Square lattice plaquette valence bond solid (pVBS) phase in SU(N) designer Hamiltonian with two-column representation

We construct an SU(N) symmetric designer four-spin Hamiltonian with a two-column representation and find that it realizes a plaquette valence bond solid (pVBS) phase on the square lattice for N ≥ 3. We add to this interaction the SU(N) Heisenberg interaction with a two-column representation which is known to realize the Néel phase for N ≤ 9 and the Haldane Nematic phase otherwise. We study the phase diagram of this model as a function of the relative strengths of the two interactions as well as the value of N. We establish direct transitions from the Néel and Haldane Nematic phases into the pVBS phase on increasing the strength of the plaquette interaction for small and large N respectively. We determine the nature of the two transitions to be first order.