  We consider a model Hamiltonian with two SU(4) fermions per site on a square
lattice, showing a competition between bilinear and biquadratic interactions.
This model has generated interest due to possible realizations in ultracold
atom experiments and existence of spin liquid ground states. Using a basis
transformation, we show that part of the phase diagram is amenable to quantum
Monte Carlo simulations without a sign problem. We find evidence for spin
nematic and valence bond crystalline phases, which are separated by a weak
first order phase transition. A U(1) symmetry is found to emerge in the valence
bond crystal histograms, suggesting proximity to a deconfined quantum critical
point. Our results are obtained with the help of a loop algorithm which allows
large-scale simulations of bilinear-biquadratic SO(N ) models on arbitrary
lattices in a certain parameter regime.
