  In magnetic topological insulators, the surface states can exhibit a gap due
to the breaking of time-reversal symmetry. Various experiments, while
suggesting the existence of the surface gap, have raised questions about its
underlying mechanism in the presence of different magnetic orderings. Here, we
demonstrate that magnon-mediated electron-electron interactions, whose effects
are not limited to the surfaces perpendicular to the magnetic ordering, can
significantly influence surface states and their effective gaps. On the
surfaces perpendicular to the spin quantization axis, many-body interactions
can enhance the band gap to a degree that surpasses the non-interacting
scenario. Then, on surfaces parallel to the magnetic ordering, we find that
strong magnon-induced fermionic interactions can lead to features resembling a
massless-like gap. These remarkable results largely stem from the fact that
magnon-mediated interactions exhibit considerable long-range behavior compared
to direct Coulomb interactions among electrons, thereby dominating the
many-body properties at the surface of magnetic topological insulators.
