Resonating plaquette phases in SU(4) Heisenberg antiferromagnet

Large spin cold atom systems can exhibit magnetic properties that do not appear in usual spin-1∕2 systems. We investigate the SU (4) resonating plaquette state in the three-dimensional cubic optical lattice with spin-3∕2 cold fermions. A gauge field formalism is constructed to describe the Rokhsar--Kivelson type of Hamiltonian and a duality transformation is used to study the phase diagram. Due to the proliferation of topological defects, the system is generally gapped for the whole phase diagram of the quantum model, which agrees with the recent numerical studies. The classical plaquette model on the cubic lattice is also studied, and a critical phase is predicted by tuning one parameter in the low energy field theory.