We consider a model Hamiltonian with two $\mathrm{SU}\left(4\right)$ 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 ultra-cold-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 $\mathrm{U}(1$) 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 $\mathrm{SO}(N$) models on arbitrary lattices in a certain parameter regime.

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• Magnetism
• Strongly correlated systems
• Quantum Monte Carlo