  In addition to low-energy spin fluctuations, which distinguish them from band
insulators, Mott insulators often possess orbital degrees of freedom when
crystal-field levels are partially filled. While in most situations spins and
orbitals develop long-range order, the possibility for the ground state to be a
quantum liquid opens new perspectives. In this paper, we provide clear evidence
that the SU(4) symmetric Kugel-Khomskii model on the honeycomb lattice is a
quantum spin-orbital liquid. The absence of any form of symmetry breaking -
lattice or SU(N) - is supported by a combination of semiclassical and numerical
approaches: flavor-wave theory, tensor network algorithm, and exact
diagonalizations. In addition, all properties revealed by these methods are
very accurately accounted for by a projected variational wave-function based on
the \pi-flux state of fermions on the honeycomb lattice at 1/4-filling. In that
state, correlations are algebraic because of the presence of a Dirac point at
the Fermi level, suggesting that the symmetric Kugel-Khomskii model on the
honeycomb lattice is an algebraic quantum spin-orbital liquid. This model
provides a good starting point to understand the recently discovered
spin-orbital liquid behavior of Ba_3CuSb_2O_9. The present results also suggest
to choose optical lattices with honeycomb geometry in the search for quantum
liquids in ultra-cold four-color fermionic atoms.
