4
Plug in the expressions (4) and (6) into (7), the Hamil- tonian reads e xplicitly as
H =
∑
j
(Jcluster/2)(Sj1 + Sj2 + Sj3 + Sj4)2 −
∑
z−links <jk>
Jz (16/9)[Sj2 · (Sj3 × Sj4)][Sk2 · (Sk3 × Sk4)]
−
∑
x−links <jk>
Jx (2Sj1 · Sj2 + 1/2)(2Sk1 · Sk2 + 1/2) −
∑
y−links <jk>
Jy (4/3)[Sj1 · (Sj3 − Sj4)][Sk1 · (Sk3 − Sk4)]
(8)
While by the represenation (4) and (5), the Hamilto- nian becomes
H =
∑
j
(Jcluster/2)(Sj1 + Sj2 + Sj3 + Sj4)2
−
∑
x−links <jk>
Jx (2Sj1 · Sj2 + 1/2)(2Sk1 · Sk2 + 1/2) −
∑
y−links <jk>
Jy (4/3)[Sj1 · (Sj3 − Sj4)][Sk1 · (Sk3 − Sk4)]
−
∑
z−links <jk>
Jz (−4/3)(2Sj3 · Sj4 + 1/2)[Sj1 · (Sj3 − Sj4)](2Sk3 · Sk4 + 1/2)[Sk1 · (Sk3 − Sk4)]
(9)
This model, in terms of physical spins S, has full
spin rotation symmetry and time-reversal symmetry. A
pseudo-magnetic ﬁeld term∑
j
⃗h · ⃗ τj term can also be
included under this mapping, however the resulting Ki-
taev model with magnetic ﬁeld is not exactly solvable.
It is quite curious that such a formidably looking Hamil-
tonian (8), with biquadratic and six-spin(or eight-spin)
terms, has an exactly solvable low energy sector.
We emphasize that because the ﬁrst intra-cluster term∑
cluster Hcluster commutes with the latter Kitaev terms
independent of the representation used, the Kitaev model
is realized as theexact low energy Hamiltonian of this
model without truncation errors of perturbation theories,
namely no (|Jx,y,z |/Jcluster)2 or higher order terms will
be generated under the projection to low energy clus-
ter singlet space. This is unlike, for example, thet/U
expansion of the half-ﬁlled Hubbard model 22,23, where
at lowest t2/U order the eﬀective Hamiltonian is the
Heisenberg model, but higher order terms ( t4/U3 etc.)
should in principle still be included in the low energy ef-
fective Hamiltonian for any ﬁnitet/U. Similar compari-
son can be made to the perturbative expansion studies of
the Kitaev-type models by Vidalet al.9, where the low
energy eﬀective Hamiltonians were obtained in certian
anisotropic (strong bond/triangle) limits. Although the
spirit of this work, namely projection to low energy sec-
tor, is the same as all previous perturbative approaches
to eﬀective Hamiltonians.
Note that the original Kitaev model (1) has three-
fold rotation symmetry around a honeycomb lattice site,
combined with a three-fold rotation in pseudo-spin space
(cyclic permutation ofτx, τy, τz ). This is not apparent
in our model (8) in terms of physical spins, under the
current representation ofτx,y,z . We can remedy this by
using a diﬀerent set of pseudo-spin Pauli matrices τ′x,y,z
in (7),
τ′x =
√
1/3τz +
√
2/3τx,
τ′y =
√
1/3τz −
√
1/6τx +
√
1/2τy,
τ′z =
√
1/3τz −
√
1/6τx −
√
1/2τy
With proper representation choice, they have a symmet-
ric form in terms of physical spins,
τ′x = −(4/3)S2 · (S3 × S4) +
√
2/3(2S1 · S2 + 1/2)
τ′y = −(4/3)S3 · (S4 × S2) +
√
2/3(2S1 · S3 + 1/2)
τ′z = −(4/3)S4 · (S2 × S3) +
√
2/3(2S1 · S4 + 1/2)
(10)
So the symmetry mentioned above can be realized by a
three-fold rotation of the honeycomb lattice, with a cyclic
permutation ofS2, S3 and S4 in each cluster. This is in
fact the three-fold rotation symmetry of the physical spin
lattice illustrated in FIG. 2. However this more symmet-
ric representation will not be used in later part of this
paper.