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

Phase transitions in coupled two dimensional XY systems with spatial anisotropy