3
Jcluster limit. So only the singlet sector remains in low
energy.
The singlet sector is then treated as a pseudo-spin-1/2
Hilbert space. From now on we denote the pseudo-spin-
1/2 operators asT = (1 /2)⃗ τ, with ⃗ τthe Pauli matri-
ces. It is convenient to choose the following basis of the
pseudo-spin
|τz = ±1⟩ = 1
√
6
(
| ↓↓↑↑⟩ + ω−τ z
| ↓↑↓↑⟩ + ωτ z
| ↓↑↑↓⟩
+ | ↑↑↓↓⟩ + ω−τ z
| ↑↓↑↓⟩ + ωτ z
| ↑↓↓↑⟩
)
(3)
where ω = e2πi/3 is the complex cubic root of unity,
| ↓↓↑↑⟩ and other states on the right-hand-side(RHS) are
basis states of the four-spin system, in terms of Sz quan-
tum numbers of physical spins 1 , . . . , 4 in sequential or-
der. This pseudo-spin representation has been used by
Harriset al. to study magnetic ordering in pyrochlore
antiferromagnets21.
We now consider the eﬀect of Heisenberg-type inter-
actions Sj · Sk inside the physical singlet sector. Note
that since any Sj · Sk within the cluster commutes with
the cluster Hamiltonian Hcluster (2), their action do not
mix physical spin singlet states with states of other total
physical spin. This property is also true for the spin-
chirality operator used later. So the pseudo-spin Hamil-
tonian constructed below will beexact low energy Hamil-
tonian, without truncation errors in typical perturbation
series expansions.
It is simpler to consider the permutation operators
Pjk ≡ 2Sj · Sk + 1 /2, which just exchange the states
of the two physical spin-1/2 moments j and k (j ̸= k).
As an example we consider the action of P34,
P34|τz = −1⟩ = 1
√
6
(
| ↓↓↑↑⟩ + ω| ↓↑↑↓⟩ + ω2| ↓↑↓↑⟩
+ | ↑↑↓↓⟩ + ω| ↑↓↓↑⟩ + ω2| ↑↓↑↓⟩
)
= |τz = +1 ⟩
and similarly P34|τz = −1⟩ = |τz = +1 ⟩. Therefore P34
is just τx in the physical singlet sector. A complete list
of all permutation operators is given in TABLE I. We
can choose the following representation ofτx and τy ,
τx = P12 = 2 S1 · S2 + 1/2
τy = ( P13 − P14)/
√
3 = (2 /
√
3)S1 · (S3 − S4)
(4)
Many other representations are possible as well, because
several physical spin interactions may correspond to the
same pseudo-spin interaction in the physical singlet sec-
tor, and we will take advantage of this later.
For τz we can use τz = −iτxτy , where i is the imagi-
nary unit,
τz = −i(2/
√
3)(2S1 · S2 + 1/2)S1 · (S3 − S4) (5)
physical spin pseudo-spin
P12, and P34 τx
P13, and P24 − (1/ 2)τx + (
√
3/ 2)τy
P14, and P23 − (1/ 2)τx − (
√
3/ 2)τy
− χ 234, χ 341, − χ 412, and χ 123 (
√
3/ 4)τz
TABLE I: Correspondence between physical spin operators
and pseudo-spin operators in the physical spin singlet sector of
the four antiferromagnetically coupled physical spins. Pjk =
2Sj ·Sk +1/ 2 are permutation operators, χ jkℓ = Sj ·(Sk × Sℓ)
are spin-chirality operators. Note that several physical s pin
operators may correspond to the same pseudo-spin operator.
However there is another simpler representation ofτz ,
by the spin-chirality operator χjkℓ = Sj · (Sk × Sℓ). Ex-
plicit calculation shows that the eﬀect of S2 · (S3 × S4) is
−(
√
3/4)τz in the physical singlet sector. This can also
be proved by using the commutation relation [ S2 ·S3, S2 ·
S4] = iS2 · (S3 × S4). A complete list of all chirality
operators is given in TABLE I. Therefore we can choose
another representation ofτz ,
τz = −χ234/(
√
3/4) = −(4/
√
3)S2 · (S3 × S4) (6)
The above representations of τx,y,z are all invariant under
global spin rotation of the physical spins.
With the machinery of equations (4), (5), and (6), it
will be straightforward to construct various pseudo-spin-
1/2 Hamiltonians on various lattices, of the Kitaev vari-
ety and beyond, as the exact low energy eﬀective Hamil-
tonian of certain spin-1/2 models with spin-rotation sym-
metry. In these constructions a pseudo-spin lattice site
actually represents a cluster of four spin-1/2 moments.
III. REALIZATION OF THE KITAEV MODEL.
In this Section we will use directly the results of the
previous Section to write down a Hamiltonian whose low
energy sector is described by the Kitaev model. The