17 0.2 0.4 0.6 0.8 1.0 T 0 2 4 6 8 10 0.2 0.4 0.6 0.8 1.0 C v T 0 1 2 3 4 5 0 10 20 30 40 1 2 3 4 5 0 100 200 300 T 2 local global T 2 χ$^{T}$ 2 local global T 2 4 6 8 12 16 L (a) (b) (c) 4 6 8 12 16 L 4 6 8 12 16 L 4 6 8 12 16 L (d) (e) 4 6 8 12 16 L ature phase cannot be described using an expansion in terms of the low–temperature order parameter. A recent very de- tailed simulation study of a family of three–dimensional di mer models with ordered ground states and high–temperature Coulomb phases found a rich variety of continuous and dis- continuous phase transitions, including double phase tran- sitions where monopole excitations condense out of the Coulomb phase to give a conventional paramagnet at inter- mediate temperatures$^{52}$. The present understanding of these phenomena is that the gauge field associated with Coulomb phase is minimally cou- pled to a matter field which condenses in the ordered phase, following an Anderson–Higgs mechanism 50–52 . In fact it is also possible to study zero temperature (quantum) phase tran- sitions from Coulomb to ordered phases in three–dimensiona l quantum dimer models$^{56}$. These can in principle be continu- ous, occurring through the condensation of monopole excita- tions in the Coulomb phase$^{57}$, but numerical simulations sug- gest that the transition is first order $^{37}$. Less is known about transitions in spin models, but one interesting scenario exists for a continuous transition in an extended Heisenberg model on a pyrochlore lattice from a Coulomb phase to a four–sublattice ordered state $^{53}$. This tran- sition is found to be in the same universality class as a uniax- ial ferroelectric with dipolar interactions, for which the upper critical dimension is three$^{58}$. This makes possible to continu- ous transitions with mean-field exponents (up to log correc- tions) — a scenario which closely resembles the transition from plateau-liquid into vector-quadrupole phase discussed below. Generically, however, transitions from Coulomb liq- uids into ordered states seem to be first order $^{53}$, a fact which may be explained by interactions between fluctuations of as- sociated gauge field $^{54}$. We conclude by noting that the complex forms of order which can occur in Heisenberg models on the pyrochlore lat- tice as a result of the interplay between farther–neighbor i n- teractions and thermal fluctuations are also a topic of cur- rent interest$^{26}$. In finite magnetic field, these lead to a half– magnetization plateau which can be tuned at will between dif- ferent forms of order$^{27}$. A similar fluctuation driven plateau, but with a uniquely defined form of order, is also expected to occur for the edge sharing tetrahedra of the FCC lattice$^{59}$. We now return to the model in question. FIG. 19: (Color online) Temperature dependence of (a) heat capacity defined by Eqs. (31), (b) and (c) the related measure of local corre- lation λ local T$_{2}$ and its susceptibility defined by Eqs. (10) and (11), and (d) and (e) the global order parameter λ global T$_{2}$ and its susceptibility de- fined by Eqs. (8) and (9) for b = 0 . 6 , h = 4 and a range of values of system size. The results are for J$_{3}$ = − 0 . 06 , showing a single first order transition into the ordered phase at T N = 0 . 70(1) (indicated by the vertical dashed line). B. Transition from plateau–liquid to ordered uuud state For h ≃ 4 , T ≲ b , Eq. (1) exhibits the plateau–liquid state described in Sec. III C [cf. Fig. 2(b)]. Inclusion of a FM third–neighbor interaction J$_{3}$ [Eq. (2)] causes it to or- der at low temperatures. We consider first the conventional limit where both | J$_{3}$ | and b are “large”, choosing parameters J$_{3}$ = − 0 . 06 and b = 0 . 6 . In this case there is strongly first or- der transition from paramagnet to four–sublattice plateaustate for T$_{N}$ ≈ 0 . 70 . This can be seen very clearly in simulation re- sults for the heat capacity and the order parameter λ global T$_{2}$ , and its susceptibility χ global T$_{2}$ , presented in Fig. 19. If we now de- crease | J$_{3}$ | , the transition temperature T$_{N}$ must also decrease, and for sufficiently small | J$_{3}$ | it will become smaller than the crossover temperature T ∗ ≈ b associated with the plateau liq- uid. In this case, there are anomalies in thermodynamic quanti- ties at two distinct temperatures as demonstrated in Fig. 20. There is a broad maximum in χ local T$_{2}$ at T ∗ ≈ 0 . 6 [Fig. 20(c)], signaling the onset of the plateau liquid state, accompanied by a broad peak in the heat capacity C$_{v}$ at a slightly lower temper-