17 [43] Note that the CF basis and corresponding CF diagonal- ization have been formulated only for the case of electrons in the LLL. To date, the application of the CF methodol- ogy to LLL bosons (see. e.g., Refs. [23, 31]) is limited to the simpler mean-field picture (see p. L276 of Ref. [42]), which omits residual two-body interactions. [44] The lack of translational invariance in the CF basis can be seen from the fact that for certain pairs, ( L , L +1), of adjacent angular momenta the CF diagonalization yields states with the same energy. In this case, the L + 1 CF state is trivially a center-of-mass excitation of the L CF state. For N = 4 e , such pairs of degenerate CF states are endemic in TABLE I of Ref. [19]; for example, L = 10 and L = 11 have the same V CF energy of 1.7849 e $^{2}$/κl B , L = 14 and L = 15 have the same V CF energy of 1.501 e $^{2}$/κl B , etc... Similar ( L , L + 1) pairs of degenerate CF states are also prominent in the CF yrast energies in all other cases with N > 4 presented in Ref. [19]. On the contrary, such pairs are missing from TABLE II in this paper, which summarizes the energies according to the RVM diagonalization. [45] T.L. Ho, Phys. Rev. Lett. 87 , 060403 (2001). [46] We note the correspondence ω c → 2Ω (see also Ref. [29]). [47] In the quantum Hall effect literature, the cusp states are commonly described as incompressible, since excitation from each of the corresponding global ground states to the corresponding lowest excited state entails an energy gap due to the enhanced stability. [48] J.K. Jain and T. Kawamura, Europhys. Lett. 29 , 321 (1995). [49] Although translationally invariant, the Hamiltonian in Eq. (3) does not separate the center-of-mass coordinate z c from the internal ones. Such a separation is always pos- sible if the external potential confinement is harmonic, and this extends also in the presence of a magnetic field (or fast rotation) even when the external potential con- finement can be neglected [case of the Hamiltonian in Eq. (3)]. For the case of N = 3 e electrons in the LLL, this separation has been carried out explicitly in Ref. [60] via the introduction of appropriate Jacobi coordinates [see Eqs. (43), (44), and (45)]. In this case, only the TI spec- trum was involved (consisting of the eigenenergies of the internal Hamiltonian). Due to this separability, the gen- eral wave-function solution of the Hamiltonian in Eq. (3) can be written as where z J i , i = 2 , . . . , N denote the internal Jacobi co- ordinates. Multiplication of this expression by z c raises the value of the center-of-mass angular momentum L c by unity (from L c to L c + 1), without modifying the energy of the original state due to the absence of an external confining potential. This accounts for the spurious de- generate energies (dark solid dots) in Figs. 1 and 2. [50] For bosons, Ref. [25] studied some aspects of the Q$_{λ}$ ex- citations in the range 0 ≤ L ≤ N . [51] E.W. Weisstein, Permanent . From MathWorld − http://mathworld.wolfram.com/Permanent.html. [52] For another mathematically different (but equivalent) definition of a normalized permanent, see Eq. (A1) in Ref. [29]. [53] K. Maki and X. Zotos, Phys. Rev. B 28 , 4349 (1983). [54] Yu.E. Lozovik, Usp. Fiz. Nauk 153 , 356 (1987) [Sov. Phys. Usp. 30 , 912 (1987)]. [55] V.M. Bedanov and F.M. Peeters, Phys. Rev. B 49 , 2667 (1994); M. Kong, B. Partoens, and F.M. Peeters, Phys. Rev. E 65 046602 (2002). [56] Y. Li, C. Yannouleas, and U. Landman, Phys. Rev. B 73 , 075301 (2006). [57] Most of the algebraic manipulations reported here were performed using MATHEMATICA; see S. Wolfram, Mathematica: A System for Doing Mathematics by Com- puter (Addison-Wesley, Reading, MA, 1991). [58] E.W. Weisstein, Permutation. From MathWorld − http://mathworld.wolfram.com/Permutation.html. See also http://en.wikipedia.org/wiki/Permutation (in Wikipedia ). [60] R.B. Laughlin, Phys. Rev. B 27 , 3383 (1983). [59] R.A. Smith and N.K. Wilkin, Phys. Rev. A 62 , 061602(R) (2000). [61] Numbers shown in parentheses are statistical uncertain- ties arising from Monte Carlo sampling. [62] F. Dalfovo, S. Giorgini, L. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71 , 463 (1999); [63] G. Baym, J. Low Temp. Phys. 138 , 601 (2005). [64] A.L. Fetter, Rev. Mod. Phys. 81 , 647 (2009). [65] I. Romanovsky, C. Yannouleas, and U. Landman, Phys. Rev. Lett. 93 , 230405 (2004). [66] In the case of two-dimensional electrons under high mag- netic field (in the LLL), it is known that many-body cor- related behavior (fractional quantum Hall effect) main- tains (and thus requires theoretical treatment beyond the mean field) as N → ∞ . In this context, the question of a transition of LLL bosons to mean-field GP behavior for large N is a challenging open topic of current re- search. Among some of the possible scenarios, we note the “heavy-center-of-mass” proposal for the yrast states with low angular momenta (0 ≤ L ≤ N ) in Sec. 2.3.1 of Ref. [31] and the time-dependent vortex-nucleation- through-a-turbulent-phase suggestion in D. Dagnino, N. Barberan, M. Lewenstein, and J. Dalibard, Nat. Phys. 5 , 431 (2009). A recent mathematically rigorous study [E.H. Lieb, R. Seiringer, and J. Yngvason, Phys. Rev. A 79 , 063626 (2009)] concerning upper and lower variational bounds finds that the exact yrast energy associated with the bosonic global Hamiltonian [see Eq. (2)] coincides for large N and small L with that of the mean-field Gross- Pitaevskii approach under the condition g << Nω , where ω = ω 0 − Ω is the difference between the trapping fre- quency and the rotational frequency. For the interaction- only LLL Hamiltonian [see Eq. (3)], one effectively has ω = 0, and the condition g << Nω seems to pre- clude the emergence of GP behavior in any region of the interaction-only LLL spectra considered in this paper, except in the trivial case of a non-interacting gas (when g = 0). Further theoretical investigations are needed to clarify these issues. ∝ z L c c exp( − z ∗ $_{c}$z c / 2)Φ $^{internal}$[ z J $_{2}$, . . . , z J $_{N}$] ,