We investigate the ground state phase diagram of an SU(
N)-symmetric antiferromagnetic spin model on a square lattice where each site hosts an irreducible representation of SU(
N) described by a square Young tableau of
N/2 rows and
2S columns. We show that negative sign free fermion Monte Carlo simulations can be carried out for this class of quantum magnets at any
S and even values of
N. In the large-
N limit, the saddle point approximation favors a four-fold degenerate valence bond solid phase. In the large
S-limit, the semi-classical approximation points to Néel state. On a line set by
N=8S+2 in the
S versus
N phase diagram, we observe a variety of phases proximate to the Néel state. At
S=1/2 and
3/2 we observe the aforementioned four fold degenerate valence bond solid state. At
S=1 a two fold degenerate spin nematic state in which the C
4 lattice symmetry is broken down to C
2 emerges. Finally at
S=2 we observe a unique ground state that pertains to a two-dimensional version of the Affleck-Kennedy-Lieb-Tasaki state. For our specific realization, this symmetry protected topological state is characterized by an SU(18),
S=1/2 boundary state, that has a dimerized ground state. These phases that are proximate to the Néel state are consistent with the notion of monopole condensation of the antiferromagnetic order parameter. In particular one expects spin disordered states with degeneracy set by
mod(4,2S).
The nature of the intermediate ground‐state phase in the spin‐1/2 frustrated square lattice model has long been debated. We investigate the phase diagram of this model using cluster density matrix embedding theory (CDMET). Two antiferromagnetic phases are directly identified from our analysis of the ground‐state wave function: the Néel antiferromagnetic phase for J 2 ≤ 0.45 and the stripe antiferromagnetic phase for J 2 ≥ 0.65. Although our analysis of the ground state finds no direct evidence of an internal phase transition within the intermediate phase, an examination of the first excited state reveals a continuous quantum phase transition therein, with a critical point at J 2 = 0.55 that separates the intermediate phase into a plaquette valence bond solid (PVBS) and a (P + C) valence bond solid (VBS).