Appendix B: Derivation of the Terms Generated by Second Order Perturbation of Inter-cluster Magnetic Interactions − λ 2 6 $_{J$_{cluster}$}$[ 9 + 9 r 2 8 + 2 r P $_{jk}$[ S$_{j}$$_{1}$ · ( S$_{j}$$_{3}$ − S$_{j}$$_{4}$ )][ S$_{k}$$_{1}$ · ( S$_{k}$$_{3}$ − S$_{k}$$_{4}$ )] P jk − (3 / 2) P $_{jk}$( S$_{k}$$_{3}$ · S$_{k}$$_{4}$ + r $^{2}$S$_{j}$$_{3}$ · S$_{j}$$_{4}$ ) P jk ] = − λ 2 6 $_{J$_{cluster}$}$[ 9 + 9 r 2 8 + 2 r (3 / 4) τ y j τ y k − (3 / 2) P $_{jk}$( S$_{k}$$_{3}$ · S$_{k}$$_{4}$ + r $^{2}$S$_{j}$$_{3}$ · S$_{j}$$_{4}$ ) P jk ] − λ 2 2 J$_{cluster}$ P $_{jk}$( H$_{perturbation}$ ) $^{2}$P jk 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}$$_{4}$ ) − λ 2 2 J$_{cluster}$ ∑ a,b [ P $_{j}$S a j $_{1}$S b j 1 P j · P $_{k}$S a k $_{1}$S b k 1 P k + 2 r P $_{j}$S a j $_{1}$S b j 2 P j · P $_{k}$S a k $_{1}$S b k 2 P k + r 2 $_{j}$S a j $_{2}$S b j 2 j $_{k}$S a k $_{2}$S b k 2 k ] λ 2 P $_{jk}$S$_{j}$$_{2}$ · ( S$_{k}$$_{3}$ × S$_{k}$$_{4}$ )(1 − P $_{jk}$) × [0 − H$_{cluster}$ j − H$_{cluster}$ $_{k}$] $^{−}$1 (1 − P $_{jk}$) S$_{j}$$_{2}$ · ( S$_{k}$$_{3}$ × S$_{k}$$_{4}$ ) P jk − λ 2 6 $_{J$_{cluster}$}$[ 9 + 9 r 2 16 + 2 r P $_{jk}$( S$_{j}$$_{1}$ · S$_{j}$$_{2}$ )( S$_{k}$$_{1}$ · S$_{k}$$_{2}$ ) P jk ] = − λ 2 6 $_{J$_{cluster}$}$[ 9 + 9 r 2 16 + ( r/ 2) τ x j τ x k − r/ 2 − r P $_{jk}$( S$_{j}$$_{1}$ · S$_{j}$$_{2}$ + S$_{k}$$_{1}$ · S$_{k}$$_{2}$ ) P jk ] . λ$_{x}$ H$_{perturbation}$$_{, 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}$ ) − λ 2 6 J$_{cluster}$ P $_{j}$S$_{j}$$_{2}$ · S$_{j}$$_{2}$ P j · P $_{k}$( S$_{k}$$_{3}$ × S$_{k}$$_{4}$ ) · ( S$_{k}$$_{3}$ × S$_{k}$$_{4}$ ) P $_{k}$. Use ( S ) 2 = 3 / 4 and ( S$_{k}$$_{3}$ × S$_{k}$$_{4}$ ) · ( S$_{k}$$_{3}$ × S$_{k}$$_{4}$ ) = ∑ a,b ( S a k $_{3}$S b k $_{4}$S a k $_{3}$S b k 4 − S a k $_{3}$S b k $_{4}$S b k $_{3}$S a k $_{4}$) = ( S$_{k}$$_{3}$ · S$_{k}$$_{3}$ )( S$_{k}$$_{4}$ · S$_{k}$$_{4}$ ) − ∑ a,b S a k $_{3}$S b k $_{3}$[ δ$_{ab}$/ 2 − S a k $_{4}$S b k $_{4}$] = 9 / 16 + ( S$_{k}$$_{3}$ · S$_{k}$$_{4}$ )( S$_{k}$$_{3}$ · S$_{k}$$_{4}$ ) − (3 / 8) 9 Similarly considering the following perturbation on y - links, λ H$_{perturbation}$ = λ [ S$_{j}$$_{1}$ · ( S$_{k}$$_{3}$ − S$_{k}$$_{4}$ ) + r S$_{k}$$_{1}$ · ( S$_{j}$$_{3}$ − S$_{j}$$_{4}$ )]. Following similar procedures we get the second order perturbation from this term In this Appendix we derive the second order pertur- bations of inter-cluster Heisenberg and spin-chirality in- teractions. The results can then be used to construct (16). First consider the perturbation λ H$_{perturbation}$ = λ [ S$_{j}$$_{1}$ · S$_{k}$$_{1}$ + r ( S$_{j}$$_{2}$ · S$_{k}$$_{2}$ )], where r is a real number to be tuned later. Due to the fact mentioned in Subsection IV B, the action of H$_{perturbation}$ on any cluster singlet state will produce a state with total spin-1 for both cluster j and k . Thus the first order perturbation in (15) van- ishes. And the second order perturbation term can be greatly simplified: operator (1 − P $_{jk}$)[0 − H$_{cluster}$ j − H$_{cluster}$ $_{k}$] $^{$^{−}$1}$(1 − P $_{jk}$) can be replaced by a c -number ( − 2 J$_{cluster}$ ) $^{$^{−}$1}$. Therefore the perturbation up to second order is So we can choose − ( r λ $^{2}$) / (4 J$_{cluster}$ ) = − J$_{y}$ , and include the last intra-cluster S$_{k}$$_{3}$ · S$_{k}$$_{4}$ + r $^{2}$S$_{j}$$_{3}$ · S$_{j}$$_{4}$ term in the first order perturbation. Therefore we can choose the following perturbation on y -links (not unique), with λ$_{y}$ = √ 4 | J$_{y}$ | · J$_{cluster}$ , r = sgn( J$_{y}$ ) is the sign of J$_{y}$ . z z The τ j τ k term is again more difficult to get. We use the representation of τ z by spin-chirality (6). And con- sider the following perturbation The first order term in (15) vanishes due to the same reason as before. There are four terms in the second order perturbation. The first one is H$_{perturbation}$ = S$_{j}$$_{2}$ · ( S$_{j}$$_{3}$ × S$_{j}$$_{4}$ ) + r S$_{k}$$_{2}$ · ( S$_{j}$$_{3}$ × S$_{j}$$_{4}$ ) This is true for other perturbations considered later in this Appendix. The cluster j and cluster k parts can be separated, this term then becomes ( a, b = x, y, z ), Then use the fact that P $_{j}$S a $_{jℓ}$S b jm P j = δ$_{ab}$ (1 / 3) P $_{j}$( S$_{jℓ}$ · S$_{jm}$ ) P j by spin rotation symmetry, the perturbation be- comes For the cluster j part we can use the same arguments as before, the H$_{cluster}$ j can be replaced by a c -number J$_{cluster}$ . For the cluster k part, consider the fact that S$_{k}$$_{3}$ × S$_{k}$$_{4}$ equals to the commutator − i [ S$_{k}$$_{4}$ , S$_{k}$$_{3}$ · S$_{k}$$_{4}$ ], the action of S$_{k}$$_{3}$ × S$_{k}$$_{4}$ on physical singlet states of k will also only produce spin-1 state. So we can replace the H$_{cluster}$ k in the denominator by a c -number J$_{cluster}$ as well. Use spin rotation symmetry to separate the j and k parts, this term simplifies to So we can choose − ( r λ $^{2}$) / (12 J$_{cluster}$ ) = − J$_{x}$ , and include the last intra-cluster S$_{j}$$_{1}$ · S$_{j}$$_{2}$ + S$_{k}$$_{1}$ · S$_{k}$$_{2}$ term in the first order perturbation. The perturbation on x -links is then (not unique), with λ$_{x}$ = √ 12 | J$_{x}$ | · J$_{cluster}$ , and r = sgn( J$_{x}$ ) is the sign of J$_{x}$ . The non-trivial terms produced by up to second order perturbation will be the τ x j τ x k term. Note that the last term in the above equation commutes with cluster Hamiltonians so it does not produce second or higher order perturbations.