2 leads to the four–sublattice long-range order (LRO) descri bed in Ref. [2], and to the finite temperature transitions shown i n Fig. 2(a)$^{14,15}$. Four sublattice order can also be stabilized by AF second neighbor interaction J$_{2}$ . More generally, however, the type of order which results depends on the details of the in- teraction H LRO $^{18,19}$. The system can therefore be tuned at will between different types of ordered state, simply by changing H $^{LRO}$. From this we conclude that, as a function of magnetic field h , for H LRO = 0 , there must be a line of second–order multicritical — or first–order multifurcative points — separat- ing a huge set of different ordered phases. The main purpose of this paper is to explore the symmetry breaking which persists in the limit of H LRO → 0 for finite biquadratic interaction b and finite temperature T . In order to make the problem accessible to large scale Monte Carlo (MC) simulation, we consider the classical S ≡ | S | → ∞ limit of Eq. (1), rescaling variables such that S ≡ 1 . Using a mixture of classical MC simulation, analytic low– T expansion, and simple field theoretical arguments, we find a set of phases in the h – T plane which exhibit power–law de- cay of spin correlation functions. Two of these phases possess long–range nematic or vector–multipole order and, most int er- estingly, the magnetization plateau persists in the absence of conventional magnetic order. We show how all of these results can be understood — and even anticipated — from a proper understanding of the geometry of the pyrochlore lattice, and the way in which a single tetrahedron behaves in magnetic field. Our findings are summarized by the h – T phase diagram shown Fig. 2(b). So far as experiment is concerned, our main conclusion will be that the thermodynamic properties of the pyrochlore anti- ferromagnet in applied magnetic field are mostly determined by symmetry breaking at the level of single tetrahedron. Local order is well–formed for H LRO = 0 , and many properties of the system are therefore insensitive to the details of the LRO order present. Thus the very simple phase diagram derived in Ref. [2] and its finite temperature generalization in Refs. [ 14] and [15] [reproduced in Fig. 2(a)], are applicable for a wide variety of different H $^{LRO}$. The paper is structured as follows: In Sec. II we briefly review the basic physics of the Heisenberg model on the py- rochlore lattice. Definitions are given of order parametersfor conventional N´eel (dipolar) order, and of rank–two tensoror- der parameters which can be used to signal multipolar order. FIG. 2: (Color online) (a) Magnetic phase diagram of the classical pyrochlore antiferromagnet with biquadratic interactions b = 0 . 1 , and additional third–neighbor interactions J$_{3}$ = − 0 . 05 , as deter- mined by classical Monte Carlo simulation$^{14,15}$. The form of four– sublattice N´eel order is illustrated, together with the irreducible rep- resentation (irrep) of the tetrahedral symmetry group T d to which it belongs. For these parameters, the model provides a good description of the half-magnetization plateau seen in CdCr$_{2}$O$_{4}$ 3–8 . (b) Equivalent magnetic phase diagram in the absence of any longer range interac- tions. Canted N´eel states are replaced by phases with multipolar or- der, while the collinear half-magnetization plateau state gives way to a collinear spin liquid. Crosses denote the crossover at temperature T ∗ from paramagnet to the plateau–liquid state, as determinedby a peak in the heat capacity. In both (a) and (b) circles with solid (red) lines denote first–order phase boundaries, while those with dashed (blue) lines denote second–order ones. Both T and h are measured in units of J$_{1}$ . Then, in Sec. III we use these tools to construct the h – T phase diagram of the pyrochlore AF with additional bi- quadratic interactions [Fig. 2(b)]. Thermal fluctuations p re- serve the extensive degeneracies present in the ground state, and fail to select any conventional long–range dipolar or- der. Despite this, the thermodynamic properties of the sys- tem and the topology of the phase diagram are essentially un- changed — the magnetization plateau survives and nematic and vector–multipole phases corresponding to the two diffe r- ent canted states are shown to exist for fields below and above the magnetization plateau [Fig. 2(a)]. In Sec. IV we explore the way in which long–range N´eel order is recovered as a FM third–neighbor interaction J$_{3}$ is which links spins in different tetrahedra, for example, where 〈 ij 〉 3 runs over the two (inequivalent) sets of third neighbor bond shown in Fig. 1. For ferromagnetic (FM) J$_{3}$ < 0 , this specific form of H LRO H LRO = J$_{3}$ ∑ 〈 ij 〉 3 S$_{i}$ · S$_{j}$ , (2)