  We study the ground state of the Hubbard model on a square lattice with two
degenerate orbitals per site and at integer fillings as a function of onsite
Hubbard repulsion $U$ and Hund's intra-atomic exchange coupling $J$. We use a
variational slave-spin mean field (VSSMF) method which allows symmetry broken
states to be studied within the computationally less intensive slave-spin mean
field formalism, thus making the method more powerful to study strongly
correlated electron physics. The results show that at half-filling, the ground
state at smaller $U$ is a Slater antiferromagnet (AF) with substantial local
charge fluctuations. As $U$ is increased, the AF state develops a Heisenberg
behavior, finally undergoing a first order transition to a Mott insulating AF
state at a critical interaction $U_c$ which is of the order of the bandwidth.
Introducing the Hund's coupling $J$ correlates the system more and reduces
$U_c$ drastically. At quarter-filling with one electron per site, the ground
state at smaller $U$ is paramagnetic metallic. At finite Hund's coupling $J$,
as interaction is increased above a lower critical value $U_{c1}$, it goes to a
fully spin polarized ferromagnetic state coexisting with an antiferro-orbital
order. The system eventually becomes Mott insulating at a higher critical value
$U_{c2}$. The results as a function of $U$ and $J$ are thoroughly discussed.
