The importance of being odd

In this Letter I mainly consider a finite XXZ spin chain with periodic boundary conditions and an odd number of sites. This system is described by the Hamiltonian H-xxz = - Sigma (N)(j=1){sigma (x)(j)sigma (x)(j+1)+sigma (y)(j)sigma (y)(j+1) + Delta sigma (z)(j)sigma (z)(j+1)}. As it turns out, the ground state energy is proportional to the number of sites E = -3N/2 for a special value of the asymmetry parameter Delta = -1/2. The trigonometric polynomial Q(u), the zeros of which are parameters of the ground state Bethe eigenvector, Is explicitly constructed. This polynomial of degree n = (N - 1)/2 satisfies the Baxter T-Q equation. Using the second independent solution of this equation that corresponds to the same eigenvalue of the transfer matrix, it is possible to find a derivative of the ground state energy w.r.t. the asymmetry parameter. This derivative is closely connected with the correlation function [sigma (z)(j)sigma (z)(j+1)] = -1/2 + 3/2N(2). This correlation function is related to the average number of spin strings for the ground state [N-string] = 3/4(N - 1/N). I would like to stress that all the above simple formulae are not applicable to the case of an even number of sites which is usually considered.