19 states for h = 0 . Once again, for “large” | J$_{3}$ | there is a strongly first order transition from the paramagnet into the N´eel phase at a unique temperature T$_{N}$ . Meanwhile, for “small” | J$_{3}$ | ≪ b , there is double transition, first from para- magnet to E –symmetry nematic phase at T$_{Q}$ ∼ b , and then into the four–sublattice N´eel order at a much lower tempera - ture T$_{N}$ , as demonstrated in Fig. 23. Within the limits of our simulation, both of these transitions appear to be first orde r in character$^{64}$. The results for varying J$_{3}$ are summarized in the phase diagram in Fig. 24. The first order transition from para - magnet to nematic phase at T$_{Q}$ at h = 0 should be compared with the crossover from paramagnet to plateau liquid T ∗ ∼ b observed for h = 4 in Fig 22. different ordered phases branch out from the point J$_{3}$ = 0 , T$_{N}$ = 0 , it can probably best be termed “multifurcative” . First–order phase transitions between different ordered phases with an infinite degeneracy at the transition occur in a number of models. Such phase transitions are first order, in the sense that neither ordered parameter collapses approach- ing the critical point. However they also exhibit one of the characteristic features of a second order transition, namely a soft excitation or set of soft excitations connecting the differ- ent ordered ground states. As far as we can tell from our present results, it seems most likely that the classical pyrochlore AF with biquadratic in- teractions exists at a multifurcative point in parameter space, with an infinite ground–state degeneracy, not at a critical end point. As shown in Fig. 21, the low temperature value of the order parameter is broadly independent of T$_{N}$ . This implies that the phase transition in fact becomes more strongly first order at T$_{N}$ → 0 , and appears to rule out a (multi)critical end point. Our collected simulation results for J$_{3}$ → 0 are sum- marized in the form of the phase diagram in Fig. 22. Perhaps the most interesting of the finite temperature trans i- tions observed in our model is the one from plateau–liquid to vector–multipole phase, already described in Sec. III D. At one level this is the most exotic phase transition we study — a continuous phase transition from a “Coulombic” state with algebraic decay of correlation functions (the plateau liq- uid) to a phase with long–range multipolar order (the vector – multipole state). But at the same time it has the simplest phenomenology of any of the phase transitions in this pa- per, with the order parameter exhibiting a simple mean–field like behavior Q ⊥ , $^{1}$( T ) ∼ √$_{T}$$_{V}$ − T with T$_{V}$ ≈ 0 . 09 , as shown in Fig. 25(a). The secondary order parameter Q ⊥ , 2 grows more slowly as expected [Fig. 25(b)]. The heat ca- pacity does not show a noticeable singularity at T = T$_{V}$ in Fig. 25(c), which is also consistent with the mean–field be- havior C$_{v}$ ∼ ( T − T$_{V}$ ) α with α = 0 . (The broad peak at T ∼ 0 . 112 again corresponds to the crossover temperature T ∗ for the plateau-liquid state.) It is amusing to note that this phase diagram bears a su- perficial resemblance to the phenomenology of a second or- der (quantum) critical point — a transition temperature which collapses to a special point with algebraic decay of correlation functions, which in turn controls a broad region of the phase diagram up to a characteristic crossover temperature T ∗ . All of this despite the fact that the only phase transition present is first order, which means that the length scale associated with fluctuations remains finite. Some of the generic feature s seen in our model — power law decay of correlations over a large, but finite, length scale — have been previously dis- cussed in the context of other models with strong local con- straints, where they were dubbed “high temperature critica l- ity” $^{60}$. The transition from a critical “Coulombic” phase described by a U (1) gauge theory into a simple ordered state as a func- tion of temperature can be studied much more cleanly in the (classical) dimer model on cubic lattice, where the constraint enforcing the dimer manifold is infinite. In this case, the ph ase transition is continuous, and exhibits interesting and unusual scaling properties$^{48,49}$. We have made a preliminary study of the “stiffness” K associated with fluctuations in a U (1) gauge theory for temperatures spanning the paramagnet and plateau liquid phases in our model (see Fig. 22), but find no clear ev- idence of a phase transition. However the way in which the dimer and loop manifolds break down at finite temperature in a model with a finite constraint is an interesting problem, and one which deserves further study. We note in passing that interesting, related, problems arising the context of quantum loop models$^{61}$. At a qualitative level, and in the spirit of this paper, it is easy to see how a continuous transition can arise between these two states. Both are built of tetrahedra with a local T$_{2}$ character, with three “up” and one “down” spin, joined at the corners. Both states will exhibit algebraic decay spin correlations at low temperatures, as a result of the infinite number of ways that these tetrahedra can be assembled to form a pyrochlore lattice. The only difference is that three “up” and one “down ” spins are canted in the vector–multipole phase, giving a fini te value of Q ⊥ , 1 and Q ⊥ , 2 [Figs. 25(a) and (b)]. As long as this canting can interpolate smoothly to zero in the collinear uuud state, the transition will be continuous. And at the level of a Ginzburg–Landau theory, nothing prevents this from happen - ing — cf. Eq. (43). However in principle it should also be possible to tran- scribe each of these phases in terms of the more sophisticated “solenoidal field” theory used to describe N´eel order in a sp in model with a high temperature Coulomb phase (cf. Ref. [53]). To the best of our knowledge, nobody has yet attempted to extend the gauge–field description of a Heisenberg type spin model to treat multipolar order. But it is interesting to note that the transition from Coulombic phase to simple N´eel or- der was found to be continuous, and in a universality class It is interesting to contrast the finite temperature phase tr an- sitions associated with the plateau states for h ≈ 4 , with the transitions into E –symmetry N´eel and nematic ordered C. Transitions from paramagnet to E –symmetry nematic phase and N´eel ordered state D. Transition from plateau–liquid to vector–multipole pha se