9 compensate for the loss of an extensive entropy of ln Ω$_{0}$ /N ∼ 0 . 5 ln 1 . 5 ≈ 0 . 20 per spin. We return to this point below. At finite temperature, the algebraic decay of spin corre- lations D ( r ) ∼ 1 /r 3 in the Coulomb phase is expected to crossover to exponential decay D ( r ) ∼ exp( − r/ξ$_{c}$ ) for r ≳ ξ$_{c}$ , where the characteristic length scale ξ$_{c}$ diverges for T → 0 . For Heisenberg spins in three dimensions, ξ$_{c}$ ∼ 1 $_{/}$√$_{T}$ $^{31,34,35}$. This is the only length scale in the sim- plest Coulomb theory, and it is therefore interesting to plot the spin correlations | D ( r ) | for a rescaled distance $_{r}$√$_{T}$$_{. This}$ is done in Fig. 8. At low temperatures T < T$_{Q}$ , the data appear to collapse onto a single power–law behavior in this range, while they collapse onto an exponential behavior above T$_{Q}$ : There is a rapid change between these behaviors, associ- ated with the discontinuous transition at T = T$_{Q}$ . The results suggest that the Coulomb–phase theory applies to the presen t bilinear–biquadratic model, and in addition, that the char acter- istic length ξ$_{c}$ suddenly changes from several lattice spacings in the nematic phase for T < T$_{Q}$ to one comparable to the lattice spacing in the paramagnetic phase for T > T$_{Q}$ . Once again, these results have a simple interpretation in terms of local, preformed order. In Fig. 6(c) we plot the ex- pectation value of the order parameter for the simplest kind of four–sublattice order, λ global E [Eq. (8)]. This clearly scales to zero with system size. However there is a sharp feature in the susceptibility associated with λ local E [Eq. (10)] at T$_{Q}$ , shown in Fig. 6(d), where tetrahedra with local E symmetry collec- tively choose collinear configurations. Indeed, as T → 0 , λ local E takes on its maximum allowed value of λ local E → 16 / 3 [Fig. 6(c)], as required for loops of perfectly collinear spins. FIG. 8: (Color online) Spin correlations spanning the nematic state and high temperature paramagnet plotted as a function of the rescaled distance $_{r}$√$_{T}$ on (a) log–log scale and (b) log–linear scales. The data are identical to those plotted in Figs. 7(b) and (c); grey lines are guides for the eye, showing ∼ 1 /r 3 and ∼ exp( − r ) , respec- tively. Temperatures should be compared with the nematic ordering temperature T Q ≈ 0 . 5 . B. Nematic phase with local E symmetry In applied magnetic field, the “up” and “down” spins of the collinear nematic phase immediately “flop” into the plan e parallel to h , and transform into the 2:2 canted coplanar con- figurations shown in Fig. 3. Such canting is entirely com- patible with the ice manifold, as is illustrated in Fig. 4(c) — entire loops of spins cant simultaneously, to give a state with smoothly evolving magnetization, but no N´eel order. realized here, all quadrupoles moments Q n are proportional to a simple scalar Q . The presence of a cubic term in the free energy Eq. (32) therefore implies that the phase transition from nematic phase to paramagnet as a function of tempera- ture must be first order — as observed in the MC results. In principle, thermal fluctuations might select a single N´e el state from the ice manifold, in which case T$_{Q}$ would mark the onset of dipolar as well as quadrupolar order. However this is not the case. Spin–spin correlations D ( r ) , defined in Eq. (5), remain short ranged [Fig. 7(a)]. At the distances accessible to simulation, they rapidly cross over from the power–law decay characteristic of the ice manifold at low temperatures [Fig. 7(b)] to the exponential decay expected for a paramag- net [Fig. 7(c)]. The correlation function P ( r ) retains a finite (reduced) value in this new canted manifold of states. However spin rotational symmetry is now explicitly broken by the magnetic field, so this does not of itself imply nematic order. Nematic order is none the less present, in the selection of a common plane within which the spins cant. This is equivalent to the se- lection of a direction (but not an orientation) in the xy plane, and long range order can now be observed in the transverse moment Q ⊥ , 2 defined by Eq. (22), as discussed in Table I. Since this director breaks the residual O (2) symmetry, the resulting nematic state must possess a branch of gapless (Goldstone) modes associated with rotations of the plane of canting about the z axis. It is worth noting that a canted N´eel state with E –type symmetry would break rotational symmetry in the same way$^{15}$. However for H LRO → 0 , simulations show The reason that the usual order–from–disorder mechanism is ineffective in selecting dipolar order is the massive degen- eracy of the ice manifold — the entropy gain of fluctuations about the favored state (relative to the average) would have to