Dielectric screening of surface states in a topological insulator

Hexagonal warping provides an anisotropy to the dispersion curves of the helical Dirac fermions that exist at the surface of a topological insulator. A sub-dominant quadratic in momentum term leads to an asymmetry between conduction and valence band. A gap can also be opened through magnetic doping. We show how these various modifications to the Dirac spectrum change the polarization function of the surface states and employ our results to discuss their effect on the plasmons. In the long wavelength limit, the plasmon dispersion retains its square root dependence on its momentum, $\boldsymbol{q}$, but its slope is modified and it can acquire a weak dependence on the direction of $\boldsymbol{q}$. Further, we find the existence of several plasmon branches, one which is damped for all values of $\boldsymbol{q}$, and extract the plasmon scattering rate for a representative case.