11 FIG. 12: (Color online) Absence of long–range magnetic orde r in the plateau liquid state for b = 0 . 6 , J$_{3}$ = 0 , and h = 4 . (a) The reduced spin correlation function D ( r ) , defined by Eq. (5), measured along the 〈 110 〉 chains, as in Fig. 7(a). (b) Characteristic 1 /r 3 power–law decay of spin correlations at low temperatures T < T ∗ ∼ 0 . 6 , asso- ciated with the dimer manifold of uuud states, plotted on a log–log scale. (c) Exponential decay of spin correlations at high temperatures T > T ∗ ∼ 0 . 6 , plotted on a log–linear scale. In (b) and (c), grey lines show guides for the eye, ∼ 1 /r 3 and ∼ exp( − r ) , respectively. In all figures the data are for temperatures ranging from T = 0 . 30 to T = 0 . 84 and the system size L = 16 . FIG. 11: (Color online) Half–magnetization plateau states( uuud states) on a pyrochlore lattice with exactly three–up and on e–down spins per tetrahedron. (a) Schematic picture of an uuud state with no long range order, associated with the “dimer” manifold; ( b) uuud state with long–range four–sublattice order with T$_{2}$ symmetry in- duced by FM J$_{3}$ , as considered in Refs. [2] and [15]; (c) 16– sublattice order. See also Fig. 15. gular features are similarly absent from the heat capacity, shown in Fig. 14(b). This smooth change is also seen in the rescaled plot of the spin correlations shown in Fig. 13. The crossover from the low– T power–law behavior to the high– T exponential decay is much more smooth compared to the case for the nematic transition at h = 0 in Fig. 8. This suggests a smooth growth of the characteristic length scale ξ$_{c}$ ∼ 1 $_{/}$√$_{T}$ at T ∼ T ∗ ≈ b . We therefore conclude that the magnetiza- tion plateau is a true spin–liquid state, continuously conn ected with the high– T paramagnet. We refer to this as the plateau liquid below. decay for high temperatures — see Fig. 12. While spins are perfectly collinear at low temperatures [Fig. 14(a)], the z axis is now singled out by magnetic field, and Q 3 z $^{2}$− r 2 contributes to P ( r ) . This means that there is no symmetry breaking associated with the smooth rise in collinearity for T ∗ ≈ b , which should be regarded as a crossover rather than a phase transition. Sin-