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

Abstract

Motivated by the recent numerical evidence[1] of a short-range resonating valence bond state in the honeycomb lattice Hubbard model, we consider Schwinger boson mean field theories of possible spin liquid states on honeycomb lattice. From general stability considerations the possible spin liquids will have gapped spinons coupled to Z$_2$ gauge field. We apply the projective symmetry group(PSG) method to classify possible Z$_2$ spin liquid states within this formalism on honeycomb lattice. It is found that there are only two relevant Z$_2$ states, differed by the value of gauge flux, zero or $\pi$, in the elementary hexagon. The zero-flux state is a promising candidate for the observed spin liquid and continuous phase transition into commensurate N\'eel order. We also derive the critical field theory for this transition, which is the well-studied O(4) invariant theory[2-4], and has an irrelevant coupling between Higgs and boson fields with cubic power of spatial derivatives as required by lattice symmetry. This is in sharp contrast to the conventional theory[5], where such transition generically leads to non-colinear incommensurate magnetic order. In this scenario the Z$_2$ spin liquid could be close to a tricritical point. Soft boson modes will exist at seven different wave vectors. This will show up as low frequency dynamical spin susceptibility peaks not only at the $\Gamma$ point (the N\'eel order wave vector) but also at Brillouin zone edge center $M$ points and twelve other points. Some simple properties of the $\pi$-flux state are studies as well. Symmetry allowed further neighbor mean field ansatz are derived in Appendix which can be used in future theoretical works along this direction.Comment: mistakes in Eq.(13) and Eq.(A17) corrected on top of published version, 14 pages, 6 figure