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

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

Realization of the Exactly Solvable Kitaev Honeycomb Lattice Model in a Spin Rotation Invariant System

The exactly solvable Kitaev honeycomb lattice model is realized as the low energy effect Hamiltonian of a spin-1/2 model with spin rotation and time-reversal symmetry. The mapping to low energy effective Hamiltonian is exact, without truncation errors in traditional perturbation series expansions. This model consists of a honeycomb lattice of clusters of four spin-1/2 moments, and contains short-range interactions up to six-spin(or eight-spin) terms. The spin in the Kitaev model is represented not as these spin-1/2 moments, but as pseudo-spin of the two-dimensional spin singlet sector of the four antiferromagnetically coupled spin-1/2 moments within each cluster. Spin correlations in the Kitaev model are mapped to dimer correlations or spin-chirality correlations in this model. This exact construction is quite general and can be used to make other interesting spin-1/2 models from spin rotation invariant Hamiltonians. We discuss two possible routes to generate the high order spin interactions from more natural couplings, which involves perturbative expansions thus breaks the exact mapping, although in a controlled manner.Comment: 11 pages, 3 figure, 1 table, RevTex4, rewritten for clarity, error corrected, references added

Progress on Hardy-type Inequalities

This paper surveys some of our recent progress on Hardy-type inequa\-lities which consist of a well-known topic in Harmonic Analysis. In the first section, we recall the original probabilistic motivation dealing with the stability speed in terms of the $L^2$-theory. A crucial application of a result by Fukushima and Uemura (2003) is included. In the second section, the non-linear case (a general Hardy-type inequality) is handled with a direct and analytic proof. In the last section, it is illustrated that the basic estimates presented in the first two sections can still be improved considerably.Comment: 13 pages, 5 figures, Festschrift Masatoshi Fukushima: In Honor of Masatoshi Fukushima's Sanju. World Scientific, 201