11 TABLE II: LLL spectra of four spin-polarized electrons interacting via the Coulomb repulsion e $^{2}$/ ( κ | z i − z j | ) . Second column: Dimensions of the full EXD and the nonspurious TI (in parenthesis) spaces (the EXD space is spanned by uncorrelated deter- minants of Darwin-Fock orbitals). Last three columns: Energy eigenvalues [in units of e $^{2}$/ ( κl B )] from the diagonalization of the Coulomb interaction in the TI subspace spanned by the trial functions Φ REM 6+4 $_{k}$(0 , 4) Q m λ and Φ REM 6+3 $_{k}$(1 , 3) Q m λ (RVM digonal- ization). Third to sixth columns: the molecular configurations ( n 1 , n 2 ) and the quantum numbers k , λ and m are indicated within brackets. There is no nonspurious state with L = 7. The full EXD spectrum at a given L is constructed by including, in addition to the listed TI energy eigenvalues [ D $^{TI}$( L ) in number], all the energies associated with angular momenta smaller than L . An integer in square brackets indicates the energy ordering in the full EXD spectrum (including both spurious and TI states), with [1] denoting an yrast state. Eight decimal digits are displayed, but the energy eigenvalues from the RVM diagonalization agree with the corresponding EXD TI ones within machine precision. L D $^{EXD}$( D $^{TI}$) [( n 1 , n 2 ) { k, λ, m } ] Energy eigenvalues (RVM diag. or EXD $^{TI}$) 6 1(1) [(0,4) { 0, λ ,0 } ] 2.22725097[1] 8 2(1) [(0,4) { 0,2,1 } ] 2.09240211[1] 9 3(1) [(1,3) { 1, λ ,0 } ] 1.93480798[1] 10 5(2) [(0,4) { 1, λ ,0 } ] [(0,4) { 0,2,2 } ] 1.78508849[1] 1.97809256[3] 11 6(1) [(1,3) { 1,2,1 } ] 1.86157215[2] 12 9(3) [(0,4) { 1,2,1 } ] [(0,4) { 0,2,3 } ] [(1,3) { 2, λ ,0 } ] 1.68518201[1] 1.76757420[2] 1.88068652[5] 13 11(2) [(1,3) { 1,2,2 } ] [(0,4) { 1,3,1 } ] 1.64156849[1] 1.79962234[5] 14 15(4) [(0,4) { 2, λ ,0 } ] [(0,4) { 1,2,2 } ] [(0,4) { 0,2,4 } ] 1.50065835[1] 1.63572496[2] 1.72910626[5] [(1,3) { 2,2,1 } ] 1.79894008[8] 15 18(3) [(1,3) { 3, λ ,0 } ] [(1,3) { 2,3,1 } ] [(1,3) { 1,3,2 } ] 1.52704695[2] 1.62342533[3] 1.74810279[8] 18 34(7) [(0,4) { 3, λ ,0 } ] [(0,4) { 2,2,2 } ] [(0,4) { 1,2,4 } ] 1.30572905[1] 1.41507954[2] 1.43427543[4] [(0,4) { 0,2,6 } ] [(1,3) { 4, λ ,0 } ] [(1,3) { 2,2,3 } ] 1.50366728[8] 1.56527615[11] 1.63564655[15] [(1,3) { 3,3,1 } ] 1.68994048[20] The expansion coefficients of the three lowest-in-energy EXD TI states (labelled [1], [2], [4]; see TABLE II) in this RVM basis are listed in TABLE III. One sees that for each case, one component (underlined) dominates this expansion; this applies for both the yrast state (No. [1]) and the two excitations (Nos. [2] and [4]). To further illustrate this, we display in Fig. 3 the conditional prob- ability (pair correlation) distributions (CPDs), relations in the EXD TI states. Full quantitative agree- ment (within machine precision) in total energies can be reached by taking into consideration all seven RVM ba- sis states [see Eq. (46)]. Naturally, a smaller number of RVM states yields intermediate degrees of high-quality quantitative agreement. The celebrated Jastrow-Laughlin ansatz [2], for these three EXD TI states (top row) and for the RVM functions (bottom row) corresponding to the dominant expansion coefficients. The similarity of the CPDs in each column is noticeable and demonstrates that the sin- gle RVM functions capture the essence of many-body cor- has been given exclusively an interpretation of a quantum-fluid state [2, 11]. However, since the RVM functions span the TI subspace, it follows that any TI trial function (including the JL ansatz above and the compact CF states) can be expanded in the RVM basis. As an example, we give in TABLE III (4th column) the RVM expansion of the JL state for N = 4 e and L = 18. One sees that, compared to the EXD yrast state (1st col- umn), the relative weight of the pure (0,4) REM (denoted by | 1 〉 ) is reduced, while the weights of higher-in-energy vibrational excitations are enhanced. In this context, the liquid characteristics are due to the stronger weight of molecular vibrations which diminish the rigidity of the molecule. Of great interest also is the L = 30 ( ν = 1 / 5) case, which in the composite-fermion picture was found to be susceptible to a competition [17] between crystalline and liquid orders. However, we found that the exact nonspu- rious states for L = 30 are actually linear superpositions TABLE III: N = 4 LLL electrons with L = 18: Expansion coefficients in the RVM basis (labelled by the | i 〉 ’s) for the three lowest-in-energy EXD TI states (labelled [1], [2], [4]; see TABLE II). The 4th column gives the RVM expansion coef- ficients of the corresponding Jastrow-Laughlin expression. RVM EXD TI [1] EXD TI [2] EXD TI [4] JL | 1 〉 0.9294 -0.3430 0.0903 0.8403 | 2 〉 -0.1188 -0.0693 0.8930 -0.1086 | 3 〉 0.0067 0.0382 -0.2596 0.0076 | 4 〉 0.0137 0.0191 -0.0968 0.0395 | 5 〉 0.2540 0.8486 0.1519 0.4029 | 6 〉 0.0211 0.0283 0.3097 0.0616 | 7 〉 -0.2387 -0.3935 0.0877 -0.3380 Φ $^{JL}$[ z ] = ∏ 1 ≤ i<j ≤ N ( z$_{i}$ − z$_{j}$ ) 2 p $^{+1}$, (48) P ( z, z$_{0}$ ) = 〈 Φ | ∑ i = j δ ( z$_{i}$ − z$_{0}$ ) δ ( z$_{j}$ − z$_{0}$ ) | Φ 〉 , (47)