6 〈| Q ⊥ , $^{2}$|$^{2}$〉 = 〈 ( Q x $^{2}$− y $^{2}$) 2 + ( Q $^{xy}$) $^{2}$〉 > 0 . (26) 〈| Q ⊥ , $^{1}$|$^{2}$〉 = 〈 ( Q $^{xz}$) 2 + ( Q $^{yz}$) $^{2}$〉 > 0 (27) Ω$_{0}$ ≈ 1 . 5 N/ 2 ≈ 1 . 22 N (28) 〈 S$_{i}$ · S$_{j}$ 〉 ∼ 1 | r$_{i}$ − r$_{j}$ | 3 , (29) tries and can generally be discarded. However the two– dimensional irreps Q ⊥ , 1 and Q ⊥ , 2 distinguish different or- dered phases. In the 2:2 canted phase the mean square value of Q ⊥ , 2 takes on a finite value lead to nematic order with broken spin–rotational symmetry , but power–law decay of spin–spin correlations $^{17}$. In what follows we use the order parameters defined in Sec. II C to identify phases of Eq. (1) which exhibit nematic order in the absence of N´eel order. We focus chiefly on dif- ferent forms of unconventional order found in magnetic field . Closely related studies in magnetic field have been made of the classical Heisenberg model on a kagome lattice$^{23}$, and classical XY model on a checkerboard lattice$^{25}$. In both these cases unconventional order is stabilized by thermal fluctua - tions. Another type of unconventional order for the pyrochlore lattice with FM second–neighbor interactions J$_{2}$ < 0 and h = 0 was recently studied in Ref. [26]. In our case the main driving force is not fluctuations but finite biquadratic inte rac- tion b ; results for order stabilized by thermal fluctuations at finite h and J$_{3}$ but b ≡ 0 will be presented elsewhere$^{27}$. This is another form of nematic order of the transverse spin moments — one transforming like e i 2 φ — and reflects the fact that spins select a common plane in which to cant. At the same time mean square value of Q ⊥ , 1 — which transforms as e $^{iφ}$, i.e., a vector in the xy plane — vanishes. Similarly, 〈| Q ⊥ , $^{2}$|$^{2}$〉 takes on a finite value in the 3:1 canted phase. However in this case the 3:1 asymmetry of the canted spin configuration defines a direction in the xy plane, and III. PARTIAL LIFTING OF DEGENERACY IN FINITE MAGNETIC FIELD A. Collinear nematic phase for h = 0 is also finite. The 3:1 canted phase therefore possess a form o f vector-multipole order. These facts are summarized in Table I. In what follows we concentrate almost exclusively on phases which do not exhibit conventional magnetic order, as defined by D ( r ) in Eq. (5), and characterize these states using the rank–two tensor order parameters listed in Table I. For further details of conventional N´eel phases, and comparison with experiment, we refer the interested reader to Ref. [15]. Rank–three tensors which also occur as order parameters in the present model are discussed in Appendix B. In the absence of magnetic field, the ground state of Eq. (1) is determined by the conditions that (i) the total magnetization of each tetrahedron be zero, to minimize the antiferromagnetic exchange interaction J$_{1}$ , and (ii) all spins be collinear, to min- imize the biquadratic interaction b . These conditions select an extensive manifold of D. General considerations Many frustrated systems with disordered ground states manage none the less to order at finite temperature. This effe ct is known as “order from disorder” and occurs where there is a net entropy gain in selecting one particular state out of the dis- ordered manifold. Entropy is gained where a given spin con- figuration (typically, collinear or coplanar) has a higher d en- sity of low–energy excitations than its peers. However this entropy gain must be sufficient to offset the entropy lost by choosing one state out of the manifold. Where the ground state manifold has an extensive degeneracy, this is a very strong constraint. Order–from–disorder effects are known to select one particular N´eel ordered ground state in e.g., the frustrated square lattice$^{22}$, but fail to do so in the case of the more frustrated kagome lattice$^{23,24}$. states with exactly two–“up” and two–“down” spins ( uudd ) in each tetrahedron. The degeneracy of this ground state man- ifold is of the same form as that encountered in Pauling’s the- ory of water ice$^{28}$, and we therefore refer to it as the “ice” manifold below. Since each spin is shared by two neighbor- ing tetrahedra, “up” and “down” spins form unbroken loops as shown in Fig. 4. We return to this point below. The fact that the direction along which “up” and “down” spins point is not determined by the Hamiltonian implies that spin rotational symmetry must be broken spontaneously (for simplicity, we none-the-less to use “up” and “‘down” to de- note the oppositely oriented spins). This can be seen in the spin collinearity Eq. (19), which takes on the maximal value P ( r ) = 1 for all states in the ice manifold, implying that the ground state manifold has nematic (i.e., quadrupolar) or- der. (This is explicitly confirmed by MC simulations below.) However, as already stated, the ground state manifold does not possess N´eel order of any form. Even where fluctuations fail to stabilize one particular N´e el ground state, they can still select a subset of states from the ground state manifold with a smaller — but none the less ex- tensive — degeneracy. This subset (submanifold) of states will not exhibit the long range spin–spin correlations whic h are the hallmark of conventional N´eel–type magnetic order . However this does not necessarily mean that the system is truly disordered — it may well exhibit long range order of a more complex type. In fact it is possible to calculate the asymptotic form of spin–spin correlations in the ice manifold by mapping them onto configurations of a notional electric (or magnetic ) field 29–31 . The condition that every tetrahedron has exactly two–“up” and exactly two–“down” spins translates into a zer o divergence condition for the electric (magnetic) field, and spin–spin correlations take on a dipolar form A good example of this second type of order–from–disorder effect is provided by the nearest–neighbor classical XY model on the pyrochlore lattice, where thermal fluctuations