content_image/426355.jpg content_image/426354.jpg content_image/426352.jpg content_image/426356.jpg
flux is conserved by all cyclic exchanges of spins on loops of alternating u and d spins, motivating a loop expansion of the fluctuation contribution to the free energy 38–42 . The leading term in such an expansion counts the number N$_{flip}$ of six–site hexagonal rings in a “flippable” u – d u – d u – d configuration. 

In Fig. 15(b) we plot the fluctuation entropy s$_{f}$ as a func- tion of N$_{flip}$ . The highest (lowest) values are achieved for the 16–sublattice (four–sublattice) ordered states with the m ost (least) flippable hexagons. The fact that randomly generate d uuud states lie extremely close to the line connecting these two states suggests that loops of more than six sites contribute little to the fluctuation entropy. 

The remaining question is why the overall difference in fluctuation entropy between different uuud states is so small? We can express the fluctuation entropy in terms of the 2 N eigenvalues { ε$_{n}$ } of M as 

The eigenvalue spectrum { ε$_{n}$ } associated with the simplest q = 0 four–sublattice uuud state can easily be calculated an- alytically; working with a four–site unit cell, there are fo ur bands, one of which is nondispersive. The associated density of states (DOS) for b = 0 . 6 is shown in Fig. 16(a), where the flat band appears as a sharp peak at ε = 16 b + h − 4 . In Figs. 16(b)–(d) we compare the integrated DOS from these four bands with numerical results for the integrated DOS of a 1024–site cluster. 

The integrated DOS, averaged within the dimer manifold of uuud states [Fig. 16(d)], is indistinguishable by eye from that of the four–sublattice state [Fig. 16(b)]. The step associa ted with the flat band survives as a set of N/ 4 localized excitations at ε = 16 b . And, critically, the gap 

<img src='content_image/426355.jpg'>

to the lowest lying excitation is set by a nodeless eigenvec- tor, whose components δS α i depend only on whether the spin S$_{i}$ points up or down. All uuud states can be made formally equivalent to four–sublattice order by renumbering the sit es in each individual tetrahedron, and the energy of this nodeless excitation is also unchanged by this renumbering of sites. It is therefore completely insensitive to whether or not the system is ordered. From the results it is clear why the thermody- namic properties of the plateau liquid state, and in particular the entropy associated with fluctuations about it, are so clo se to those of the ordered plateau state. 

From these results, it is also possible to understand why the numerically determined entropy gain Δ s$_{f}$ increases as b → 0 for h = 4 (cf. Table II). This singular behavior can be traced back to a band of excitations above the spin-wave gap Δ ≈ 4 b , with bandwidth Δ ε ∼ b , which collapses to become a strict set of zero modes for b → 0 . Since zero modes are excluded from the sum which determines Δ s$_{f}$ , while the collapsing band con- tributes as ∼ ln b , b acts as a singular perturbation, and in- finitesimal b may drive the system to order. This is despite the fact that it is disordered for b = 0 , and for the relatively large b used in our simulations. 

<img src='content_image/426354.jpg'>

<img src='content_image/426352.jpg'>

## D. vector–multipole phase with local T$_{2}$ symmetry 

At the upper critical field of the magnetization plateau, the collinear spins of the uuud configurations cant away from the z axis. This instability occurs at the level of a single tetrahe- dron (Fig. 3), where it is continuous. On a lattice, it is associ- ated with the closing of the gap Δ [Eq. (42)] in the excitation spectrum of the plateau liquid. Because of the special struc- ture of this excitation, discussed above, the gap closes at the same value of h$_{c}$ = 4 + 8 b for all uuud states, and the transi- tion is once again continuous — at least for T = 0 . However, since the spin configurations in question are simply 3:1 cant ed versions of the uuud states, with local T$_{2}$ symmetry, all of the entropic arguments presented above for the plateau liquid still hold. Thermal fluctuations alone cannot restore (canted) N´ eel order, and spin–spin correlations exhibit a power–law deca y of 1 /r 3 for T → 0 . The resulting state does however exhibit long range order 

in both the rank–two tensor order parameters Q ⊥ , 1 and Q ⊥ , 2 [Eqs. (22) and (23), and Table I]. The 3:1 canting of the uuud spins selects a direction in the xy plane, and the primary order parameter is therefore the lower–symmetry irrep, Q ⊥ , $^{1}$. The finite value of the nematic order parameter Q ⊥ , 2 reflects the fact that this canting is coplanar. Since Q ⊥ , 1 transforms like a vector under rotations about the z axis, we classify this state as a vector–multipole phase with local T$_{2}$ symmetry. 

Within the framework of a Ginzburg–Landau theory, the contribution to the free energy from this pair of order param- eters is 

<img src='content_image/426356.jpg'>

