content_image/442163.jpg content_image/442164.jpg content_image/442170.jpg content_image/442171.jpg content_image/442176.jpg
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$_{sat}$ 

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 

the system remains “disordered”. The nature of this degener - acy, and its consequences, are explored in some detail below. 

Where N´eel order is present, it can be detected in the re- duced spin–spin correlation function 

Here m 2 is the expectation value of the squared magnetization per spin, 

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 : 

## B. Bond order parameters 

<img src='content_image/442163.jpg'>

<img src='content_image/442164.jpg'>

<img src='content_image/442170.jpg'>

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 associated generalized susceptibilities 

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, 

<img src='content_image/442171.jpg'>

<img src='content_image/442176.jpg'>

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 

