  We investigate the correlation effects on spin-orbit coupling (SOC) in a
two-orbital Hubbard model on a square lattice by applying the variational Monte
Carlo method. We consider an effective SOC constant $\lambda_{\text{eff}}$ in
the one-body part of the variational wave function and mainly discuss the cases
of the electron number per site $n=1$, that is, quarter filling. We find that
$\lambda_{\text{eff}}$ is proportional to the bare value $\lambda$ and depends
on the electron-electron interactions through $(U'-J')$ in a relatively wide
parameter range in the paramagnetic (PM) phase, where $U'$ is the interorbital
Coulomb interaction and $J'$ is the pair hopping interaction. Increasing the
electron-electron interactions in the PM phase leads to a transition to an
effective one-band state, in which the upper band becomes empty due to the
enhanced $\lambda_{\text{eff}}$. We also construct phase diagrams considering
magnetic order. The carrier doping effect on $\lambda_{\text{eff}}$ is also
investigated. We find that $\lambda_{\text{eff}}$ enhances in a strongly
correlated region around the Mott transition point and it is necessary to
include the correlation effects beyond the Hartree-Fock approximation to
describe the enhanced SOC properly.
