13 P ( z ; z 0 $_{1}$, z 0 $_{1}$, z 0 $_{3}$, z 0 $_{4}$) ∝ | Φ( z ; z 0 $_{1}$, z 0 $_{2}$, z 0 $_{3}$, z 0 $_{4}$) | $^{2}$, (52) functions for this EXD TI state, i.e., the quantity which gives the probability distribution of finding the fifth boson at a position z under the condition that the remaining four bosons are fixed at positions z 0 $_{j}$, j = 1 , 2 , 3 , 4. The N -body correlations exhibit an (1,4) con- figuration that corresponds to the dominant Φ RBM 8 (1 , 4) RVM component [component | 1 〉 , defined in Eq. (51)]. In particular, when the four fixed points are forming a complete square inscribed in a circle of radius r$_{0}$ = 1 . 3Λ [see Fig. 4(a)], the fifth boson is localized around the ori- gin. When three fixed points coincide with only three vertices of this square, with the fourth being at the ori- gin, the fifth boson is localized around the fourth vertex of the square, as expected from the classical (1,4) config- uration. In contrast to the above, we found that the Moore- Read state [see Eq. (50)] exhibits a drastically different behavior in its N -body correlation function [portrayed in Fig. 4(c)]. To analyze this behavior, we have expanded the Moore-Read state (for N = 5 bosons and L = 8) in the RVM basis of Eq. (51). As was the case with the Jastrow-Laughlin 1 / 3 function for N = 4 electrons, such an expansion is possible due to the fact that the MR state is translationally invariant. The corresponding expansion coefficients are listed in TABLE V. It is remarkable that one dominant coefficient (underlined) does appear, but it is associated with the RVM basis state Φ RBM 5 (0 , 5) Q$_{3}$ (denoted as | 3 〉 ), instead of the RVM state | 1 〉 that domi- nates the expansion of the EXD TI yrast state. This basis state | 3 〉 corresponds to a octupolar single-phonon vi- bration of a (0,5) polygonal configuration, and this is reflected in the N -body correlation plotted in Fig. 4(c). Indeed, with the four fixed points positioned at the ver- tices of a regular pentagon inscribed in a circle of radius r$_{0}$ = 1 . 02Λ, the probability of finding the remaining bo- son is concentrated in the neighborhood of the fifth ver- tex that completes the pentagon (denoted by X ), but in addition it exhibits a prominent node precisely at X . Fig. 4(d) contrasts the radial single-particle densities of the EXD TI and MR states. Naturally, the radial single- particle densities provide a reduced amount of informa- FIG. 4: (Color online) N -body correlation functions for N = 5 LLL bosons with total angular momentum L = 8. (a- b): The yrast EXD TI state (labeled [1]; see TABLE IV). (c): The Moore-Read state given by Eq. (50). (d): Correspond- ing radial single-particle densities for the yrast EXD TI state (solid line; red online) and the Moore-Read state (dashed line; blue online). Solid dots denote the four fixed points placed at r 0 exp[ jπi/ 2], j = 1 , 2 , 3 , 4 (with r 0 = 1 . 3Λ) in (a), (0 , 0) and r 0 exp[ jπi/ 2], j = 1 , 2 , 3 (with r 0 = 1 . 3Λ) in (b), and r 0 exp[2 jπi/ 5], j = 1 , 2 , 3 , 4 (with r 0 = 1 . 02Λ) in (c). Crosses denote the expected position of the fifth localized bo- son according to the classical (1,4) polygonal-ring configura- tion in (a-b) and the (0,5) single-ring configuration in (c). For the (1,4) configuration, there are two non-equivalent ways of choosing the four fixed points. In (c), a node, associated with an octupole vibration (see text), develops at the expected position of the fifth particle. In the radial single-particle den- sities in (d), note the maximum (local minimum) at the origin of the solid line (dashed line) in agreement with the underly- ing (1,4)[(0,5)] configuration. TABLE V: N = 5 LLL bosons with L = 8: Expansion co- efficients in the RVM basis [Eq. (51)] (labelled by the | i 〉 ’s). The 2nd column gives the yrast EXD TI state (labelled [1]; see TABLE IV). The 3rd column gives the RVM expansion coefficients of the corresponding Moore-Read expression [Eq. (50)]. However, for N = 5 and L = 8, we found (see TABLE IV) that the exact TI solutions are linear superpositions of the following five RVM states [involving both the (0,5), (1,4), and (2,3) configurations]: | 1 〉 = Φ RBM 8 (1 , 4) , | 2 〉 = Φ RBM 4 (1 , 4) Q 2 $_{2}$, 3 = Φ RBM 5 (0 , 5) Q$_{3}$, 4 = Φ RBM 0 (0 , 5) Q 4 $_{2}$, | 5 〉 = Φ RBM 6 (2 , 3) Q$_{2}$. (51) ( N − 1) $^{2}$/ 2 for odd N and L = N ( N − 2) / 2 for even N [24]. The expansion coefficients of the yrast EXD TI state (labelled [1]; see TABLE IV) in this RVM basis are listed in TABLE V. One sees that one component (labelled | 1 〉 , underlined) dominates this expansion. To further illustrate this, we display in Fig. 4 the N -body correlation RVM EXD TI [1] MOORE-READ | 1 〉 0.7879 -0.5159 | 2 〉 -0.1162 0.1502 | 3 〉 -0.6005 0.7999 | 4 〉 -0.0684 0.1873 | 5 〉 -0.0198 -0.1908 -2 -1  0  1  2 -2 -1  0  1  2 -2 -1  0  1  2 -2 -1  0  1  2 x x -2 -1  0  1  2 -2 -1  0  1  2 x  0  0.1  0.2  0.3  0.4  0  1  2  3  4 (a) (b) (c) (d) EXD MR N=5 bosons L=8