Quantum phases of collective SU(3) spin systems with bipartite symmetry

We study a bipartite collective spin-\(1\) model with exchange interaction between the spins. The bipartite nature of the model manifests itself by the spins being divided into two equal-sized subsystems; within each subsystem the spin-spin interactions are of the same strength, across the subsystems they are also equal, but the two coupling values within and across the subsystem are different. Such a set-up is inspired by recent experiments with ultracold atoms. Using the \(\mathrm{SU}(3)\)-symmetry of the exchange interaction and the permutation symmetry within the subsystems, we can employ representation theoretic methods to diagonalize the Hamiltonian of the system in the entire parameter space of the two coupling-strengths. These techniques then allow us to explicitly construct and explore the ground-state phase diagram. The phase diagram turns out to be rich containing both gapped and gapless phases. An interesting observation is that one of the five phases features a strong bipartite symmetry breaking, meaning that the two subsystems in the ground states are in different \(\mathrm{SU}(3)\) representations.