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provide a useful measure of the correlation which survives in the absence of long range order, a central question for this pa- per. To this end, we introduce a measure of local correlation 

and its associated generalized susceptibility 

For a single tetrahedron, λ local ν and λ global ν are identical. On a lattice, λ local ν lacks crossterms between different tetrahedra present in λ global ν , and is therefore a measure of correlation in the absence of long range order. We return to these points below. 

Not all of the phases supported by the Hamiltonian Eq. (1) can be described using the bond order parameters Eq. (7). In Appendix A we formally classify the different types of sym- metry breaking which can arise in this model at the level of a single site. Here we restrict ourselves to the simplest possible generalization from N´eel to multipolar order; both the T$_{2}$ and E symmetry canted states possess order of transverse (i.e., x and y ) spin components which vanishes in the collinear uuud state, and which can survive even in the absence of conven- tional (canted) N´eel order. 

To describe this, it is convenient to introduce the rank–two tensor order parameters 

where the local quadrupole moments 

are summed over all lattice sites i . 

Where spin rotational symmetry is not already broken by magnetic field, i.e., for h = 0 , spins may select a common axis without selecting a direction on it. This is conventional nematic order, of the type exhibited by uniaxial molecules, and can be detected using the order parameter 

which is invariant under O (3) rotations. This order param- eter takes on its maximal value 〈 Q $^{2}$〉 → 4 / 3 in a perfectly collinear state, such as the 2:2 state for T 0 . 

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## C. Rank–two tensor order parameters 

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In what follows we will also make use of the correlation function measuring collinearity 

considered in Ref. [17]. As defined, − 1 / 2 ≤ 〈 P ( r$_{ij}$ ) 〉 ≤ 1 , taking on the value 〈 P ( r$_{ij}$ ) 〉 = 0 for uncorrelated spins. In fact P ( r$_{ij}$ ) can also be expressed in terms of quadrupolar op- erators as 

and it follows that 

At finite h , the O (3) invariant correlation function Eq. (19) still provides a useful measure of collinearity, but does not by itself signal a broken symmetry. In this case it is convenient to group quadrupoles according to way in which they trans- form under the remaining O (2) rotations about the direction of magnetic field — conventionally the z axis. We therefore consider 

where the magnetic field is assumed to be parallel to the z axis. Each of the separate irreps Q ⊥ ,n transforms like { cos nφ, sin nφ } — or equivalently, e inφ — where φ is the polar angle in the plane perpendicular to the magnetic field. They can therefore be used as order parameters to detect the n –fold breaking of rotational symmetry in the xy plane. The conventional nematic order parameter with full O (3) symme- try, Eq. (18), is given by the sum of squares 

In finite magnetic field, the one–dimensional irrep Q ⊥ , 0 does not contain any information about broken symme- 

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