  A key feature of topological insulators is the robustness of the electron
energy spectrum. At a surface of a topological insulator, Dirac point is
protected by the characteristic symmetry of the system. The breaking of the
symmetry opens a gap in the energy spectrum. Therefore, topological insulators
are very sensitive to magnetic fields, which can open a gap in the electronic
spectrum. Concerning "internal" magnetic effects, for example the situation
with doped magnetic impurities, is not trivial. A single magnetic impurity is
not enough to open the band gap, while in the case of a ferromagnetic chain of
deposited magnetic impurities the Dirac point is lifted. However, a much more
interesting case is when localized magnetic impurities form a chiral spin
order. Our first principle density functional theory calculations have shown
that this is the case for Fe deposited on the surface of Bi2Se3 topological
insulator. But not only magnetic impurities can form a chiral helicoidal spin
texture. An alternative way is to use chiral multiferroics (prototype material
is LiCu2O2) that induce a proximity effect. The theoretical approach we present
here is valid for both cases. We observed that opposite to a ferromagnetically
ordered case, a chiral spin order does not destroy the Dirac point. We also
observed that the energy gap appears at the edges of the new Brillouin zone.
Another interesting result concerns the spin dynamics. We derived an equation
for the spin density dynamics with a spin current and relaxation terms. We have
shown that the motion of the conductance electron generates a magnetic torque
and exerts a certain force on the helicoidal texture.
