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.

Figure 1

(Color online) Three flippable configurations in one cube. The resonance is represented in the $t$ term in Eq. (1).

Figure 2

(Color online) The duality transformation defined in Eq. (14). On dual sites 1 and 2, there are three components of dual vector height $h_{\mu }$, and the dual transformation for the shaded face is $E_{yz}={(h_y-h_z)}_2-{(h_y-h_z)}_1={\nabla }_x(h_y-h_z)$.

Figure 3

(Color online) (a) The conserved quantity $m_{xy}$ is defined as the summation of all $E_{xy}$ on all the shaded squares along one $z$ axis. (b) The view of the 3D lattice from the top. If the quantity $m_{xy}$ is fixed on all the shaded squares shown in this figure, $m_{xy}$ is determined on the whole lattice.