Ground state of the one-dimensional antiferromagnetic Heisenberg model

Abstract

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