4 the system remains “disordered”. The nature of this degener - acy, and its consequences, are explored in some detail below. In order to understand how nearest–neighbor biquadratic interactions b select among this manifold of states, it is suf- ficient to solve the problem of a single tetrahedron embed- ded in the 3D lattice. This problem was considered in Ref. [2]. For b > 0 , biquadratic interactions select coplanar (and collinear) configurations from the larger ground–stat e manifold of Eq. (3). There are four dominant phases, illus- trated in Fig. 3 : (i) a 2:2 coplanar canted state for low field (ii) a 3:1 collinear ( uuud ) half–magnetization plateau state for intermediate field (iii) a 3:1 coplanar canted state for fields approaching satu - ration (iv) a saturated ( uuuu ) state for large magnetic field h > h$_{sat}$ B. Bond order parameters Where N´eel order is present, it can be detected in the re- duced spin–spin correlation function D ( r$_{ij}$ ) = 〈 S$_{i}$ · S$_{j}$ 〉 − m $^{2}$. (5) m 2 = 〈 1 N ( ∑ i S$_{i}$ ) 2 〉 , (6) Here m 2 is the expectation value of the squared magnetization per spin, An exhaustive enumeration of possible states is given in Ref. [20]. Up to this point, we have not been specific about how the tetrahedron was embedded in the lattice. It could, trivially, form part of a state with N´eel order, e.g., the simple four– sublattice order favored by FM J$_{3}$ . However, there are in- finitely many other ways of joining 2:2 or 3:1 tetrahedra to- gether at the corners, and not all of them correspond to N´eel ordered states. In fact the ground state manifold retains an ex- tensive Ising–like degeneracy for all h < h$_{sat}$ , and as a result which vanishes in the absence of magnetic field. N is the total number of spins. The simplest form of order supported by the pyrochlore lattice is the four–sublattice N´eel order favo red by FM J$_{3}$ , as illustrated in Fig. 2(a). Written in terms of the minimal four–site unit cell of the py- rochlore lattice, four–sublattice order has momentum q = 0 , and different states can easily be classified using the A$_{1}$ , E , and T$_{2}$ irreducible representations (irreps) of the symmetry group T d for a single tetrahedron :    Λ$_{A}$$_{1}$ Λ$_{E}$$_{,}$$_{1}$ Λ$_{E}$$_{,}$$_{2}$ Λ$_{T}$$_{2}$$_{,}$$_{1}$ Λ$_{T}$$_{2}$$_{,}$$_{2}$ Λ$_{T}$$_{2}$$_{,}$$_{3}$    =     1 √$_{6}$ 1 √$_{6}$ 1 √$_{6}$ 1 √$_{6}$ 1 √$_{6}$ 1 √$_{6}$ 1 √$_{3}$ − 1 $_{2}$√$_{3}$ − 1 $_{2}$√$_{3}$ − 1 $_{2}$√$_{3}$ − 1 $_{2}$√$_{3}$ 1 √$_{3}$ 0 1 2 − 1 2 − 1 2 1 2 0 0 0 1 √$_{2}$ 1 √$_{2}$ 0 0 0 − 1 √$_{2}$ 0 0 1 √$_{2}$ 0 − 1 √$_{2}$ 0 0 0 0 1 √$_{2}$        S$_{1}$ · S$_{2}$ S$_{1}$ S$_{3}$ S$_{1}$ · S$_{4}$ S$_{2}$ S$_{3}$ S$_{2}$ · S$_{4}$ S$_{3}$ S$_{4}$    , (7) λ global ν = 4 N ( ∑ tetra Λ$_{ν}$ ) $^{2}$, (8) where the spins S$_{i}$ belong to a single tetrahedron 2,20,21 — cf. right panel of Fig. 3. We can use these irreps to define bond order parameters and the transition from the A$_{1}$ to T$_{2}$ states at high field — cf. Fig. 2(a) — but not the more subtle second–order transition between the T$_{2}$ symmetry uuud plateau state and the T$_{2}$ sym- metry 3:1 canted state. These are none the less distinct phases — the collinear and canted T$_{2}$ states are connected by a zone– center (i.e., q = 0 ) excitation which is gapped in the collinear uuud state, and becomes soft at the critical field marking the onset of the 3:1 canted state$^{20}$. The condensation of this soft spin mode corresponds to the emergence of order in the trans- verse spin components 〈 S $^{x}$〉 , 〈 S $^{y}$〉 — i.e., canting of spins away from the z axis (the direction of applied magnetic field). The bond order parameters defined in Eq. (7) couple di- rectly to the lattice, through changes in bond length$^{2}$. They are therefore particularly well suited to describing simple N´eel or- dered states, where the magnetic ordering is driven by the lat- tice effects. However the irreps on which they are based also and associated generalized susceptibilities χ global ν = N T [ 〈 ( λ global ν ) $^{2}$〉 − 〈 λ global ν 〉 $^{2}$] , (9) where the sum ∑ tetra runs over all N/ 4 independent A– sublattice tetrahedra, and Λ$_{ν}$ is the vector associated with the ν = { E , T$_{2}$ } irreps of the tetrahedral symmetry group T $_{d}$, namely, Λ$_{E}$ = (Λ$_{E}$$_{,}$$_{1}$ , Λ$_{E}$$_{,}$$_{2}$) and Λ$_{T}$$_{2}$ = (Λ$_{T}$$_{2}$$_{,}$$_{1}$ , Λ$_{T}$$_{2}$$_{,}$$_{2}$ , Λ$_{T}$$_{2}$$_{,}$$_{3}$) . These order parameters allow us to distinguish the sharp first–order transition between orders in the T$_{2}$ and E irreps,