6 as (within a proportionality constant) In Eq. (18), the common factor | 0 〉 represents the prod- uct of Gaussians defined in Eq. (6). To simplify the no- tation, this common factor is usually omitted. (II) Second step: In the following, we will continue with the details of the complete derivation in the simplest case of a single (0 , N ) ring. Thus we consider the special case Φ N $_{L}$[ z ] = l$_{1}$ + ··· + l$_{N}$ = L ∑ l$_{1}$,...,l$_{N}$ P ( l$_{1}$, . . . , l$_{N}$ ) l$_{1}$ ! . . . l$_{N}$ ! e i ( φ$_{1}$l$_{1}$ + ··· + φ$_{N}$l$_{N}$ $^{)}$, (21) Z$_{j}$ = Re 2 πi (1 − j ) $^{/N}$, 1 ≤ j ≤ N, (19) where R is the radius of the single ring. The Slater permanent Ψ $^{N}$[ z ] breaks the rotational symmetry and thus it is not an eigenstate of the total angular momentum ¯ h ˆ L = ¯ h ∑ N j $_{=1}$ˆ l$_{j}$ . However, one can restore [12, 15] the rotational symmetry by applying onto Ψ $^{N}$[ z ] the projection operator where ¯ hL are the eigenvalues of the total angular mo- mentum. When applied onto Ψ $^{N}$[ z ], the projection operator P L acts as a Kronecker delta: from the unrestricted sum in Eq. (18) it picks up only those terms having a given total angular momentum L (henceforth we drop the constant prefactor ¯ h when referring to angular momenta). As a re- sult the projected wave function Φ N L = P $_{L}$Ψ N is written P L ≡ 1 2 π ∫ 2 π 0 dγe iγ ( ˆ L − L $^{)}$, (20) e iπ ( N − 1) $^{L/N}$, (23) which is independent of the individual l$_{j}$ ’s. The final result for the (0 , N ) RBM wave function is (within a proportionality constant): The second permanent in Eq. (22) can be shown [57] to be equal (within a proportionality constant) to a sum of cosine terms times a phase factor Φ N $_{L}$[ z ] = l$_{1}$ + l$_{2}$ + ··· + l$_{N}$ = L ∑ 0 ≤ l$_{1}$ ≤ l$_{2}$ ≤···≤ l$_{N}$ P ( l$_{1}$, . . . , l$_{N}$ ) l$_{1}$ ! . . . l$_{N}$ ! × perm[ e iφ$_{1}$$^{l$_{1}$}$, eiφ$_{2}$$^{l$_{2}$}$, . . . , eiφ$_{N}$ $^{l$_{N}$}$] p$_{1}$ ! p$_{2}$ ! . . . p$_{M}$ ! . (22) with φ$_{j}$ = 2 π ( j − 1) /N . We further observe that it is advantageous to rewrite Eq. (21) by restricting the summation to the ordered ar- rangements l$_{1}$ ≤ l$_{2}$ ≤ . . . ≤ l$_{N}$ , in which case we get where the coefficients are given by different expressions for even or odd number of bosons N . (i) For even N , one has Φ RBM L (0 , N )[ z ] = l$_{1}$ + l$_{2}$ + ... + l$_{N}$ = L ∑ 0 ≤ l$_{1}$ ≤ l$_{2}$ ≤ ... ≤ l$_{N}$ C$_{b}$ ( l$_{1}$, l$_{2}$, . . . , l$_{N}$ )perm[ z l$_{1}$ 1 , z l$_{2}$ 2 , . . . , z l$_{N}$ N ] , (24) C$_{b}$ ( l$_{1}$, l$_{2}$, . . . , l$_{N}$ ) = ( N ∏ i =1 l$_{i}$ ! ) − 1 ( M ∏ k =1 p$_{k}$ ! ) − 1 × ∑ σ cos { [( N − 1) l$_{σ}$$_{1}$ + ( N − 3) l$_{σ}$$_{2}$ + . . . + l$_{σ}$$_{(}$$_{N/}$$_{2)}$ − l$_{σ}$$_{(}$$_{N/}$$_{2+1)}$ − . . . − ( N − 3) l$_{σ}$$_{(}$$_{N}$$_{−}$$_{1)}$ − ( N − 1) l$_{σ}$$_{N}$ ] π N } , (25) (ii) With the notation K = N − 1, the corresponding coefficients for odd N are: where the summation ∑ σ runs over the permutations of the set of N indices { 1 , 2 , . . . , N } .