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Magnetic topological insulators (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_n$ wereanticipated to exhibit magnetic energy gaps while recent spectroscopic studiesdid not observe them. Thus, magnetism on the surface is under debate. In thiswork, we propose another symmetry criterion to probe the surface magnetism.Because of both time-reversal symmetry-breaking and inversionsymmetry-breaking, we demonstrate that the surface band structure violatesmomentum-inversion symmetry and leads to a three-fold rather than six-foldrotational symmetry on the Fermi surface if corresponding surface states couplestrongly to the surface magnetism. Such a momentum-inversion symmetry violationis significant along the $\Gamma-K$ direction for surface bands on the (0001)plane.

Momentum-inversion symmetry breaking on the Fermi surface of magnetic topological insulators