5
Another note to take is that it is not necessary to have
such a highly symmetric cluster Hamiltonian (2). The
mappings to pseudo-spin-1/2 should work as long as the
ground states of the cluster Hamiltonian are the two-fold
degenerate singlets. One generalization, which conforms
the symmetry of the lattice in FIG. 2, is to have
Hcluster = ( Jcluster/2)(r · S1 + S2 + S3 + S4)2 (11)
with Jcluster > 0 and 0 < r < 3. However this is not
convenient for later discussions and will not be used.
We brieﬂy describe some of the properties of (8). Its
low energy states are entirely in the space that each of the
clusters is a physical spin singlet (called cluster singlet
subspace hereafter). Therefore physical spin correlations
are strictly conﬁned within each cluster. The excitations
carrying physical spin are gapped, and their dynamics
are ‘trivial’ in the sense that they do not move from one
cluster to another. But there are non-trivial low energy
physical spin singlet excitations, described by the pseudo-
spins deﬁned above. The correlations of the pseudo-spins
can be mapped to correlations of their corresponding
physical spin observables (the inverse mappings are not
unique, c.f. TABLE I). For exampleτx,y correlations
become certain dimer-dimer correlations, τz correlation
becomes chirality-chirality correlation, or four-dimer cor-
relation. It will be interesting to see the corresponding
picture of the exotic excitations in the Kitaev model, e.g.
the Majorana fermion and the Ising vortex. However this
will be deferred to future studies.
It is tempting to call this as an exactly solved spin liq-
uid with spin gap ( ∼ Jcluster), an extremely short-range
resonating valence bond(RVB) state, from a model with
spin rotation and time reversal symmetry. However it
should be noted that the unit cell of this model contains
an even number of spin-1/2 moments (so does the orig-
inal Kitaev model) which does not satisfy the stringent
deﬁnition of spin liquid requiring odd number of elec-
trons per unit cell. Several parent Hamiltonians of spin
liquids have already been constructed. See for example,
Ref.24–27 .
IV. GENERATE THE HIGH ORDER PHYSICAL
SPIN INTERACTIONS BY PERTURBATIVE
EXPANSION.
One major drawback of the present construction is that
it involves high order interactions of physical spins[see
(8) and (9)], thus is ‘unnatural’. In this Section we will
make compromises between exact solvability and natu-
ralness. We consider two clustersj and k and try to
generate the Jx,y,z interactions in (7) from perturbation
series expansion of more natural(lower order) physical
spin interactions. Two diﬀerent approaches for this pur-
pose will be laid out in the following two Subsections. In
Subsection IV A we will consider the two clusters as two
tetrahedra, and couple the spin system to certain opti-
cal phonons, further coupling between the phonon modes
(a) (b) (c) (d)
(b) (c) (d)
Q E
2
Q E
1
(a)
1
1 1 1
1 1 1
2
2
2
2
2
2 2
2
3
3
3 3 3
3 3 4 4
4 4
3
4
4
4
4
1
FIG. 3: Illustration of the tetragonal to orthorhombic
QE
1(top) and QE
2(bottom) distortion modes. (a) Perspective
view of the tetrahedron. 1 , . . . , 4 label the spins. Arrows in-
dicate the motion of each spin under the distortion mode. (b)
Top view of (a). (c)(d) Side view of (a).
of the two clusters can generate at lowest order the de-
sired high order spin interactions. In Subsection IV B we
will introduce certain magnetic, e.g. Heisenberg-type, in-
teractions between physical spins of diﬀerent clusters, at
lowest order(second order) of perturbation theory the de-
sired high order spin interactions can be achieved. These
approaches involve truncation errors in the perturbation
series, thus the mapping to low energy eﬀect Hamilto-
nian will no longer be exact. However the error intro-
duced may be controlled by small expansion parameters.
In this Section we denote the physical spins on cluster
j(k) as j1, . . . , j 4 ( k1, . . . , k 4), and denote pseudo-spins