  It is shown that, in a reasonable approximation, the quantum state of
$p$-bosons in a bi-partite square two-dimensional optical lattice is governed
by the nonlinear boson model describing tunneling of \textit{boson pairs}
between two orthogonal degenerate quasi momenta on the edge of the first
Brillouin zone. The interplay between the lattice anisotropy and the atomic
interactions leads to the second-order phase transition between the
number-squeezed and coherent phase states of the $p$-bosons. In the isotropic
case of the recent experiment, Nature Physicis 7, 147 (2011), the $p$-bosons
are in the coherent phase state, where the relative global phase between the
two quasi momenta is defined only up to mod($\pi$): $\phi=\pm\pi/2$. The
quantum phase diagram of the nonlinear boson model is given.
