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, kto label four-spin clusters (pseudo-spin-1/2 sites), the
physical spins in cluster j are labeled as Sj1, . . . , Sj4.
Apply the mappings developed in Section II, we have
the desired Hamiltonian in short notation,
H =
∑
cluster
Hcluster −
∑
x−links <jk>
Jxτx
j τx
k
−
∑
y−links <jk>
Jyτy
j τy
k −
∑
z−links <jk>
Jzτz
j τz
k
(7)
where j, k label the honeycomb lattice sites thus the four-
spin clusters, Hcluster is given by (2), τx,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.