content_image/401915.jpg content_image/401916.jpg content_image/401927.jpg content_image/401928.jpg content_image/401912.jpg
<img src='content_image/401915.jpg'>

form 

where E$_{0}$ is the ground state energy, Ω$_{0}$ its degeneracy, and 〈 ... 〉 Ω 0 the average over all degenerate ground states. For a generic ordered phase, Ω$_{0}$ is finite, and det takes 

M on the same value for all (symmetry related) ground states. In this case ln Ω$_{0}$ /N → 0 for N → ∞ . However for the dimer manifold, Ω$_{0}$ ≈ 1 . 14 $^{N}$, which means that the ground state has a finite entropy per site 

In this case, different ground states are not related by simple lattice symmetries and the fluctuation entropy per site 

takes on a range of values. 

We have studied the distribution of values of s$_{f}$ within the dimer manifold for a range of values of b , by numeri- cally calculating det M for 10000 randomly generated uuud states in a cluster of N = 1024 sites ( L = 4 ), using a Monte Carlo algorithm based on loop updates of spins. We found that the highest value of s$_{f}$ is achieved by an eight– fold degenerate, 16–sublattice “ R -state” $^{13}$, in which the four A–sublattice tetrahedra within the 16–site cubic unit cell of the pyrochlore lattice take on all four possible uuud config- urations [Fig. 11(c)]. This state has overall cubic symmetry, and is actually observed in the plateau phase of HgCd$_{2}$O$_{4}$$^{6}$. The lowest value of s$_{f}$ is achieved by the four–sublattice or- der shown in Fig. 11(b). The calculated values of the maxi- mum and minimum values s max f and s min f are listed in Table II together with the mean value 〈 s$_{f}$ 〉 and the difference between s max f and the mean 〈 s$_{f}$ 〉 , Δ s$_{f}$ . From these results it is immediately clear why thermal fluc- 

tuations alone fail to select a unique ground state for any value 

<img src='content_image/401916.jpg'>

<img src='content_image/401927.jpg'>

<img src='content_image/401928.jpg'>

<img src='content_image/401912.jpg'>

of b considered in this paper. The fluctuation entropy per site gained by choosing the cubic 16–sublattice state is miserly , for example, Δ s$_{f}$ = 0 . 00183 for b = 0 . 1 and Δ s$_{f}$ = 0 . 00051 for b = 0 . 6 . These numbers must be compared with the ex- tensive entropy S$_{0}$/N ≈ 0 . 13 of the liquid phase, all of which is lost if the system orders. So for the values of b considered here, thermal fluctuations cannot drive the system to order. 

However it is amusing to note that the entropy gain Δ s$_{f}$ in- creases as b decreases, scaling approximately as ln b , as shown in Table II. This raises the intriguing possibility that b acts as a singular perturbation, and that for sufficiently small b , fluctu- ations might overcome the extensive entropy S$_{0}$/N ≈ 0 . 13 of the dimer manifold, driving the system order — even though it is disordered for b = 0 . Such an order-from disorder ef- fect would presumably favor the cubic 16–sublattice R -state, which is also believed to be selected by quantum fluctuations at T = 0 $^{12,13,37}$. However in the present model, it would occur only for vanishingly low temperatures, and would therefore be extremely difficult to access in simulation. This questio n remains for future study. 

The result above explains why the system does not order at finite temperature, but not why the fluctuation entropy fa- vors the 16–sublattice state? We can answer this question by looking at the distribution of the fluctuation entropies s$_{f}$ within the dimer manifold of uuud states. Figure 15 shows the distribution for b = 0 . 6 . The uuud states can be broken up into classes of states with a different net flux of an effec- tive magnetic (or, equivalently, electric) field 12,13,29–31,37 . This 

