content_image/397782.jpg content_image/397787.jpg content_image/397781.jpg content_image/397792.jpg content_image/397788.jpg content_image/397797.jpg content_image/397798.jpg
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 

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 

<img src='content_image/397782.jpg'>

<img src='content_image/397787.jpg'>

as (within a proportionality constant) 

<img src='content_image/397781.jpg'>

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 

<img src='content_image/397792.jpg'>

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 

which is independent of the individual l$_{j}$ ’s. 

The final result for the (0 , N ) RBM wave function is (within a proportionality constant): 

<img src='content_image/397788.jpg'>

<img src='content_image/397797.jpg'>

where the coefficients are given by different expressions for even or odd number of bosons N . 

(i) For even N , one has 

<img src='content_image/397798.jpg'>

where the summation ∑ σ runs over the permutations of the set of N indices { 1 , 2 , . . . , N } . 

(ii) With the notation K = N − 1, the corresponding coefficients for odd N are: 

