FIG. 1: The honeycomb lattice for the Kitaev model. Filled and open circles indicate two sublattices. x, y, z label the links along three different directions used in (1). FIG. 2: Left: the physical spin lattice for the model (8). The dash circles are honeycomb lattice sites, each of which is ac- tually a cluster of four physical spins. The dash straight lines are honeycomb lattice bonds, with their type x, y, z labeled. The interaction between clusters connected by x, y, z bonds are the J x,y,z terms in (8) or (9) respectively. Note this is not the 3-12 lattice used in Ref. $^{9,10}$. Right: enlarged picture of the clusters with the four physical spins labeled as 1 , . . . , 4. Thick solid bonds within one cluster have large antiferromag- netic Heisenberg coupling J cluster . 2 4 2 3 1 z z x x y y 2 3 4 1 y x z z z z z z z z z y y y x x x x x x y y y y x y x z z y y x x x y z x x derived as well. There have been several proposals to open the fermion gap for the non-Abelian phase without spoiling exact solvability $^{4,6}$. And many generalizations to other(even 3D) lattices have been developed in the last few years $^{10–16}$. All these efforts have significantly enriched our knowledge of exactly solvable models and quantum phases of matter. However, in the original Kitaev model and its later generalizations in the form of spin models, spin rotation symmetry is explicitly broken. This makes them harder to realize in solid state systems. There are many pro- posals to realized the Kitaev model in more controllable situations, e.g. in cold atom optical lattices $^{17,18}$, or in superconducting circuits $^{19}$. But it is still desirable for theoretical curiosity and practical purposes to realize the Kitaev-type models in spin rotation invariant systems. tion III the Kitaev model will be explicitly constructed using this formalism, and some properties of this con- struction will be discussed. In Section IV we will discuss two possible ways to generate the high order spin in- teractions involved in the construction of Section III by perturbative expansions. Conclusions and outlook will be summarized in Section V. In this paper we realize the Kitaev honeycomb lattice model as the low energy Hamiltonian for a spin rotation invariant system. The trick is not to use the physical spin as the spin in the Kitaev model, instead the spin-1/2 in Kitaev model is from some emergent two-fold degener- ate low energy states in the elementary unit of physical system. This type of idea has been explored recently by Jackeli and Khaliullin $^{20}$, in which the spin-1/2 in the Ki- taev model is the low energy Kramers doublet created by strong spin-orbit coupling of t$_{2}$$_{g}$ orbitals. In the model presented below, the Hilbert space of spin-1/2 in the Ki- taev model is actually the two dimensional spin singlet sector of four antiferromagnetically coupled spin-1/2 mo- ments, and the role of spin-1/2 operators(Pauli matrices) in the Kitaev model is replaced by certain combinations of S$_{j}$ · S$_{k}$ [or the spin-chirality S$_{j}$ · ( S$_{k}$ × S$_{ℓ}$ )] between the four spins. In this Section we will construct the pseudo-spin-1/2 from a cluster of four physical spins, and map the phys- ical spin operators to pseudo-spin operators. The map- ping constructed here will be used in later Sections to construct the effective Kitaev model. In this Section we will work entirely within the four-spin cluster, all unspec- ified physical spin subscripts take values 1 , . . . , 4. II. FORMULATION OF THE PSEUDO-SPIN-1/2 FROM FOUR-SPIN CLUSTER. One major drawback of the model to be presented is that it contains high order spin interactions(involves up to six or eight spins), thus is still unnatural. However it opens the possibility to realize exotic (exactly solvable) models from spin-1/2 Hamiltonian with spin rotation in- variant interactions. We will discuss two possible routes to reduce this artificialness through controlled perturba- tive expansions, by coupling to optical phonons or by magnetic couplings between the elementary units. The outline of this paper is as follows. In Section II we will lay out the pseudo-spin-1/2 construction. In Sec- Consider a cluster of four spin-1/2 moments(called physical spins hereafter), labeled by S$_{1}$$_{,...,}$$_{4}$ , antiferro- magnetically coupled to each other (see the right bot- tom part of FIG. 2). The Hamiltonian within the clus- ter(up to a constant) is simply the Heisenberg antiferro- magnetic(AFM) interactions, The energy levels should be apparent from this form: one group of spin-2 quintets with energy 3 J$_{cluster}$ , three groups of spin-1 triplets with energy J$_{cluster}$ , and two spin singlets with energy zero. We will consider large positive H$_{cluster}$ = ( J$_{cluster}$/ 2) ( S$_{1}$ + S$_{2}$ + S$_{3}$ + S$_{4}$ ) 2 (2)