13 P (a.u.) 0 50 100 150 200 250 N$_{flip}$ -4.1988 -4.1986 -4.1984 -4.1982 -4.1980 s f 0 300 600 P (a.u.) 16 subl. 4 subl. b=0.6 N=1024 (a) (b) (c) TABLE II: Fluctuation entropy per site calculated for randomly gen- erated uuud states in an N = 1024 cluster. Here s min f , s max f , and 〈 s$_{f}$ 〉 are the lowest, highest, and mean value of the entropy, respectively, and Δ s$_{f}$ = s max f − 〈 s$_{f}$ 〉 measures the deviation of the highest value of entropy from the mean. Statistical errors on all numbers are less than 10 − $^{6}$. form b s min f s max f 〈 s$_{f}$ 〉 Δ s$_{f}$ 0.05 -0.79931 -0.79608 -0.79837 0.00228 0.1 -1.63451 -1.63191 -1.63374 0.00183 0.2 -2.54944 -2.54758 -2.54888 0.00130 0.3 -3.12864 -3.12725 -3.12823 0.00098 0.4 -3.56055 -3.55948 -3.56025 0.00077 0.5 -3.90757 -3.90672 -3.90734 0.00062 0.6 -4.19874 -4.19806 -4.19857 0.00051 F N = E$_{0}$ N − T ln T + T 2 N 〈 ln det M〉 Ω 0 − T N ln Ω$_{0}$ + O ( T $^{2}$) , (38) FIG. 15: (Color online) (a) Probability distribution of the flippable hexagons within the dimer manifold. (b) The fluctuation entr opy per site s$_{f}$ [Eq. (40)] as a function of the number of “flippable” hexagons . The lower bound s$_{f}$ = − 4 . 19874 is set by the four–sublattice state shown in Fig. 11(b), which has no flippable hexagons. The uppe r bound s$_{f}$ = − 4 . 19806 is set by the 16–sublattice state with the maximum number of flippable hexagons [see Fig. 11(c)]. The bl ue dots represent a sample of 10000 random configurations. (c) P rob- ability distribution of the fluctuation entropy per site s$_{f}$ within the dimer manifold. All results are for a cluster of N = 1024 sites with b = 0 . 6 . 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 M takes 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 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. 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. 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. 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 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 S$_{0}$ N ≈ ln 1 . 14 ≈ 0 . 13 . (39) s$_{f}$ = − ln det M 2 N (40)