  We non-perturbatively investigate the ground state magnetic properties of the
2D half-filled SU($2N$) Hubbard model in the square lattice by using the
projector determinant quantum Monte Carlo simulations combined with the method
of local pinning fields. Long-range Neel orders are found for both the SU(4)
and SU(6) cases at small and intermediate values of $U$. In both cases, the
long-range Neel moments exhibit non-monotonic behavior with respect to $U$,
which first grow and then drop as $U$ increases. This result is fundamentally
different from the SU(2) case in which the Neel moments increase monotonically
and saturate. In the SU(6) case, a transition to the columnar dimer phase is
found in the strong interaction regime.
