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that spin–spin correlations retain their power–law charac ter at low temperatures, implying the absence of long–range N´eel order. 

The transition from local– E symmetry nematic state to collinear state as h → 0 is completely smooth, and the finite T properties of nematic state at finite h are qualitatively iden- tical to those shown in Figs. 6 and 7, with the obvious caveats that P ( r ) < 1 for T → 0 , and the collinear order parameter Q must be replaced by Q ⊥ , 2 = | Q ⊥ , $^{2}$| . Once again the on- set of nematic order at T$_{Q}$ is associated with a sharp peak in heat capacity, a rise in the local fluctuations with E symmetry, λ local E , and the absence of long range order of the form λ global E . A suitable free energy to describe this nematic state is 

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which permits both first and second order phase transitions into a paramagnetic phase as a function of temperature, de- pending on the sign of c$_{22}$ . However MC simulations sug- gest that the transition remains first order. Figure 9 shows the temperature dependences of the nematic order parameter Q ⊥ , 2 = | Q ⊥ , $^{2}$| [Eq. (22)] and the heat capacity [Eq. (31)] for h = 1 . 2 and h = 2 . 4 . For both cases, the order parameter exhibits a sharp onset and the heat capacity shows a jump, in- dicating that the E –symmetry nematic transition is of the first order, as for h = 0 in Fig. 6. As noted by comparing the results for h = 1 . 2 and h = 2 . 4 , the discontinuity becomes clearer as h increases. 

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In principle the E –symmetry nematic state could interpo- late to saturation, simply by canting all spins until they are aligned with the magnetic field. However this is not energeti - cally favorable at the level of a single tetrahedron (Fig. 3), and for a magnetic field h ≈ 3 , the system undergoes a first order transition into a state with magnetization m = 1 / 2 , seen as the plateau in Fig. 10. This state is discussed in detail in the section below. 

## C. Plateau liquid with local T$_{2}$ symmetry 

The half–magnetization plateaux observed in Cr spinels are associated with collinear states with three–up and one–dow n spin per tetrahedron. There are in fact an extensive number 

of such uuud states — a manifold isomorphic to hard–core dimer coverings of the diamond lattice formed by joining the centers of tetrahedra$^{36}$. (Dimers on bonds of the diamond lat- tice correspond to the down spins in uuud states on the py- rochlore lattice.) We therefore refer to it as the “dimer” ma n- ifold below. Collinear uuud states with and without simple N´eel order are illustrated in Fig. 11. 

It is possible to construct a field theory for the dimer man- ifold at T = 0 by exact analogy with the treatment of the ice manifold above. The condition that every tetrahedron has exactly three–up and exactly one–down spins translates int o a zero divergence condition for an electric (magnetic) field , modified to include a source term $^{13}$. 

Once again, thermal fluctuations are ineffective in restor- ing long–range N´eel order. Reduced spin–spin correlation s Eq. (5) at finite distance exhibit a crossover between a dipo- lar form [cf. Eq. (29)] for low temperatures, and exponential 

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