9
Appendix B: Derivation of the Terms Generated by
Second Order Perturbation of Inter-cluster
Magnetic Interactions
In this Appendix we derive the second order pertur-
bations of inter-cluster Heisenberg and spin-chirality in-
teractions. The results can then be used to construct
(16).
First consider the perturbation λ Hperturbation = λ[Sj1 ·
Sk1 + r(Sj2 · Sk2)], where r is a real number to be tuned
later. Due to the fact mentioned in Subsection IV B,
the action ofHperturbation on any cluster singlet state
will produce a state with total spin-1 for both cluster j
and k. Thus the ﬁrst order perturbation in (15) van-
ishes. And the second order perturbation term can be
greatly simpliﬁed: operator (1− P jk )[0 − Hcluster j −
Hcluster k]−1(1 − P jk ) can be replaced by a c-number
(−2Jcluster)−1. Therefore the perturbation up to second
order is
− λ2
2Jcluster
Pjk (Hperturbation)2Pjk
This is true for other perturbations considered later in
this Appendix. The clusterj and cluster k parts can be
separated, this term then becomes ( a, b = x, y, z ),
− λ2
2Jcluster
∑
a,b
[
Pj Sa
j1Sb
j1Pj · PkSa
k1Sb
k1Pk
+ 2r PjSa
j1Sb
j2Pj · PkSa
k1Sb
k2Pk
+ r2 Pj Sa
j2Sb
j2Pj · PkSa
k2Sb
k2Pk
]
Then use the fact that Pj Sa
jℓSb
jm Pj = δab(1/3)Pj(Sjℓ ·
Sjm )Pj by spin rotation symmetry, the perturbation be-
comes
− λ2
6Jcluster
[ 9 + 9 r2
16 + 2r Pjk (Sj1 · Sj2)(Sk1 · Sk2)Pjk
]
= − λ2
6Jcluster
[ 9 + 9 r2
16 + (r/2)τx
j τx
k − r/2
− r Pjk (Sj1 · Sj2 + Sk1 · Sk2)Pjk
]
.
So we can choose −(r λ2)/(12Jcluster) = −Jx, and include
the last intra-cluster Sj1 · Sj2 + Sk1 · Sk2 term in the ﬁrst
order perturbation.
The perturbation on x-links is then (not unique),
λx Hperturbation, x =λx[Sj1 · Sk1 + sgn(Jx) · (Sj2 · Sk2)]
− Jx(Sj1 · Sj2 + Sk1 · Sk2)
with λx =
√
12|Jx| · Jcluster, and r = sgn( Jx) is the sign
of Jx. The non-trivial terms produced by up to second
order perturbation will be the τx
j τx
k term. Note that the
last term in the above equation commutes with cluster
Hamiltonians so it does not produce second or higher
order perturbations.
Similarly considering the following perturbation on y-
links, λ Hperturbation = λ[Sj1 · (Sk3 − Sk4) + r Sk1 · (Sj3 −
Sj4)]. Following similar procedures we get the second
order perturbation from this term
− λ2
6Jcluster
[ 9 + 9 r2
8
+ 2r Pjk [Sj1 · (Sj3 − Sj4)][Sk1 · (Sk3 − Sk4)]Pjk
− (3/2) Pjk (Sk3 · Sk4 + r2 Sj3 · Sj4)Pjk
]
= − λ2
6Jcluster
[ 9 + 9 r2
8 + 2r (3/4)τy
j τy
k
− (3/2) Pjk (Sk3 · Sk4 + r2 Sj3 · Sj4)Pjk
]
So we can choose −(r λ2)/(4Jcluster) = −Jy, and include
the last intra-cluster Sk3 · Sk4 + r2 Sj3 · Sj4 term in the
ﬁrst order perturbation.
Therefore we can choose the following perturbation on
y-links (not unique),
λy Hperturbation, y
=λy[Sj1 · Sk1 + sgn(Jy) · (Sj3 − Sj4) · (Sk3 − Sk4)]
− |Jy|(Sj3 · Sj4 + Sk3 · Sk4)
with λy =
√
4|Jy| · Jcluster, r = sgn( Jy ) is the sign of Jy.
The τz
j τz
k term is again more diﬃcult to get. We use
the representation of τz by spin-chirality (6). And con-
sider the following perturbation
Hperturbation = Sj2 · (Sj3 × Sj4) + r Sk2 · (Sj3 × Sj4)
The ﬁrst order term in (15) vanishes due to the same
reason as before. There are four terms in the second
order perturbation. The ﬁrst one is
λ2 Pjk Sj2 · (Sk3 × Sk4)(1 − Pjk )
× [0 − Hcluster j − Hcluster k]−1
× (1 − Pjk )Sj2 · (Sk3 × Sk4)Pjk
For the cluster j part we can use the same arguments
as before, the Hcluster j can be replaced by a c-number
Jcluster. For the cluster k part, consider the fact that
Sk3 × Sk4 equals to the commutator −i[Sk4, Sk3 · Sk4],
the action of Sk3 × Sk4 on physical singlet states of k will
also only produce spin-1 state. So we can replace the
Hcluster k in the denominator by a c-number Jcluster as
well. Use spin rotation symmetry to separate the j and
k parts, this term simpliﬁes to
− λ2
6Jcluster
Pj Sj2 · Sj2Pj · Pk(Sk3 × Sk4) · (Sk3 × Sk4)Pk.
Use ( S)2 = 3 /4 and
(Sk3 × Sk4) · (Sk3 × Sk4)
=
∑
a,b
(Sa
k3Sb
k4Sa
k3Sb
k4 − Sa
k3Sb
k4Sb
k3Sa
k4)
= ( Sk3 · Sk3)(Sk4 · Sk4) −
∑
a,b
Sa
k3Sb
k3[δab/2 − Sa