3 a given magnetic field B . This special form takes advan- tage of the simplifications at the limit of large B , i.e., when the relevant Hilbert space can be restricted to the lowest Landau level, given that ¯ hω$_{0}$ << ¯ hω$_{c}$/ 2; the fre- quency ω$_{0}$ specifies the external harmonic confinement and ω$_{c}$ = eB/ ( mc ) is the cyclotron frequency. Then the many-body hamiltonian reduces to where L = ∑ N i $_{=1}$l$_{i}$ is the total angular momentum and z = x + iy . In the case of N rapidly rotating bosons (with ω$_{0}$ spec- ifying the external confinement of the two-dimensional harmonic trap and Ω denoting the rotational frequency), the corresponding Hamiltonian [24, 25, 29, 45] (in the limit Ω /ω$_{0}$ → 1) is written as [46] H e,global LLL = N ¯ hω$_{c}$ 2 + ¯ h ( √ ω 2 0 + ω 2 $_{c}$/ 4 − ω$_{c}$/ 2) L + N ∑ i<j e 2 κ | z$_{i}$ − z$_{j}$ | , (1) H b,global LLL = N ¯ hω$_{0}$ + ¯ h ( ω$_{0}$ − Ω) L + N ∑ i<j gδ ( z$_{i}$ − z$_{j}$ ) . (2) H$_{LLL}$ = N ∑ i<j v ( z$_{i}$ − z$_{j}$ ) , (3) Since we will consider many-body energy eigenstates that are eigenstates of the total angular momentum as well, it follows that only the interaction terms are non- trivial in both Hamiltonians (1) and (2). As a result, we will henceforth follow the practice of focussing on the simpler interaction-only LLL Hamiltonian where v ( z$_{i}$ − z$_{j}$ ) denotes the two-body interaction (Coulomb for electrons and repulsive contact potential for bosons). The “ground states” of the Hamiltonian in Eq. (3) coincide with the “yrast” band [36]. We remind the reader that the EXD method is based on the fact that the full LLL Hilbert space at a given total angular momentum L is spanned by the set of all possible uncorrelated permanents (for bosons) or Slater determinants (for electrons) made out from the Darwin- Fock zero-node single-particle levels (referred to also as orbitals) to a description in terms of an excited rotating/vibrating quantal molecule. Specific examples of the reducibility of the JL and Moore-Read states to the molecular descrip- tion introduced in this paper are provided in Sections III C and III D. This is a surprising result, since these Jastrow based trial functions are widely described in the previous literature as being liquid-like in an essential way. FIG. 2: (Color online) LLL spectra for N = 4 spin-polarized electrons calculated using exact diagonalization. Only the Hamiltonian term containing the two-body Coulomb interac- tion, e $^{2}$/ ( κ | z i − z j | ), [see Eq. (3)] was considered in the ex- act diagonalization. The gray solid dots (marked by arrows; green online) denote the translationally invariant states. The dark solid dots are the spurious states (see text). The dashed line denotes the yrast band, while the cusp states are marked by a circle. Energies in units of e $^{2}$/ ( κl B ). The number of translationally invariant states is much smaller than the total number of LLL states.  1.5  1.7  1.9  2.1  2.3  6  8  10  12  14 N=4 f er mions L Energy B. Characteristic properties of the lowest-Landau-level spectra ψ$_{l}$$_{i}$ ( z ) = z l$_{i}$ √$_{πl}$$_{i}$$_{!}$ exp( − zz $^{∗}$/ 2) , (4) with l$_{i}$ ≥ 0. The position variable z is given in units of Λ = √ ¯ h/ ( mω$_{0}$ ) in the case of a rotating harmonic trap (with lateral confinement frequency ω$_{0}$ ) or $_{l$_{B}$}$√$_{2 in the}$ case of an applied magnetic field B , with l$_{B}$ = √ ¯ h/ ( mω$_{c}$ ) being the magnetic length and ω$_{c}$ the cyclotron frequency [46]. (For details concerning the EXD method, see, e.g., Refs. [15, 29].) In the following, we use the convention For completeness and clarity in the presentation, we briefly provide in this section a graphical illustration of some of the main characteristics of the LLL spectra, cal- culated via the exact-diagonalization approach. First we describe here the special form [11, 15, 30] of the many-body Hamiltonian used for calculating the global ground state of a finite number N of electrons at