5 λ local ν = 4 N ∑ tetra Λ 2 $_{ν}$, (10) χ local ν = N T [ 〈 ( λ local ν ) $^{2}$〉 − 〈 λ local ν 〉 $^{2}$] . (11) TABLE I: Classification of tensor order operators accordingto ro- tational symmetry about a z axis defined by magnetic field: Each forms an irrep transforming like e $^{inφ}$, where n is an integer and φ is the polar angle in the xy plane. Also indicated are the finite values of the order parameters in the 2:2 and 3:1 canted states. order par. tensor operators 2:2 1:3 e 2 iφ { Q x $^{2}$− y $^{2}$, Q $^{xy}$} finite finite e iφ { Q $^{xz}$, Q$^{yz}$} 0 finite { S $^{x}$, S$^{y}$} 0 0 1 Q 3 z $^{2}$− r 2 finite finite S z finite finite P ( r$_{ij}$ ) = 3 2 [ ( S$_{i}$ · S$_{j}$ ) 2 − 1 3 ] , (19) P ( r$_{ij}$ ) = 3 4 ∑ α Q α i Q α j , (20) Q 2 = 1 N 2 ∑ ij 4 3 P ( r$_{ij}$ ) . (21) Q α = 1 N N ∑ i =1 Q α i , (12) Q 3 z $^{2}$− r 2 i = 1 √$_{3}$ [ 2( S z $_{i}$) 2 − ( S x i ) 2 − ( S y i ) $^{2}$] , (13) Q x $^{2}$− y 2 i = ( S x i ) 2 − ( S y i ) $^{2}$, (14) Q xy i = 2 S x i S y i , (15) Q xz i = 2 S x i S z i , (16) Q yz i = 2 S y $_{i}$S z i , (17) Q ⊥ , 2 = { Q x $^{2}$− y $^{2}$, Q $^{xy}$} , (22) , 1 Q ⊥ , 1 = Q $^{xz}$, Q$^{yz}$} , (23) Q ⊥ , 0 = Q 3 z $^{2}$− r $^{2}$, (24) Q 2 = ( Q 3 z $^{2}$− r $^{2}$) 2 + ( Q x $^{2}$− y $^{2}$) 2 + ( Q $^{xy}$) 2 + ( Q $^{xz}$) 2 + ( Q $^{yz}$) 2 (18) Q 2 = ( $_{Q}$⊥ , $^{0}$)2 + | Q ⊥ , $^{1}$|2 + | Q ⊥ , $^{2}$|$^{2}$. (25) C. Rank–two tensor order parameters 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. 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 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. and it follows that To describe this, it is convenient to introduce the rank–two tensor order parameters 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 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 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- 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 .