  We study phase transitions of coupled two dimensional XY systems with spatial
anisotropy and $U(1) \times \mathbb{Z}_2$ symmetry, motivated by spinless
bosonic atoms trapped in square optical lattice on the metastable first excited
$p-$level orbitals with anisotropic hopping amplitudes. The phase transitions
of the system are generally split into an Ising transition and an XY
transition, but the sequence and the nature of the transitions depend on the
ratio between the anisotropic couplings. In the isotropic limit the XY
variables are expected to be disordered before the Ising variables when thermal
or quantum fluctuations are turned on gradually. In the anisotropic limit with
zero perpendicular hoppings, the finite temperature transition is a
Kosterlitz-Thouless transition driven by proliferation of hybrid half vortices,
and the zero temperature quantum phase transition is split into a bond order
transition and a 3D XY transition, which can be driven by the condensation of
either single vortices or half vortices. After the condensation of half
vortices the resultant state is a Mott Insulator of paired bosons. A small
perpendicular hopping $J_b$ leads to a 2D Ising transition at low temperature
and a 2+1d quantum Ising transition with a small charging energy at zero
temperature. Global phase diagrams for both classical and quantum phase
transitions are drawn. The analytical results obtained in this work are
expected to be checked both numerically and experimentally.
