10 Using this result we can choose the following pertur- bation on z -links, this term becomes 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}$$_{4}$ ) − λ 2 6 J$_{cluster}$ · (3 / 4)[3 / 16 + ( τ $^{x}$/ 2 − 1 / 4) $^{2}$] = − ( λ $^{2}$) / (32 J$_{cluster}$ ) · (2 − τ x $_{k}$) . r λ 2 P $_{jk}$S$_{j}$$_{2}$ · ( S$_{k}$$_{3}$ × S$_{k}$$_{4}$ )(1 − P $_{jk}$) × [0 − H$_{cluster}$ j − H$_{cluster}$ $_{k}$] $^{−}$1 (1 − P $_{jk}$) S$_{k}$$_{2}$ · ( S$_{j}$$_{3}$ × S$_{j}$$_{4}$ ) P jk H$_{magnetic}$ = ∑ j H$_{cluster}$ j + ∑ x $^{−}$links <jk> λ$_{x}$H$_{perturbation}$ x + ∑ y $^{−}$links <jk> λ$_{y}$H$_{perturbation}$ y + ∑ z $^{−}$links <jk> λ$_{z}$H$_{perturbation}$ z − r λ 2 2 J$_{cluster}$ P $_{jk}$[ S$_{j}$$_{2}$ · ( S$_{k}$$_{3}$ × S$_{k}$$_{4}$ )][ S$_{k}$$_{2}$ · ( S$_{j}$$_{3}$ × S$_{j}$$_{3}$ )] P $_{jk}$. − r λ 2 16 J$_{cluster}$ · τ z j τ z k + λ 2 32 J$_{cluster}$ ( τ x k + r 2 τ x j − 2 r 2 − 2) . Another second order perturbation term r $^{2}$λ2 P $_{jk}$S$_{k}$$_{2}$ · ( S$_{j}$$_{3}$ × S$_{j}$$_{4}$ )(1 − P $_{jk}$)[0 − H$_{cluster}$ j − H$_{cluster}$ $_{k}$] $^{$^{−}$1}$(1 − P $_{jk}$) S$_{k}$$_{2}$ · ( S$_{j}$$_{3}$ × S$_{j}$$_{4}$ ) P jk can be computed in the similar way and gives the result − ( r 2 λ $^{2}$) / (32 J$_{cluster}$ ) · (2 − τ x j ). For one of the cross term with λ$_{z}$ = 4 √ | $^{J$_{z}$}$|J$_{cluster}$ , r = sgn( J$_{z}$ ) is the sign of J$_{z}$ . The last term on the right-hand-side is to cancel the non- trivial terms ( r 2 τ x j + τ x $_{k}$) λ 2 $_{z}$/ (32 J$_{cluster}$ ) from the second order perturbation of the first term. Up to second order perturbation this will produce − J$_{z}$τ z j τ z k interactions. Finally we have been able to reduce the high order interactions to at most three spin terms, the Hamiltonian H$_{magnetic}$ is We can use the previous argument for both cluster j and k , so (1 −P $_{AB}$)[0 − H$_{cluster}$ j − H$_{cluster}$ $_{k}$] $^{$^{−}$1}$(1 −P $_{jk}$) can be replace by c -number ( − 2 J$_{cluster}$ ) $^{$^{−}$1}$. This term becomes Spin rotation symmetry again helps to separate the terms for cluster j and k , and we get − ( r λ $^{2}$) / (32 J$_{cluster}$ ) · τ z j τ z $_{k}$. The other cross term r λ 2 P $_{jk}$S$_{k}$$_{2}$ · ( S$_{j}$$_{3}$ × S$_{j}$$_{4}$ )(1 − P $_{jk}$)[0 − H$_{cluster}$ j − H$_{cluster}$ $_{k}$] $^{$^{−}$1}$(1 − P $_{jk}$) S$_{j}$$_{2}$ · ( S$_{k}$$_{3}$ × S$_{k}$$_{4}$ ) P jk gives the same result. In summary the second order perturbation from λ [ S$_{j}$$_{2}$ · ( S$_{j}$$_{3}$ × S$_{j}$$_{4}$ ) + r S$_{k}$$_{2}$ · ( S$_{j}$$_{3}$ × S$_{j}$$_{4}$ )] is where H$_{cluster}$ j are given by (2), λ$_{x,y,z}$ H$_{perturbation}$ x,y,z are given above. Plug in relevant equations we get (16) in Subsection IV B. (2009). 1 Alexei Kitaev, Ann. Phys. (N.Y.) 321 , 2 (2006). 2 Xiao-Yong Feng, Guang-Ming Zhang, Tao Xiang, Phys. Rev. Lett. 98 , 087204 (2007). 3 Han-Dong Chen, Zohar Nussinov, J. Phys. A: Math. Theor. 41 , 075001 (2008). 4 Dung-Hai Lee, Guang-Ming Zhang, Tao Xiang, Phys. Rev. Lett. 99 , 196805 (2007). 5 Yue Yu, Nucl. Phys. B 799 , 345 (2008). 6 Yue Yu, Ziqiang Wang, Europhys. Lett. 84 , 57002 (2008). 14 Congjun Wu, Daniel Arovas, Hsiang-Hsuan Hung, Phys. Rev. B 79 , 134427 (2009). 15 Shinsei Ryu, Phys. Rev. B 79 , 075124 (2009). 16 G. Baskaran, G. Santhosh, R. Shankar, arXiv:0908.1614 (2009). 17 L.-M. Duan, E. Demler, M. D. Lukin, Phys. Rev. Lett. 91 , 090402 (2003). 18 A. Micheli, G. K. Brennen, P. Zoller, Nature Physics 2 , 341 (2006). 7 G. Kells, J. K. Slingerland, J. Vala, Phys. Rev. B 80 , 125415 (2009). 8 Han-Dong Chen, B. Wang, S. Das Sarma, arXiv:0906.0017 (2009). 19 J. Q. You, Xiao-Feng Shi, Xuedong Hu, Franco Nori, Phys. Rev. B 81 , 014505 (2010). 20 G. Jackeli, G. Khaliullin, Phys. Rev. Lett. 102 , 017205 (2009). 9 K.P. Schmidt, S. Dusuel, and J. Vidal, Phys. Rev. Lett. 100 , 057208 (2008); J. Vidal, K.P. Schmidt, and S. Dusuel, Phys. Rev. B 78 , 245121 (2008); S. Dusuel, K.P. Schmidt, J. Vidal, and R.L. Zaffino, Phys. Rev. B 78 , 125102 (2008). 21 A. B. Harris, A. J. Berlinsky, C. Bruder, J. Appl. Phys. 69 , 5200 (1991). 22 K. A. Chao, J. SpaGLYPH<suppress>lek, A. M. Ole´s, Phys. Rev. B 18 , 3453 (1978). 10 Hong Yao, Steven A. Kivelson, Phys. Rev. Lett. 99 , 247203 (2007). 23 A. H. MacDonald, S. M. Girvin, D. Yoshioka, Phys. Rev. B 37 , 9753 (1988). 24 J. T. Chayes, L. Chayes, S. A. Kivelson, Commun. Math. Phys. 123 , 53 (1989). 25 C. D. Batista, S. A. Trugman, Phys. Rev. Lett. 93 , 217202 (2004). 11 S. Yang, D. L. Zhou, C. P. Sun, Phys. Rev. B 76 , 180404(R) (2007). 12 Hong Yao, Shou-Cheng Zhang, Steven A. Kivelson, Phys. Rev. Lett. 102 , 217202 (2009). 13 Zohar Nussinov, Gerardo Ortiz, Phys. Rev. B 79 , 214440