6
form
J′(QE
1fE
1 + QE
2fE
2 )
where J′ is the derivative of Heisenberg coupling Jcluster
between two spins ℓ and m with respect to their distance
rℓm, J′ = d Jcluster/drℓm; QE
1,2 are the generalized coor-
dinates of these two modes; and the functions fE
1,2 are
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).
According to TABLE I we have fE
1 = −(
√
3/2)τx and
fE
2 = (
√
3/2)τy. Then the coupling becomes
(
√
3/2)J′(−QE
1τx + QE
2τy ) (12)
The spin-lattice(SL) Hamiltonian on a single cluster j
is [equation (1.8) in Ref. 35],
Hcluster j, SL =Hcluster j + k
2 (QE
1j )2 + k
2 (QE
2j )2
−
√
3
2 J′(QE
1j τx
j − QE
2jτy
j ),
(13)
where k > 0 is the elastic constant for these phonon
modes, J′ is the spin-lattice coupling constant, QE
1j and
QE
2j are the generalized coordinates of the QE
1 and QE
2
distortion modes of cluster j, Hcluster j is (2). As al-
ready noted in Ref. 35, this model does not really break
the pseudo-spin rotation symmetry of a single cluster.
Now we put two clusters j and k together, and in-
clude a perturbation λ Hperturbation to the optical phonon
Hamiltonian,
Hjk,SL =Hcluster j, SL + Hcluster k, SL
+ λ Hperturbation[QE
1j , QE
2j , QE
1k, QE
2k]
where λ (in fact λ/k) is the expansion parameter.
Consider the perturbation Hperturbation = QE
1j · QE
1k,
which means a coupling between the QE
1 distortion
modes of the two tetrahedra. Integrate out the optical
phonons, at lowest non-trivial order, it produces a term
(3J′2 λ)/(4 k2) τx
j · τx
k . This can be seen by minimizing
separately the two cluster Hamiltonians with respect to
QE
1, which gives QE
1 = (
√
3 J′)/(2 k)τx, then plug this
into the perturbation term. Thus we have produced the
Jx term in the Kitaev model with Jx = −(3 J′2 λ)/(4 k2).
Similarly the perturbation Hperturbation = QE
2j · QE
2k
will generate (3 J′2 λ)/(4 k2) τy
j · τy
k at lowest non-trivial
order. So we can make Jy = −(3 J′2 λ)/(4 k2).
The τz
j · τz
k coupling is more diﬃcult to get. We
treat it as −τx
j τy
j · τx
k τy
k . By the above reasoning, we
need an anharmonic coupling Hperturbation = QE
1jQE
2j ·
QE
1kQE
2k. It will produce at lowest non-trivial or-
der (9 J′4 λ)/(16 k4) τx
j τy
j · τx
k τy
k . Thus we have Jz =
(9 J′4 λ)/(16 k4).
Finally we have made up a spin-lattice model HSL,
which involves only Sℓ ·Sm interaction for physical spins,
HSL =
∑
cluster
Hcluster, SL +
∑
x−links <jk>
λx QE
1j · QE
1k
+
∑
y−links <jk>
λy QE
2j · QE
2k
+
∑
z−links <jk>
λz QE
1j QE
2j · QE
1kQE
2k
where QE
1j is the generalized coordinate for the QE
1mode
on cluster j, and QE
1k, QE
2j, QE
2k are similarly deﬁned;
λx,y = −(4Jx,yk2)/(3J′2) and λz = (16 Jzk4)/(9J′4); the
single cluster spin-lattice Hamiltonian Hcluster, SL is (13).
Collect the results above we have the spin-lattice
Hamiltonian HSL explicitly written as,
HSL =
∑
cluster j
[
(Jcluster/2)(Sj1 + Sj2 + Sj3 + Sj4)2 + k
2 (QE
1j )2 + k
2 (QE
2j )2
+ J′
(
QE
1j
Sj1 · Sj4 + Sj2 · Sj3 + Sj2 · Sj4 + Sj1 · Sj3 − 2Sj1 · Sj2 − 2Sj3 · Sj4
√
12
+ QE
2j
Sj2 · Sj4 + Sj1 · Sj3 − Sj1 · Sj4 − Sj2 · Sj3
2
)]
−
∑
x−links <jk>
4Jxk2
3J′2 QE
1j · QE
1k −
∑
y−links <jk>
4Jyk2
3J′2 QE
2j · QE
2k +
∑
z−links <jk>
16Jzk4
9J′4 QE
1j QE
2j · QE
1kQE
2k
(14)
The single cluster spin-lattice Hamiltonian [ﬁrst three
lines in (14)] is quite natural. However we need some
harmonic(on x- and y-links of honeycomb lattice) and an-
harmonic coupling (on z-links) between optical phonon