7 modes of neighboring tetrahedra. And these coupling constants λ$_{x,y,z}$ need to be tuned to produce J$_{x,y,z}$ of the Kitaev model. This is still not easy to implement in solid state systems. At lowest non-trivial order of pertur- bative expansion, we do get our model (9). Higher order terms in expansion destroy the exact solvability, but may be controlled by the small parameters λ$_{x,y,z}$/k . the desired J$_{x,y,z}$ terms in (8) from the first and second order of perturbations. The calculation can be dramatically simplified by the following fact that any physical spin-1/2 operator S x,y,z ℓ converts the cluster spin singlet states | τ z = ± 1 〉 into spin-1 states of the cluster. This can be checked by explicit calculations and will not be proved here. For all the perturbations to be considered later, the above mentioned fact can be exploited to replace the factor [0 − H$_{cluster}$ j − H$_{cluster}$ $_{k}$] $^{−}$1 in the second order pertur- bation to a c -number ( − 2 J$_{cluster}$ ) $^{$^{−}$1}$. The detailed calculations are given in Appendix B. We will only list the results here. The perturbation on x -links is given by In this Subsection we consider more conventional per- turbations, magnetic interactions between the clusters, e.g. the Heisenberg coupling S$_{j}$ · S$_{k}$ with j and k belong to different tetrahedra. This has the advantage over the previous phonon approach for not introducing additional degrees of freedom. But it also has a significant disad- vantage: the perturbation does not commute with the cluster Heisenberg Hamiltonian (2), so the cluster sin- glet subspace will be mixed with other total spin states. In this Subsection we will use the spin-chirality represen- tation (6) for τ $^{z}$. B. Generate the High Order Terms by Magnetic Interactions between Clusters. λ$_{x}$ H$_{perturbation}$$_{, x}$ = λ$_{x}$ [ S$_{j}$$_{1}$ · S$_{k}$$_{1}$ + sgn( J$_{x}$ ) · ( S$_{j}$$_{2}$ · S$_{k}$$_{2}$ )] − J$_{x}$ ( S$_{j}$$_{1}$ · S$_{j}$$_{2}$ + S$_{k}$$_{1}$ · S$_{k}$$_{2}$ ) . where λ$_{x}$ = √ 12 | J$_{x}$ | · J$_{cluster}$ , sgn( J$_{x}$ ) = ± 1 is the sign of J$_{x}$ . The perturbation on y -links is λ$_{y}$ H$_{perturbation}$$_{, y}$ = λ$_{y}$ [ S$_{j}$$_{1}$ · S$_{k}$$_{1}$ + sgn( J$_{y}$ ) · ( S$_{j}$$_{3}$ − S$_{j}$$_{4}$ ) · ( S$_{k}$$_{3}$ − S$_{k}$$_{4}$ )] − | J$_{y}$ | ( S$_{j}$$_{3}$ · S$_{j}$$_{4}$ + S$_{k}$$_{3}$ S$_{k}$$_{4}$ ) with λ$_{y}$ = √ 4 | J$_{y}$ | · J$_{cluster}$ . The perturbation on z -links is λ$_{z}$ H$_{perturbation}$$_{, z}$ = λ$_{z}$ [ S$_{j}$$_{2}$ · ( S$_{k}$$_{3}$ × S$_{k}$$_{4}$ ) + sgn( J$_{z}$ ) · S$_{k}$$_{2}$ · ( S$_{j}$$_{3}$ × S$_{j}$$_{4}$ )] − | J$_{z}$ | ( S$_{j}$$_{3}$ · S$_{j}$$_{4}$ + S$_{k}$$_{3}$ · S$_{k}$$_{4}$ ) . with λ$_{z}$ = 4 √ | J$_{z}$ | · J$_{cluster}$ . The entire Hamiltonian H$_{magnetic}$ reads explicitly as, Again consider two clusters j and k . For simplicity of notations define a projection operator P jk = P j P $_{k}$, where P j,k is projection into the singlet subspace of clus- ter j and k , respectively, P j,k = ∑ s = $^{±}$1 | τ z j,k = s 〉〈 τ z j,k = s | . For a given perturbation λ H$_{perturbation}$ with small parameter λ (in factor λ/J$_{cluster}$ is the expansion param- eter), lowest two orders of the perturbation series are With proper choice of λ and H$_{perturbation}$ we can generate λ P $_{jk}$H$_{perturbation}$ P jk + λ 2 P $_{jk}$H$_{perturbation}$ (1 − P $_{jk}$) × [0 − H$_{cluster}$ j − H$_{cluster}$ $_{k}$] $^{$^{−}$1}$(1 − P $_{jk}$) H$_{perturbation}$ P jk (15) H$_{magnetic}$ = ∑ cluster j ( J$_{cluster}$/ 2)( S$_{j}$$_{1}$ + S$_{j}$$_{2}$ + S$_{j}$$_{3}$ + S$_{j}$$_{4}$ ) 2 + ∑ x $^{−}$links $_{<jk>}${ √ 12 | J$_{x}$ | · J$_{cluster}$ [ S$_{j}$$_{1}$ · S$_{k}$$_{1}$ + sgn( J$_{x}$ ) · ( S$_{j}$$_{2}$ · S$_{k}$$_{2}$ ) ] − J$_{x}$ ( S$_{j}$$_{1}$ · S$_{j}$$_{2}$ + S$_{k}$$_{1}$ · S$_{k}$$_{2}$ ) } + ∑ y $^{−}$links $_{<jk>}${ √ 4 | J$_{y}$ | · J$_{cluster}$ [ S$_{j}$$_{1}$ · ( S$_{k}$$_{3}$ − S$_{k}$$_{4}$ ) + sgn( J$_{y}$ ) S$_{k}$$_{1}$ · ( S$_{j}$$_{3}$ − S$_{j}$$_{4}$ ) ] − | J$_{y}$ | ( S$_{j}$$_{3}$ · S$_{j}$$_{4}$ + S$_{k}$$_{3}$ · S$_{k}$$_{4}$ ) } + ∑ z $^{−}$links $_{<jk>}${ 4 √ | J$_{z}$ | · J$_{cluster}$ [ S$_{j}$$_{2}$ · ( S$_{k}$$_{3}$ × S$_{k}$$_{4}$ ) + sgn( J$_{z}$ ) S$_{k}$$_{2}$ · ( S$_{j}$$_{3}$ × S$_{j}$$_{4}$ ) ] − | J$_{z}$ | ( S$_{j}$$_{3}$ · S$_{j}$$_{4}$ + S$_{k}$$_{3}$ · S$_{k}$$_{4}$ ) } . (16) In (16), we have been able to reduce the four spin in- teractions in (8) to inter-cluster Heisenberg interactions, and the six-spin interactions in (8) to inter-cluster spin- chirality interactions. The inter-cluster Heisenberg cou- plings in H$_{perturbation}$ x,y may be easier to arrange. The inter-cluster spin-chirality coupling in H$_{perturbation}$ z ex- plicitly breaks time reversal symmetry and is probably harder to implement in solid state systems. However spin-chirality order may have important consequences in frustrated magnets $^{36,37}$, and a realization of spin-