based on our results obtained from the density matrix renormalization group (DMRG) method [@White; @Uli] implemented for systems with periodic boundary conditions.

Directly from the computation of the reduced bilinear spin correlation function Eq. (\[totalbl\]) it is hardly possible to analyze the space dependence of its uniform part. The reason is that the reduced correlation function $G^{a,a}_{r}$, is strongly dominated by the leading term, its staggered part, which is expected to behave as $G^{a,a}_{s}(r)\sim  1/(r \ln^{\frac{3}{2}}r)$ [@Affleck+89], and such decay is much slower than that of the uniform part ($\sim \ln^2r/r^4$). We have performed numerical simulations of $G^{a,a}_{r}$ for different system sizes, ranging from $L=24$ (Lanczos) to $L=48, 96$ and $L=144$ sites (DMRG), assuming periodic boundary conditions. In Fig. 1 we present the behavior of the reduced correlation function of the bilinear spin operators for the Heisenberg spin-$\frac{1}{2}$ AFM chain with $L=96$ and $L=144$ sites. We use conformal mapping of an infinite 2-dimensional plane on a cylinder [@comment01] with finite circumference in the spatial direction to compare the analytic results for the thermodynamic limit with finite-size calculations for the systems with periodic boundary conditions. This implies that the distances are replaced by the chord distances on the circle, $\rho= L/\pi\sin(r\pi/L)$.

