8 1.6 2.4 3.2 4.0 0.0 0.2 0.4 0.6 0.8 1.0 0.08 0.10 0.12 0.14 0.16 0.18 4 6 8 12 16 L Q T (a) 0 5 10 15 20 C v (b) 0.00 0.02 0.04 0.06 0.08 0.10 (c) 4 6 8 12 16 L 4 6 8 12 16 L 0 200 400 600 800 0.08 0.10 0.12 0.14 0.16 0.18 T (d) 4 6 8 12 16 L FIG. 7: (Color online) (a) Spin correlations D [Eq. (5)] at h = 0 for b = 0 . 6 , for r ≤ L measured along 〈 110 〉 chains in the py- rochlore lattice and averaged over different chain directions as in Fig. 5. r is in units of the distance between nearest neighbor spins. (b) Characteristic 1 /r 3 power–law decay of spin correlations in the low–temperature nematic phase for T < T Q ∼ 0 . 5 associated with the ice manifold of 2:2 states, plotted on a log–log scale. (c ) Expo- nential decay of spin correlations in the high–temperatureparamag- netic phase for T > T Q ∼ 0 . 5 , plotted on a log–linear scale. In (b) and (c), grey lines are guides for the eye, ∼ 1 /r 3 and ∼ exp( − r ) , respectively. In all figures the data are for temperatures ra nging from T = 0 . 30 to T = 0 . 84 and the system size L = 16 . FIG. 6: (Color online) (a) Nematic order parameter Q [Eq. (18)], showing the onset of nematic order at T Q ≈ 0 . 13 ; (b) Heat capacity; (c) Absence of long range four–sublattice order λ global E is accompa- nied by well–formed local correlations λ local E ; (d) The associated lo- cal susceptibility shows a sharp jump at T $_{Q}$, where spins in tetrahe- dra with preformed local order gain energy by selecting long range collinearity. All data are for h = 0 , b = 0 . 1 , and for system size ranging from L = 4 to L = 16 . is given by The sharp onset of order and jump in heat capacity imply a first order phase transition at T$_{Q}$ ≈ 0 . 13 . Treating this nematic order at the level of a Ginzburg- Landau theory, the free-energy terms allowed by lattice and spin rotational symmetries are where Q 2 is defined in Eq. (18), the third order invariant Q 3 In a three–dimensional uniaxial nematic state, such as that Q 3 = 2( Q 3 z $^{2}$− r $^{2}$) 3 + 3 Q 3 z $^{2}$− r 2 [ ( Q $^{xz}$) 2 + ( Q $^{yz}$) $^{2}$] − 6 Q 3 z $^{2}$− r 2 [ ( Q x $^{2}$− y $^{2}$) 2 + ( Q $^{xy}$) 2 ] $^{$_{+3}$√$_{3}$}$( Q x $^{2}$− y 2 [ ( Q $^{xz}$) 2 − ( Q $^{yz}$) $^{2}$] +2 Q $^{xy}$Q$^{xz}$Qyz ) . (33) F = aQ $^{2}$+ bQ $^{3}$+ cQ 4 + ... , (32)