The functions fT2
1,2,3 for the T2 modes are
fT2
1 = ( S2 · S3 − S1 · S4),
fT2
2 = ( S1 · S3 − S2 · S4),
fT2
3 = ( S1 · S2 − S3 · S4)
Now we can use TABLE I to convert the above cou-
plings into pseudo-spin. It is easy to see that fA and
fT2
1,2,3 are all zero when converted to pseudo-spins, namely
projected to the physical spin singlet sector. But fE
1 =
(P14 +P23 +P24 +P13 −2P12 −2P34)/(4
√
3) = −(
√
3/2)τx
and fE
2 = ( P24 + P13 − P14 − P23)/4 = (
√
3/2)τy. This
has already been noted by Tchernyshyov et al.28, only
the E modes can lift the degeneracy of the physical spin
singlet ground states of the tetrahedron. Therefore the
general spin lattice coupling is the form of (12) given in
the main text.