  The antiferromagnetic (AFM) topological insulator MnBi$_{2}$Te$_{4}$ was
believed to have a topological surface state (TSS) with large band gap due to
the ferromagnetic (FM) order on the surface, and be able to host the
long-sought axion states. However, recent angle-resolved photoemission
spectroscopy (ARPES) experiments indicate that the TSS is gapless,
contradicting the theoretical predictions. Meanwhile, several experiments have
suggested that there is robust out-of-plane FM order on the surface of
MnBi$_{2}$Te$_{4}$. To understand these seemingly contradictory results, we
carry out comprehensive first-principles calculations to investigate the
interplay between the surface magnetism and the TSS. Our calculations provide
direct evidence that in a wide range of parameters, the (nearly) gapless TSS
can coexist with the surface FM order, therefore solving the paradox of the
surface magnetism and the gapless TSS. We further show that proximity effects
can be a promising route to open the gap in the TSS of MnBi$_{2}$Te$_{4}$. Our
research deepens the understanding of the relationship between surface
magnetism and TSS.
