  Recently, the lattice shaking technique has been used to couple different
Bloch bands resonantly. For the one-dimensional (1D) case, in which shaking is
along only one direction, experimental observation of domain-wall formation has
been explained by superfluid Ising transition. Inspired by these, we generalize
to a 2D case in which shaking is along two orthogonal directions. Analogous to
the 1D case, we find three different phases, the normal superfluid (NSF) phase,
the $D_4$ symmetry-breaking superfluid ($D_4$SF) phase and the Mott insulator
(MI) phase. Furthermore, we demonstrate that the interaction effect induced by
inhomogeneous band mixing can modify the critical shaking amplitude. Unlike in
the 1D case, shaking types also can modify the critical shaking amplitude.
Unlike in the 1D case, shaking types also can modify the critical shaking
amplitude. We also construct a low-energy effective field theory to study the
quantum criticality of bosons near the tricritical point of NSF, $D_4$SF and MI
phases. Moreover, we find a Bose liquid with anisotropically algebraic order
and propose to change the Bose-Einstein condensation (BEC) into a non-condensed
Bose liquid by tuning the shaking amplitude approaching the critical value.
