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146 L. Jaubert, unpublished.

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154 In contrast, it has recently been found that Dzyaloshinskii-Moriya interactions in the “breathing-pyrochlore” material Ba$_{3}$Yb$_{2}$Zn$_{5}$O$_{11}$ are rather large$^{59,60}$.

155 The first line of Eq. (24) is a quadratic invariant of T$_{d}$, while the next three form a basis for the T$_{1}$ irreducible representation.