10 0.0 0.2 0.4 0.6 0.8 0.06 0.08 0.10 0.12 0.14 0.16 4 6 8 12 16 L T (a) h=1.2 h=2.4 0 4 8 12 16 0.06 0.08 0.10 0.12 0.14 0.16 C v T 4 6 8 12 16 L (b) h=1.2 h=2.4 Q ,2 FIG. 10: (Color online) Dependence of the magnetization m [Eq. (6)] on magnetic field h for b = 0 . 1 and J$_{3}$ = 0 , showing the existence of the magnetization plateau in the absence of long–range N´eel order. Symbols show the result of Monte Carlo simulations for temperatures ranging from T = 0 . 04 to T = 0 . 24 and the system size L = 8 . The dashed line is the result obtained by minimizing the energy for T = 0 . FIG. 9: (Color online) (a) Nematic order parameter Q ⊥ , 2 = | Q ⊥ , $^{2}$| [Eq. (22)], signaling the nematic long–range order in finite magnetic field; (b) Heat capacity [Eq. (31)]. Data are at h = 1 . 2 and h = 2 . 4 for b = 0 . 1 , and for system size ranging from L = 4 to L = 16 . 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. that spin–spin correlations retain their power–law charac ter at low temperatures, implying the absence of long–range N´eel order. 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 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 C. Plateau liquid with local T$_{2}$ symmetry Ω$_{0}$ ≈ 1 . 7 N/ 4 ≈ 1 . 14 N (35) F = a$_{2}$ | Q ⊥ , $^{2}$|$^{2}$+ c$_{22}$ | Q ⊥ , $^{2}$|$^{4}$+ e$_{222}$ | Q ⊥ , $^{2}$|$^{6}$, (34) 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. 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. 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