duced may be controlled by small expansion parameters.
In this Section we denote the physical spins on cluster
j(k) as j1, . . . , j 4 ( k1, . . . , k 4), and denote pseudo-spins
on cluster j(k) as ⃗ τj (⃗ τk).
A. Generate the High Order Terms by Coupling to
Optical Phonon.
In this Subsection we regard each four-spin cluster
as a tetrahedron, and consider possible optical phonon
modes(distortions) and their couplings to the spin sys-
tem. The basic idea is that the intra-cluster Heisen-
berg couplingJcluster can linearly depend on the dis-
tance between physical spins. Therefore certain distor-
tions of the tetrahedron couple to certain linear combi-
nations ofSℓ · Sm. Integrating out phonon modes will
then generate high order spin interactions. This idea has
been extensively studied and applied to several magnetic
materials28–34. More details can be found in a recent
review by Tchernyshyov and Chern 35. And we will fre-
quently use their notations. In this Subsection we will
use the representation (5) forτz .
Consider ﬁrst a single tetrahedron with four spins
1, . . . , 4. The general distortions of this tetrahedron can
be classiﬁed by their symmetry (see for example Ref. 35).
Only two tetragonal to orthorhombic distortion modes,
QE
1and QE
2(illustrated in FIG. 3), couple to the pseudo-
spins deﬁned in Section II. A complete analysis of all
modes is given in Appendix A. The coupling is of the