Interaction between spin and Abrikosov vortices in doped topological insulators

In the topological superconductor with the nematic superconductivity in the ${E}_{u}$ representation, it is possible to have different types of vortices. One is associated with the vorticity in the particle-hole space and corresponds to the Abrikosov vortex. Another type corresponds to the vorticity in the spin space and is called the spin vortex. We study the interaction of the Abrikosov vortex with the spin vortices. We derive the free energy of the sample with the Abrikosov and the strain-induced spin vortices using the Ginzburg-Landau approach for the two-component superconducting order parameter. We calculate the critical strain at which the spin vortex is formed. We show that the spin vortex and the Abrikosov vortex attract each other, and as a result, they have a common core. We apply Bogoliubov--de Gennes equations to study electronic states in a combined vortex structure. We show that no zero-energy states (Majorana fermions) are localized near the common vortex core of the Abrikosov vortex and the spin vortex of any type. Possible experimental realization is discussed.