7
modes of neighboring tetrahedra. And these coupling
constantsλx,y,z need to be tuned to produce Jx,y,z of
the Kitaev model. This is still not easy to implement in
solid state systems. At lowest non-trivial order of pertur-
bative expansion, we do get our model (9). Higher order
terms in expansion destroy the exact solvability, but may
be controlled by the small parametersλx,y,z /k.
B. Generate the High Order Terms by Magnetic
Interactions between Clusters.
In this Subsection we consider more conventional per-
turbations, magnetic interactions between the clusters,
e.g. the Heisenberg couplingSj · Sk with j and k belong
to diﬀerent tetrahedra. This has the advantage over the
previous phonon approach for not introducing additional
degrees of freedom. But it also has a signiﬁcant disad-
vantage: the perturbation does not commute with the
cluster Heisenberg Hamiltonian (2), so the cluster sin-
glet subspace will be mixed with other total spin states.
In this Subsection we will use the spin-chirality represen-
tation (6) forτz .
Again consider two clusters j and k. For simplicity
of notations deﬁne a projection operator Pjk = Pj Pk,
where Pj,k is projection into the singlet subspace of clus-
ter j and k, respectively, Pj,k = ∑
s=±1 |τz
j,k = s⟩⟨τz
j,k =
s|. For a given perturbation λ Hperturbation with small
parameter λ (in factor λ/Jcluster is the expansion param-
eter), lowest two orders of the perturbation series are
λPjk HperturbationPjk + λ2 Pjk Hperturbation(1 − Pjk )
× [0 − Hcluster j − Hcluster k]−1(1 − Pjk )HperturbationPjk
(15)
With proper choice of λ and Hperturbation we can generate
the desired Jx,y,z terms in (8) from the ﬁrst and second
order of perturbations.
The calculation can be dramatically simpliﬁed by the
following fact that any physical spin-1/2 operator Sx,y,z
ℓ
converts the cluster spin singlet states |τz = ±1⟩ into
spin-1 states of the cluster. This can be checked by
explicit calculations and will not be proved here. For
all the perturbations to be considered later, the above
mentioned fact can be exploited to replace the factor
[0− Hcluster j − Hcluster k]−1 in the second order pertur-
bation to a c-number ( −2Jcluster)−1.
The detailed calculations are given in Appendix B. We
will only list the results here.
The perturbation on x-links is given by
λx Hperturbation, x = λx[Sj1 · Sk1 + sgn(Jx) · (Sj2 · Sk2)]
− Jx(Sj1 · Sj2 + Sk1 · Sk2).
where λx =
√
12|Jx| · Jcluster, sgn( Jx) = ±1 is the sign
of Jx.
The perturbation on y-links is
λy Hperturbation, y
=λy[Sj1 · Sk1 + sgn(Jy) · (Sj3 − Sj4) · (Sk3 − Sk4)]
− |Jy|(Sj3 · Sj4 + Sk3 · Sk4)
with λy =
√
4|Jy| · Jcluster.
The perturbation on z-links is
λz Hperturbation, z
= λz [Sj2 · (Sk3 × Sk4) + sgn( Jz) · Sk2 · (Sj3 × Sj4)]
− |Jz|(Sj3 · Sj4 + Sk3 · Sk4).
with λz = 4
√
|Jz| · Jcluster.
The entire Hamiltonian Hmagnetic reads explicitly as,
Hmagnetic =
∑
cluster j
(Jcluster/2)(Sj1 + Sj2 + Sj3 + Sj4)2
+
∑
x−links <jk>
{√
12|Jx| · Jcluster
[
Sj1 · Sk1 + sgn(Jx) · (Sj2 · Sk2)
]
− Jx(Sj1 · Sj2 + Sk1 · Sk2)
}
+
∑
y−links <jk>
{√
4|Jy| · Jcluster
[
Sj1 · (Sk3 − Sk4) + sgn( Jy)Sk1 · (Sj3 − Sj4)
]
− |Jy|(Sj3 · Sj4 + Sk3 · Sk4)
}
+
∑
z−links <jk>
{
4
√
|Jz| · Jcluster
[
Sj2 · (Sk3 × Sk4) + sgn( Jz )Sk2 · (Sj3 × Sj4)
]
− |Jz|(Sj3 · Sj4 + Sk3 · Sk4)
}
.
(16)
In (16), we have been able to reduce the four spin in-
teractions in (8) to inter-cluster Heisenberg interactions,
and the six-spin interactions in (8) to inter-cluster spin-
chirality interactions. The inter-cluster Heisenberg cou-
plings inHperturbation x,y may be easier to arrange. The
inter-cluster spin-chirality coupling in Hperturbation z ex-
plicitly breaks time reversal symmetry and is probably
harder to implement in solid state systems. However
spin-chirality order may have important consequences
in frustrated magnets36,37, and a realization of spin-