3 physical spin pseudo-spin P 12 , and P 34 τ x P 13 , and P 24 − (1 / 2) τ x $_{+ (}$√$_{3}$$_{/}$$_{2)}$$_{τ}$ y P 14 , and P 23 − (1 / 2) τ x − $_{(}$√$_{3}$$_{/}$$_{2)}$$_{τ}$ y − χ 234 , χ 341 , − χ 412 , and χ 123 $_{(}$√$_{3}$$_{/}$$_{4)}$$_{τ}$ z TABLE I: Correspondence between physical spin operators and pseudo-spin operators in the physical spin singlet sector of the four antiferromagnetically coupled physical spins. P$_{jk}$ = 2 S j · S$_{k}$ +1 / 2 are permutation operators, χ$_{jkℓ}$ = S j · ( $^{S$_{k}$}$×S$_{ℓ}$ ) are spin-chirality operators. Note that several physical spin operators may correspond to the same pseudo-spin operator. J$_{cluster}$ limit. So only the singlet sector remains in low energy. The singlet sector is then treated as a pseudo-spin-1/2 Hilbert space. From now on we denote the pseudo-spin- 1/2 operators as T = (1 / 2) ⃗ τ , with ⃗ τ the Pauli matri- ces. It is convenient to choose the following basis of the pseudo-spin | τ z = ± 1 〉 = 1 √$_{6}$ ( | ↓↓↑↑〉 + ω − τ $^{z}$|  ↓↑↓↑〉 + ω τ $^{z}$|  ↓↑↑↓〉 + | ↑↑↓↓〉 + ω − τ $^{z}$|  ↑↓↑↓〉 + ω τ $^{z}$|  ↑↓↓↑〉 ) (3) τ z = − χ$_{234}$/ $_{(}$√$_{3}$$_{/}$$_{4) =}$ − (4 $_{/}$√$_{3)}$$_{S}$$_{2}$ · ( S$_{3}$ × S$_{4}$ ) (6) P$_{34}$ | τ z = − 1 〉 = 1 √$_{6}$ ( | ↓↓↑↑〉 + ω | ↓↑↑↓〉 + ω $^{2}$|  ↓↑↓↑〉 + | ↑↑↓↓〉 + ω | ↑↓↓↑〉 + ω $^{2}$|  ↑↓↑↓〉 ) = | τ z = +1 〉 τ x = P$_{12}$ = 2 S$_{1}$ · S$_{2}$ + 1 / 2 τ y = ( P$_{13}$ − P$_{14}$ ) $_{/}$√$_{3 = (2}$$_{/}$ √$_{3)}$$_{S}$$_{1}$ · ( S$_{3}$ − S$_{4}$ ) (4) H = ∑ cluster H$_{cluster}$ − ∑ x $^{−}$links <jk> J$_{x}$τ x j τ x k − ∑ y $^{−}$links <jk> J$_{y}$τ y j τ y k − ∑ z $^{−}$links <jk> J$_{z}$τ z j τ z k (7) τ z = − i (2 $_{/}$√$_{3)(2}$$_{S}$$_{1}$ · S$_{2}$ + 1 / 2) S$_{1}$ · ( S$_{3}$ − S$_{4}$ ) (5) where ω = e 2 πi/ 3 is the complex cubic root of unity, | ↓↓↑↑〉 and other states on the right-hand-side(RHS) are basis states of the four-spin system, in terms of S z quan- tum numbers of physical spins 1 , . . . , 4 in sequential or- der. This pseudo-spin representation has been used by Harris et al. to study magnetic ordering in pyrochlore antiferromagnets $^{21}$. However there is another simpler representation of τ $^{z}$, by the spin-chirality operator χ$_{jkℓ}$ = S$_{j}$ · ( S$_{k}$ × S$_{ℓ}$ ). Ex- plicit calculation shows that the effect of S$_{2}$ · ( S$_{3}$ × S$_{4}$ ) is − $_{(}$√$_{3}$$_{/}$$_{4)}$$_{τ}$ z in the physical singlet sector. This can also be proved by using the commutation relation [ S$_{2}$ · S$_{3}$ , S$_{2}$ · S$_{4}$ ] = i S$_{2}$ · ( S$_{3}$ × S$_{4}$ ). A complete list of all chirality operators is given in TABLE I. Therefore we can choose another representation of τ $^{z}$, We now consider the effect of Heisenberg-type inter- actions S$_{j}$ · S$_{k}$ inside the physical singlet sector. Note that since any S$_{j}$ · S$_{k}$ within the cluster commutes with the cluster Hamiltonian H$_{cluster}$ (2), their action do not mix physical spin singlet states with states of other total physical spin. This property is also true for the spin- chirality operator used later. So the pseudo-spin Hamil- tonian constructed below will be exact low energy Hamil- tonian, without truncation errors in typical perturbation series expansions. The above representations of τ x,y,z are all invariant under global spin rotation of the physical spins. With the machinery of equations (4), (5), and (6), it will be straightforward to construct various pseudo-spin- 1/2 Hamiltonians on various lattices, of the Kitaev vari- ety and beyond, as the exact low energy effective Hamil- tonian of certain spin-1/2 models with spin-rotation sym- metry. In these constructions a pseudo-spin lattice site actually represents a cluster of four spin-1/2 moments. It is simpler to consider the permutation operators P$_{jk}$ ≡ 2 S$_{j}$ · S$_{k}$ + 1 / 2, which just exchange the states of the two physical spin-1/2 moments j and k ( j = k ). As an example we consider the action of P$_{34}$ , In this Section we will use directly the results of the previous Section to write down a Hamiltonian whose low energy sector is described by the Kitaev model. The Hamiltonian will be constructed on the physical spin lat- tice illustrated in FIG. 2. In this Section we will use j, k to label four-spin clusters (pseudo-spin-1/2 sites), the physical spins in cluster j are labeled as S$_{j}$$_{1}$ , . . . , S$_{j}$$_{4}$ . Apply the mappings developed in Section II, we have the desired Hamiltonian in short notation, and similarly P$_{34}$ | τ z = − 1 〉 = | τ z = +1 〉 . Therefore P$_{34}$ is just τ x in the physical singlet sector. A complete list of all permutation operators is given in TABLE I. We can choose the following representation of τ x and τ $^{y}$, III. REALIZATION OF THE KITAEV MODEL. where j, k label the honeycomb lattice sites thus the four- spin clusters, H$_{cluster}$ is given by (2), τ x,y,z should be replaced by the corresponding physical spin operators in (4) and (5) or (6), or some other equivalent representa- tions of personal preference. Many other representations are possible as well, because several physical spin interactions may correspond to the same pseudo-spin interaction in the physical singlet sec- tor, and we will take advantage of this later. For τ z we can use τ z = − iτ $^{x}$τ$^{y}$, where i is the imagi- nary unit,