Phase diagrams of S= 1/2 Bilayer Models of SU(2) symmetric antiferromagnets

We study the T=0 phase diagrams of models of bilayers of S=1/2 square lattices antiferromagnets with SU(2) Heisenberg symmetry that have 2, 4 and 6 spin exchanges. We study two families of models, a traditional bilayer model in which the interlayer interaction is Heisenberg so that the two layers can exchange spin (and energy) with each other making it possible to achieve a simple dimerized liquid like state. The resulting phase diagrams is rich with Neel, valence bond solid and simple dimer phases and both first order and continuous transitions, which we demonstrate our consistent with the conventional Landau theory or order prameters. In a second family of models in which the layers can exchange only energy but no spin (reminiscent of the Ashkin-Teller coupling), the simple dimer state cannot occur. Interestingly, we find that the phase transition between N\'eel and VBS is first order in both the spin-spin and energy-energy coupled models, although they have strikingly distinct finite size scaling behavior.