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Motivated by the recent numerical evidence[1] of a short-range resonatingvalence bond state in the honeycomb lattice Hubbard model, we considerSchwinger boson mean field theories of possible spin liquid states on honeycomblattice. From general stability considerations the possible spin liquids willhave gapped spinons coupled to Z$_2$ gauge field. We apply the projectivesymmetry group(PSG) method to classify possible Z$_2$ spin liquid states withinthis formalism on honeycomb lattice. It is found that there are only tworelevant Z$_2$ states, differed by the value of gauge flux, zero or $\pi$, inthe elementary hexagon. The zero-flux state is a promising candidate for theobserved spin liquid and continuous phase transition into commensurate N\'eelorder. We also derive the critical field theory for this transition, which isthe well-studied O(4) invariant theory[2-4], and has an irrelevant couplingbetween Higgs and boson fields with cubic power of spatial derivatives asrequired by lattice symmetry. This is in sharp contrast to the conventionaltheory[5], where such transition generically leads to non-colinearincommensurate magnetic order. In this scenario the Z$_2$ spin liquid could beclose to a tricritical point. Soft boson modes will exist at seven differentwave vectors. This will show up as low frequency dynamical spin susceptibilitypeaks not only at the $\Gamma$ point (the N\'eel order wave vector) but also atBrillouin zone edge center $M$ points and twelve other points. Some simpleproperties of the $\pi$-flux state are studies as well. Symmetry allowedfurther neighbor mean field ansatz are derived in Appendix which can be used infuture theoretical works along this direction.

Schwinger Boson Mean Field Theories of Spin Liquid States on Honeycomb Lattice: Projective Symmetry Group Analysis and Critical Field Theory