  The study of SU(N) quantum spin models is relevant to a variety of physical
systems including ultracold atoms in optical lattices, and also leads to
insights into novel quantum phases and phase transitions of SU(2) spin models.
We use Gutzwiller projected fermionic variational wavefunctions to explore the
phase diagram and correlation functions of SU(N) spin models in the
self-conjugate representation, with Heisenberg bilinear and biquadratic
interactions. In 1D, the variational phase diagram of the SU(4) spin chain is
constructed by examining instabilities of the Gutzwiller projected free fermion
ground state to various broken symmetries, and it agrees well with exact
results.The spin and dimer correlations of the Gutzwiller projected free
fermion state with N flavors of fermions are also in good agreement with exact
and 1/N calculations for the critical points of SU(N) spin chains. In 2D, the
variational phase diagram on the square lattice is obtained by studying
instabilities of the Gutzwiller projected pi-flux state. The variational ground
state of the pure Heisenberg model is found to exhibit long range Neel order
for N=2,4 and spin Peierls order for N > 4. For N=4 and 6, biquadratic
interactions lead to a complex phase diagram which includes an extended valence
bond crystal in both cases, as well as a stable pi-flux phase for N=6. The spin
correlations of the projected pi-flux state at N=4 are in good agreement with
1/N calculations. We find that this state also shows strongly enhanced dimer
correlations, in qualitative accord with the large-N results. We compare our
results with a recent QMC study of the SU(4) Heisenberg model.
