Ground state of the one-dimensional antiferromagnetic Heisenberg model

We have calculated the two-point correlation functions ₈₋ (N) = (4/3) 〈S₈₈+₋ 〉 and their averages over i, ₋ (N), in the ground state of the one-dimensional antiferromagnetic Heisenberg model for N=4, 6, 8,. . . , 16 spins. Both periodic (rings) and free-end (chains) boundary conditions are considered. Surprisingly tight lower and upper bounds have been obtained for ₋ () under reasonable assumptions. In addition to showing the rather strong even-l--odd-l alternation in ₋ (N), known from earlier results of Bonner and Fisher for rings with N up to 10, our bounds indicate a smooth behavior in l₋ () for l odd and l even, with, surprisingly, a broad maximum attained within the odd-l values. The bounds obtained from the chain results were essential to seeing this maximum (because of the larger l values available for given N). The quantity l₋ (N) for chains with fixed N also shows such a maximum, and in addition shows a similar maximum for even l's. If the indicated trends for large l and N continue in ₋ () and in S₍, the structure factor at wave vector, then finite-size contributions to ₋ (N) will have to contribute to the (seemingly) logarithmic divergence of S₍ as N. We are not aware of any models where a similarly weak divergence shows such a finite-size contribution.