Spin-S Designer Hamiltonians and the Square Lattice S=1 Haldane Nematic

We introduce a strategy to write down lattice models of spin rotational symmetric Hamiltonians with arbitrary spin S that are Marshall positive and can be simulated efficiently using world-line Monte Carlo methods. As an application of our approach we consider a square lattice S = 1 model for which we design a (3 x 3)-spin plaquette interaction. By numerical simulations we establish that our model realizes a novel Haldane nematic phase that breaks lattice rotational symmetry by the spontaneous formation of Haldane chains, while preserving spin rotations, time reversal, and lattice translations. By supplementing our model with a two-spin Heisenberg interaction, we present a study of the transition between Neel and Haldane nematic phase, which we find to be of first order.