  The surface state of a three dimensional strong topological insulator (TI) is
well described in the independent particle picture (IPP) by an isotropic Dirac
cone at the $\Gamma$-point and perpendicular spin-momentum locking. Away from
this point, the crystal point group symmetry causes anisotropic effects on the
surface spectrum where a number of unusual effects are experimentally observed.
In particular, the perturbative violations of the perpendicular spin-momentum
locking frequently observed in many experiments remains to be a poorly
understood feature theoretically. In parallel, the existence of electron-phonon
interaction has been unquestionably verified by a number of experimental
groups. In this article, we device an interacting theory of the spin texture
using the spin-dependent self-energy formalism. We observe that the
interactions lead to the observable spin-texture anomalies in the presence of a
Fermi surface anisotropy while weakly affecting the energy bands. In
particular, the experimental observation of the six-fold symmetric modulation
of the in-plane spin in the $Bi_{2-y}Sb_ySe_{3-x}Te_x$ family and the resulting
violation of the spin-momentum locking is explained using the coupling of an
optical surface phonon to the surface electrons reported in earlier
experiments. We also discuss recent puzzling results of the out-of-plane spin
polarization experiments in this context.
  Our results introduce an interacting approach to the spin-related anomalies
where the anisotropy of the Dirac cone around the Fermi surface proves to have
an unconventional role. New experiments reporting unusual spin orientations in
other materials with different symmetries signify that the theory introduced
here may be relevant to a larger set of Dirac materials.
