Anisotropic surface transport in topological insulators in proximity to a helical spin density wave

We study the effects of spatially localized breakdown of time reversal symmetry on the surface of a topological insulator (TI) due to proximity to a helical spin density wave (HSDW). The HSDW acts like an externally applied one-dimensional periodic (magnetic) potential for spins on the surface of the TI, rendering the Dirac cone on the TI surface highly anisotropic. The decrease in group velocity along the direction (x) over cap of the applied spin potential is twice as great as that perpendicular to (x) over cap. At the Brillouin zone boundaries, it also gives rise to new semi-Dirac points that have a linear dispersion along (x) over cap but a quadratic dispersion perpendicular to (x) over cap. The group velocity of electrons at these new semi-Dirac points is also shown to be highly anisotropic. Experiments using TI systems on multiferroic substrates should realize our predictions. We further discuss the effects of other forms of spin density waves on the surface transport property of TIs.