Ground state of the one-dimensional antiferromagnetic Heisenberg model

We have calculated the two-point correlation functions ωᵢₗ(N)=(4/3)〈S→ᵢ{·}S→ᵢ₊ₗ 〉 and their averages over i,ωₗ (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 ωₗ({∞}) under reasonable assumptions. In addition to showing the rather strong even-l--odd-l alternation in {}ωₗ(N){}, known from earlier results of Bonner and Fisher for rings with N up to 10, our bounds indicate a smooth behavior in l{}ωₗ({∞}){} 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{}ωₗ(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 ωₗ({∞}) and in SN, the structure factor at wave vector {π}, then finite-size contributions to ωₗ(N) will have to contribute to the (seemingly) logarithmic divergence of SN as N{→}{∞}. We are not aware of any models where a similarly weak divergence shows such a finite-size contribution.