Ground state of the one-dimensional antiferromagnetic Heisenberg model

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