  We investigate the magnetic properties of a repulsive fermionic SU($3$)
Hubbard model on the Lieb lattice from weak to strong interaction by means of
the mean-field approximation. To validate the method we employed, we first
discuss the SU($2$) Hubbard model at the mean-field level, and find that our
results are consistent with known rigorous theorems. We then extend the
calculation to the case of SU($3$) symmetry. We find that, at $4/9$ filling,
the SU$(3)$ symmetry spontaneously breaks into the SU$(2)\times$U$(1)$ symmetry
in the ground state, leading to a staggered ferromagnetic state for any
repulsive $U$ at zero temperature. We then investigate the stability of the
ferromagnetic state by relaxing the filling away from $4/9$, and conclude that
the ferromagnetic state is sensitive but robust to fillings, as it can persist
within a certain filling regime. We also apply the mean-field approximation to
finite temperature to calculate the critical temperature and the critical
entropy of the ferromagnetic state. As the resulting critical entropy per
particle is significantly greater than that can be realized in experiments, we
expect some quasi-long-range-ordered features of such a ferromagnetic state can
be realized and observed with fermionic alkaline-earth-metal(-like) atoms
loaded into optical lattices.
