  We consider the phase diagram of the most general SU(4)-symmetric two-site
Hamiltonian for a system of two fermions per site (ie self-conjugate $\bf 6$
representation) on the square lattice. It is known that this model hosts
magnetic phases breaking SU(4) symmetry and quantum disordered dimer-like
phases breaking lattice translation symmetry. Motivated by a previous work [O.
Gauth\'e, S. Capponi and D. Poilblanc, Phys. Rev. B $\textbf{99}$, 241112(R)
(2019)], we investigate the possibility of the existence of SU(4) quantum spin
liquid phases in this model, using SU(4)-symmetric Projected Entangled Pair
States (PEPS) of small bond dimensions, which can be classified according to
point group and charge (C) symmetries. Among several (disconnected) families of
SU(4)-symmetric PEPS, breaking or not C-symmetry, we identify critical or
topological spin liquids which may be stable in some regions of the phase
diagram. These results are confronted to exact diagonalisation (ED) and density
matrix renormalisation group (DMRG) calculations.
