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where, following Eqs. (22) and (23), $_{Q}$⊥ , 2 1 = Q x $^{2}$− y $^{2}$, $_{Q}$⊥ , 2 2 = Q $^{xy}$, $_{Q}$⊥ , 1 1 = Q $^{xz}$, and $_{Q}$⊥ , 1 2 = Q $^{yz}$. 

Eq. (43) should be contrasted with the form of free en- ergy in the absence of magnetic field, Eq. (32). The cubic invariant Q 3 survives as an interaction b$_{12}$ between Q ⊥ , 1 and Q ⊥ , $^{2}$, which transforms like e 2 $^{iφ}$e − $^{iφ}$e − iφ ∼ 1 under rota- tions about the z axis. This means that components of Q ⊥ , 2 couple linearly to a quadratic combination of the components of Q ⊥ , $^{1}$. Because of this, a finite value of the (lower symme- try) order parameter Q ⊥ , $^{1}$, immediately induces a finite value of the (higher symmetry) order parameter Q ⊥ , $^{2}$. 

In principle Eq. (43) permits both first and second order phase transitions into the vector–multipole phase from dis or- dered (paramagnetic) or pure nematic phases, depending on the sign of the coefficients c$_{11}$ , c$_{12}$ , and c$_{22}$ . The full solution for Q ⊥ , 1 and Q ⊥ , 2 is further complicated by the fact that these order parameters also couple to octupolar spin moments (see Appendix B for details). However the relationship between Q ⊥ , 1 and Q ⊥ , 2 is clear at the level of a single-tetrahedron theory (cf. Ref. [2]). 

Within the theory for a single, embedded tetrahedron — 

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which is exact for T = 0 — the primary order parameter Q ⊥ , 1 = | Q ⊥ , $^{1}$| grows as 

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while the secondary order parameter Q ⊥ , 2 = | Q ⊥ , $^{2}$| grows 

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