20 Acknowledgments with upper critical dimension three, i.e., one where the critical behavior is mean–field like, up to log corrections $^{53,58}$. We are pleased to acknowledge stimulating discussions with F. Alet, J. Chalker, G. Kriza, G. Misguich, T. Momoi, H. Shiba, H. Takagi, O. Tchernyshyov, S. Trebst, H. Tsunet- sugu, and H. Ueda. We are particularly indebted to R. Moess- ner and M. E. Zhitomirsky for valuable comments about mul- ticritical points and the classification of multipolar orde r. This work was supported under EPSRC Grants No. EP/C539974/1 and EP/G031460/1, and SFB 463 of the DFG (NS); Hungarian OTKA T049607 and K62280 (KP); Grant– in–Aid for Scientific Research No. 16GS50219, 17740244, and 19052008 from MEXT, Japan; Global COE Program “the Physical Sciences Frontier”, MEXT, Japan, and Next Gener- ation Super Computing Project, Nanoscience Program (YM). Part of this work was done while KP and YM were visitors at KITP Santa Barbara. KP and NS also acknowledge the hos- pitality of MPI–PKS Dresden, where a part of this work was completed. We have studied the ordered and disordered phases of the classical, bilinear–biquadratic Heisenberg model on the p y- rochlore lattice at finite temperature and in applied magnet ic field. We find a rich collection of unconventional states — ne- matic and vector–multipole phases with distinct and differ ent local symmetries, separated by a half–magnetization plate au with spin–liquid character. All of these phases show an unde r- lying “Coulombic” character with algebraic decay of spin co r- relation functions over distances r ≲ ξ$_{c}$ ∼ 1 $_{/}$√$_{T}$$_{. Interest-}$ ingly, the transition from plateau–liquid to vector–multi pole phase is continuous, and appears to be well–described by mean field theory. V. SUMMARY AND CONCLUSIONS Appendix A: Classification of symmetry breaking at the levelof a single site While this behavior is undeniably exotic, all of these states can be understood — and even anticipated — from a proper understanding of the geometry of the pyrochlore lattice, and the properties of a single tetrahedron. Strong local fluctua- tions of N´eel order are present in all of these phases, and the zero temperature phase diagram can be understood sim- ply from the “self assembly” of these ordered tetrahedra int o complex states with higher symmetry. In order to identify the different possible forms of magnetic order which can survive where conventional N´eel order breaks down, it is helpful to classify the different forms of symmetry breaking which exist at the level of a single site. This analysis is in the spirit of the detailed classification for the nemati cs in liquid crystals undertaken in Ref. [62], and motivates the rank–two and rank–three tensor order parameters introduce d in Sec. II C and Appendix B. In order to keep contact with quantum spins, which are axial rather than polar vectors, we must keep track of time reversal symmetry. It is therefore unsurprising that conventional N´eel order (with four–sublattice structure) is immediately restoredby the introduction of a ferromagnetic third–neighbor coupli ng J$_{3}$ . However for small | J$_{3}$ | , the unconventional states survive above the N´eel transition temperature T$_{N}$ . In particular, the spin–liquid plateau survives above T$_{N}$ , up to a crossover tem- perature T ∗ ≈ b . The transition between liquid and ordered plateaux is first order in nature, and remains so for T$_{N}$ → 0 . For small | J$_{3}$ | , the system also exhibits a first–order transition between the N´eel and nematic phases, in addition to the first – order transition from the nematic phase into high-temperature paramagnet. In Table III we show the transformation rules for the spins under selected symmetry operations, including time reversal Θ S = − S . In contrast to the usual polar vectors, inver- sion leaves the axial vectors invariant – as a consequence, a ll the usual (reflection, rotation, and inversion) symmetry op er- ation can be represented by an orthogonal matrix belonging to SO (3), with determinant equal to +1. The role of inversion in the case of polar vectors is taken over by the time reversal operator Θ . So far as experiment is concerned, our main finding is that the physics of a pyrochlore antiferromagnet in magnetic fiel d can be largely determined by the properties of a single tetra- hedron. In the simple models which we have considered it is possible to tune between states with entirely different point group symmetries at will, simply by changing the form of (weak) long range interactions present. This is an oversimpli- fication, in the sense that magnetostriction in real systemsis likely single out a particular phonon (or family of phonons) with definite symmetry, which will then drive the system towards collinearity ( b , in our model) and select the low– temperature ordering pattern (long range interactions, e.g., J$_{3}$ , in our model). However, as long as there is a strong coupling to phonons within individual tetrahedra, the form of the mag- netization plateau and associated phases may be largely inde- pendent of these (system dependent) details. All of the symmetry operations, extended with the time re- versal, can be represented by orthogonal matrices with deter- minant -1. In the Table III we also check if the collinear and coplanar states are invariant under those symmetry operations. Since we are interested in the symmetry breaking which can occur in the absence of broken translational symmetry, we do not apply the symmetry elements to the lattice points (i.e., we treat all the spins as they were at the origin). We find that the invariant operations of the 2:2 state include a C$_{2}$ ( z ) rotation in addition to the symmetry operations of the 3:1 state. In Table IV we show the symmetry group of each of the spin states. In order to facilitate comparison with Ref. [62], we also show the symmetry group of the states if the spins were polar vectors. We can see that as the time reversal does not play a role for the 2:2 collinear state, its symmetry group