions leads to new many-body phenomena in topological insulators. in this paper, we obtained the polarization function and dynamical dielectric function of the surface state fermions by using an approximation up to second-order in the wave vector. by applying this approach, we calculated the plasmon dispersion from zeros of the real and imaginary parts of dielectric function. in addition, we performed the full numerical calculation and first-order approximation and compare their results with those of second-order one. we found that the plasmon modes calculated from second-order approximation are in agreement with full numerical results in a wide range of wave vectors, unlike the first order approximation data. moreover, it is was obtained that by increasing the warping strength, the dispersion curve gets much closer to the full numerical one.