Phase Diagram of square lattice S=1 Mott insulators

We study the phase diagram of a spin-1 Heisenberg Model with both 2 and 4 spin bilinear-biquadratic interactions on the square lattice. This model can be simulated sign-problem free with means of a Stochastic Series Expansion Quantum Monte Carlo (SSE QMC). We study the competition between the N\’eel, quadrupolar, valence bond solid and Haldane phases, and the nature of the quantum phase transitions between these states of matter.