Even-odd effect in short antiferromagnetic Heisenberg chains

Motivated by recent experiments on chemically synthesized magnetic molecular chains, we investigate the lowest-lying energy band of short spin-$s$ antiferromagnetic Heisenberg chains focusing on effects of open boundaries. By numerical diagonalization we find that the Land\'e pattern in the energy levels, i.e., $E(S)\ensuremath{\propto}S(S+1)$ for total spin $S$, known from, e.g., ring-shaped nanomagnets, can be recovered in odd-membered chains, while strong deviations are found for the lowest excitations in chains with an even number of sites. This particular even-odd effect in the short Heisenberg chains cannot be explained by simple effective Hamiltonians and symmetry arguments. We go beyond these approaches, taking into account quantum fluctuations by means of a path-integral description and the valence bond basis, but the resulting quantum edge-spin picture which is known to work well for long chains does not agree with the numerical results for short chains and cannot explain the even-odd effect. Instead, by analyzing also the classical chain model, we show that spatial fluctuations dominate the physical behavior in short chains, with length $N\ensuremath{\lesssim}{e}^{\ensuremath{\pi}s}$, for any spin $s$. Such short chains are found to display a unique behavior, which is not related to the thermodynamic limit and cannot be described well by theories developed for this regime.