15 0.0 0.5 1.0 1.5 2.0 2.5 0 200 400 600 0.016 0.032 0.048 0.064 T 0 4 8 12 0.0 0.1 0.2 0.3 0.4 0.5 0.016 0.032 0.048 0.064 T C v 4 6 8 12 16 L (a) (b) (c) 4 6 8 12 16 L 4 6 8 12 16 L (d) 4 6 8 12 16 L Q ,1 0.00 0.02 0.04 0.06 0.08 0.10 FIG. 17: (Color online) Magnetic field dependence of (a) prim ary order parameter Q ⊥ , 1 = | Q ⊥ , $^{1}$| [Eq. (23)] and (b) secondary order parameter Q ⊥ , 2 = | Q ⊥ , $^{2}$| [Eq. (22)] in the vector–multipole phase for b = 0 . 1 . The continuous transition from the plateau liquid state into the vector–multipole phase at lower magnetic fields, an d the di- rect transition from the vector–multipole phase into the (s aturated) paramagnet at higher fields are clearly visible. Dashed line s show behavior at T = 0 in a single-tetrahedron theory. Points show results of MC simulations for the system size N = 8 from T = 0 . 02 to T = 0 . 08 . FIG. 18: (Color online) Temperature dependence of (a) the primary order parameter Q ⊥ , 1 = | Q ⊥ , $^{1}$| defined by Eq. (23), showing the onset of the vector–multipole order at T = T V ≈ 0 . 042 , (b) heat capacity Eq. (31), (c) the related measure of local correlation λ local T$_{2}$ defined by Eq. (10), and the global order parameter λ global T$_{2}$ defined by Eq. (8), and (d) the associated local susceptibility Eq. (11). Simula- tions were performed for h = 6 , b = 0 . 1 , in clusters with L = 4 to L = 16 . 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]). which is exact for T = 0 — the primary order parameter Q ⊥ , 1 = | Q ⊥ , $^{1}$| grows as Within the theory for a single, embedded tetrahedron — while the secondary order parameter Q ⊥ , 2 = | Q ⊥ , $^{2}$| grows Q ⊥ , $^{1}$( h ≳ h$_{c}$ ) | ∼ = 3 √ 2(3 − 2 b ) √ h − h$_{c}$ , (44)