  Using infinite projected entangled pair states (iPEPS), exact
diagonalization, and flavor-wave theory, we show that the SU(4) Heisenberg
model undergoes a spontaneous dimerization on the square lattice, in contrast
to its SU(2) and SU(3) counterparts, which develop Neel and three-sublattice
stripe-like long-range order. Since the ground state of a dimer is not a
singlet for SU(4) but a 6-dimensional irrep, this leaves the door open for
further symmetry breaking. We provide evidence that, unlike in SU(4) ladders,
where dimers pair up to form singlet plaquettes, here the SU(4) symmetry is
additionally broken, leading to a gapless spectrum in spite of the broken
translational symmetry.
