content_image/384687.jpg
more slowly as 

The results of this theory for the T$_{2}$ vector–multipole phase are shown by the dashed lines in Fig. 17. For the value of b used in the present study, the zero temperature transition from T$_{2}$ vector–multipole phase to paramagnet at high field is strongly first order, even at the level of a single-tetrahe dron theory — cf. Fig. 3 — and remains so throughout. 

The nature of the finite temperature transition from the vector–multipole phase into the paramagnet is harder to de- termine. However, as shown in Fig. 17, it appears to be first order for all h > h ∗ , where ( T ∗ , h ∗ ) ≈ (0 . 1 , 5 . 2) marks the point at which the crossover line T joins the boundary of the vector–multipole phase, T$_{V}$ , as shown in Fig. 2(b). On the basis of our results, we consider that there is a tricritical point at ( T ∗ , h ∗ ) where the nature of the phase transition into the T$_{2}$ vector–multipole phase changes from continuous to first order. 

In Fig. 18 we present MC simulation results for the finite– temperature transition into the vector–multipole phase fo r h = 6 . 0 > h ∗ . The primary order parameter Q ⊥ , 1 be- comes nonzero with a sharp jump at a transition temperature T$_{V}$ ≃ 0 . 042 [Fig. 18(a)]. Both heat capacity and local T$_{2}$ susceptibility show a jump at the transition T$_{V}$ [Figs. 18(b) and (d)], but no sign of long range order in the bond–order parameter given by Eq. (7) [Fig. 18(c)]. 

The finite temperature transition from the plateau liquid to T$_{2}$ vector–multipole phase for h < h ∗ deserves special atten- tion, since the plateau liquid exhibits algebraic decay of cor- relations for intermediate distances. Monte Carlo simulations suggest that the transition has a continuous character with (ap- proximately) mean–field exponents. We return to this below in Sec. IV D. 

Our results for the h – T phase diagram of the antiferromag- netic nearest–neighbor Heisenberg model with additional b i- quadratic interactions b [Eq. (1)] are summarized in Fig. 2(b). There are two ordered phases, a nematic phase with local E symmetry and a vector–multipole phase with local T$_{2}$ sym- metry, both of which break spin rotational symmetry about the direction of the magnetic field. These are separated by a plateau–liquid state with all the symmetries of a paramagne t in magnetic field. 

This phase diagram bears a very strong resemblance to that of the corresponding model with weak FM third–neighbor in- teraction J$_{3}$ = − 0 . 05 , which enforces four–sublattice order, as shown in Fig. 2(a) (cf. Ref. [15]). So far as the topology of the phase diagram is concerned the only change is the re- placement of a line of first order phase transitions terminat ing the four–sublattice plateau state (which breaks lattice sy mme- tries), by a crossover in the case of the plateau liquid (which does not). 

<img src='content_image/384687.jpg'>

## E. Global structure of the h – T phase diagram 

Throughout this paper, we have argued that preformed lo- cal order at the level of a single tetrahedron exists in all of these phases. Moreover, in the case of the half–magnetizati on plateau, we have seen in Sec. III C that conventional magnetic order has very little impact on the excitation spectrum, and therefore on the thermodynamic properties of the system. 

Viewed in this way, the correspondence between the two h – T phase diagrams is not at all surprising — the role of sec- ondary interactions like J$_{3}$ is merely to select between an in- finite set of different ordered ground states. Precisely how enforcement of long range order works at finite temperature i s a complex and very interesting question, to which we provide only a partial answer below. 

## IV. THERMAL TRANSITIONS BETWEEN DIFFERENT ORDERED AND DISORDERED STATES 

None of the phases described above possess conventional magnetic order of the form 〈 S$_{i}$ 〉 ̸ = 0 . However they all con- tain the seeds of such order in the form of well formed local orders λ local E and λ local T$_{2}$ . Long range order can easily be re- stored by adding additional terms to the Hamiltonian Eq. (1). The simplest possible choice is a FM third–neighbor interac - tion J$_{3}$ < 0 in Eq. (2), leading to four–sublattice order of the form considered in Refs. [2] and [15]. In what follows we study how FM | J$_{3}$ | ≪ b precipitates an ordered uuud state from the plateau liquid for h = 4 , and contrast this with the way in which N´eel order emerges from the nematic phase with local E symmetry for h = 0 . We also discuss the continuous transition from plateau liquid to vector–multipole phase f or h < h ∗ , T < T ∗ . 

We study these phase transitions as a function of tempera- ture T which also gives us access to the high temperature para- magnetic phase. This is interesting because, for intermediate distances r < ξ$_{c}$ ∼ √$_{T}$ the nematic phases exhibit the alge- braic decay of spin correlations characteristic of a Coulomb phase, rather than the exponential decay of correlations more usually associated with a paramagnet. Transitions between a disordered phase subject to an ice–rule type constraint anda phase with conventional order have been discussed for a long time in the context of hydrogen–bonded ferroelectrics 43–46 . More recently such questions have arisen again in the con- text of experiments of many highly frustrated magnets$^{47}$, and in the past few years there has been a theoretical effort to un- derstand how order can emerge from a Coulomb phase in clas- sical dimer$^{48–52}$and spin models$^{53,54}$. 

A strong motivation for this work has been the possibility of observing an unusual continuous phase transitions, includ- ing transitions lying outside the Landau–Ginzburg–Wilson paradigm$^{55}$. Indeed the (classical) dimer model on cubic lattice does exhibit a continuous transition from a Coulomb phase at high temperatures to a simple crystalline ordered phase as a function of temperature$^{48}$. This transition has unusual scaling properties$^{49}$, and does not naively admit a Landau–Ginzburg–Wilson description, since the high tempe r- 

## A. General context 

