  We study the instabilities caused by four fermion interactions on algebraic
spin liquids. Renormalization group (RG) is used for three types of previously
proposed spin liquids on the square lattice: the staggered flux state of SU(2)
spin system, the $\pi-$flux state of SU(4) spin system, and the $\pi-$flux
state of SU(2) spin system. The low energy field theories of the first two
types of spin liquids are QED3 with emerged SU(4) and SU(8) flavor symmetries,
the low energy theory of the $\pi-$flux SU(2) spin liquid is the QCD3 with
SU(2) gauge field and emergent Sp(4) (SO(5)) flavor symmetry. Suitable large-N
generalization of these spin liquids are discussed, and a systematic 1/N
expansion is applied to the RG calculations. The most relevant four fermion
perturbations are identified, and the possible phases driven by relevant
perturbations are discussed.
