Phase diagram of the SU(N) antiferromagnet of spin S on a square lattice

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 fourfold degenerate valence bond solid phase. In the large S limit, the semiclassical approximation points to the Neel 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 Neel state. At S = 1/2 and 3/2, we observe the aforementioned fourfold degenerate valence bond solid state. At S = 1, a twofold 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 Neel 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).