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RVM trial functions. 

It is remarkable that the numerical results of the present theory were found in all tested cases to be amenable (if so desired) to an agreement within machine precision with exact-diagonalization (EXD) results, in- cluding energies, wave functions, and overlaps. This nu- merical behavior points toward a deeper mathematical finding, i.e., that the RVM trial functions for both statis- tics provide a complete and correlated basis (see below) that spans the translationally invariant (TI) subspace [5] of the LLL spectrum. An uncorrelated basis, with- out physical meaning, built out of products of elemen- tary symmetric polynomials is also known to span the (bosonic) TI subspace [38]. 

For the sake of clarity, we comment here on the use of the terms “correlated functions” and/or “correlated ba- sis.” Indeed, the exact many-body eigenstates are cus- tomarily called correlated when interactions play a domi- nant role. Consequently, a basis is called correlated when its members incorporate/anticipate effects of the strong two-body interaction a priori (before the explicit use of the two-body interaction in an exact diagonalization). In this respect, Jastrow-type basis wavefunctions (e.g., the Feenberg-Clark method of correlated-basis functions [39– 41] and/or the composite-fermion basis [18, 19, 42, 43]) are described as correlated, since the Jastrow factors in- corporate the effect of a strong two-body repulsion in keeping the interacting particles apart on the average. Our RVM basis is referred to as correlated since, in addi- tion to keeping the interacting particles away from each other, the RVM functions incorporate the strong-two- body-repulsion effect of particle localization in concentric polygonal rings and formation of Wigner molecules; this localization effect has been repeatedly demonstrated via EXD calculations in the past decade (see, e.g., the review in Ref. [15] and references therein). In this spirit, we de- scribe the basis of elementary symmetric polynomials as “uncorrelated,” since the elementary symmetric polyno- mials do not incorporate/anticipate this dominant effect of a strong two-body repulsion, i.e., that of keeping the interacting particles apart. 

We are unaware of any other strongly-correlated func- tions which span the TI subspace. Indeed, although the Jastrow-Laughlin function (used for describing yrast states) is translationally invariant, its quasi-hole and quasi-electron excitations are not [5]. Similarly, the com- pact composite-fermion trial functions are translationally invariant [30], but the CF excitations which are needed to complete the CF basis are not [18, 19, 42, 44]. The short- coming of the above well known correlated LLL theo- ries to satisfy fundamental symmetries of the many-body Hamiltonian represents an unsatisfactory state of affairs, and the present paper provides a remedy to this effect. In this context, we note that although the Moore-Read functions [7] are also translationally invariant, they ad- dress only certain specific LLL states and they do not form a basis spanning the TI subspace. 

Our introduction of a correlated basis that spans the 

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TI subspace is of importance in the following two ways: (1) From a practical (and calculational) viewpoint, one can perform controlled and systematic stepwise improve- ments of the original strongly-correlated variational wave function, e.g., the pure REM or RBM. (For detailed il- lustrative examples of the rapid-convergence properties of the RVM basis, see the Appendix.) This calculational viewpoint was also the motivation behind the introduc- tion of other correlated bases in many-body physics; see, e.g., the treatment of quantum liquids and nuclear mat- ter in Refs. [39–41] and the composite-fermion correlated basis in Refs. [18, 19, 42, 43]. (2) Conceptually, it guar- antees that the properties of the RVM functions, and in particular the molecular point-group symmetries, are irrevocably incorporated in the properties of the exact LLL wave functions. Furthermore, it follows that all other translationally invariant trial functions (e.g., the JL, compact CF, or Moore-Read functions), are reducible 

