ical spin operators to pseudo-spin operators. The map-
ping constructed here will be used in later Sections to
construct the eﬀective Kitaev model. In this Section we
will work entirely within the four-spin cluster, all unspec-
iﬁed physical spin subscripts take values 1, . . . , 4.
Consider a cluster of four spin-1/2 moments(called
physical spins hereafter), labeled by S1,...,4, antiferro-
magnetically coupled to each other (see the right bot-
tom part of FIG. 2). The Hamiltonian within the clus-
ter(up to a constant) is simply the Heisenberg antiferro-
magnetic(AFM) interactions,
Hcluster = ( Jcluster/2) (S1 + S2 + S3 + S4)2 (2)
The energy levels should be apparent from this form:
one group of spin-2 quintets with energy 3Jcluster, three
groups of spin-1 triplets with energy Jcluster, and two spin
singlets with energy zero. We will consider large positive