8
chirality interactions in cold atom optical lattices has
been proposed38.
Our model (8) is achieved at second order of the per-
turbation series. Higher order terms become trunca-
tion errors but may be controlled by small parameters
λx,y,z /Jcluster ∼
√
|Jx,y,z |/Jcluster.
V. CONCLUSIONS.
We constructed the exactly solvable Kitaev honeycomb
model1 as the exact low energy eﬀective Hamiltonian of
a spin-1/2 model [equations (8) or (9)] with spin-rotation
and time reversal symmetry. The spin in Kitaev model is
represented as the pseudo-spin in the two-fold degenerate
spin singlet subspace of a cluster of four antiferromag-
netically coupled spin-1/2 moments. The physical spin
model is a honeycomb lattice of such four-spin clusters,
with certain inter-cluster interactions. The machinery
for the exact mapping to pseudo-spin Hamiltonian was
developed (see e.g. TABLE I), which is quite general
and can be used to construct other interesting (exactly
solvable) spin-1/2 models from spin rotation invariant
systems.
In this construction the pseudo-spin correlations in the
Kitaev model will be mapped to dimer or spin-chirality
correlations in the physical spin system. The correspond-
ing picture of the fractionalized Majorana fermion exci-
tations and Ising vortices still remain to be clariﬁed.
This exact construction contains high order physical
spin interactions, which is undesirable for practical im-
plementation. We described two possible approaches to
reduce this problem: generating the high order spin in-
teractions by perturbative expansion of the coupling to
optical phonon, or the magnetic coupling between clus-
ters. This perturbative construction will introduce trun-
cation error of perturbation series, which may be con-
trolled by small expansion parameters. Whether these
constructions can be experimentally engineered is how-
ever beyond the scope of this study. It is conceivable that
other perturbative expansion can also generate these high
order spin interactions, but this possibility will be left for
future works.
Acknowledgments
The author thanks Ashvin Vishwanath, Yong-Baek
Kim and Arun Paramekanti for inspiring discussions, and
Todadri Senthil for critical comments. The author is sup-
ported by the MIT Pappalardo Fellowship in Physics.
Appendix A: Coupling between Distortions of a
Tetrahedron and the Pseudo-spins
In this Appendix we reproduce from Ref. 35 the cou-
plings of all tetrahedron distortion modes to the spin
system. And convert them to pseudo-spin notation in
the physical spin singlet sector.
Consider a general small distortion of the tetrahedron,
the spin Hamiltonian becomes
Hcluster, SL = ( Jcluster/2)(
∑
ℓ
Sℓ)2 + J′ ∑
ℓ<m
δrℓm(Sℓ · Sm)
(A1)
where δrℓm is the change of bond length between spins
ℓ and m, J′ is the derivative of Jcluster with respect to
bond length.
There are six orthogonal distortion modes of the tetra-
hedron [TABLE 1.1 in Ref. 35]. One of the modes A is the
trivial representation of the tetrahedral group Td; two E
modes form the two dimensional irreducible representa-
tion ofTd; and three T2 modes form the three dimen-
sional irreducible representation. The E modes are also
illustrated in FIG. 3.
The generic couplings in (A1) [second term] can be
converted to couplings to these orthogonal modes,
J′(QAfA + QE
1fE
1 + QE
2fE
2 + QT2
1 fT2
1 + QT2
2 fT2
2 + QT2
3 fT2
3 )
where Q are generalized coordinates of the corresponding
modes, functions f can be read oﬀ from TABLE 1.2 of
Ref.35. For the A mode, δrℓm =
√
2/3QA, so fA is
fA =
√
2/3 (S1 · S2 + S3 · S4 + S1 · S3
+ S2 · S4 + S1 · S4 + S2 · S3).
The functions fE
1,2 for the E modes have been given before
but are reproduced here,
fE
2 = (1 /2)(S2 · S4 + S1 · S3 − S1 · S4 − S2 · S3),
fE
1 =
√
1/12(S1 · S4 + S2 · S3 + S2 · S4 + S1 · S3
− 2S1 · S2 − 2S3 · S4).
The functions fT2
1,2,3 for the T2 modes are
fT2
1 = ( S2 · S3 − S1 · S4),
fT2
2 = ( S1 · S3 − S2 · S4),
fT2
3 = ( S1 · S2 − S3 · S4)
Now we can use TABLE I to convert the above cou-