  The SU(4) Heisenberg model can serve as a low energy model of the Mott
insulating state in materials where the spins and orbitals are highly
symmetric, or in systems of alkaline-earth atoms on optical lattice. Recently,
it has been argued that on the honeycomb lattice the model exhibits a unique
spin-orbital liquid phase with an algebraic decay of correlations [P. Corboz et
al., Phys. Rev. X 2, 041013 (2012)]. Here we study the instability of the
algebraic spin-orbital liquid toward spontaneous formation of SU(4) singlet
plaquettes (tetramerization). Using a variational Monte Carlo approach to
evaluate the projected wave-function of fermions with $\pi$-flux state, we find
that the algebraic liquid is robust, and that a finite value of the next
nearest exchange is needed to induce tetramerization. We also studied the phase
diagram of a model which interpolates between the nearest neighbor Heisenberg
model and a Hamiltonian for which the singlet-plaquette product state is an
exact ground state.
