The even-odd effect in short antiferromagnetic Heisenberg chains

TL;DR

This study investigates the energy level patterns of short antiferromagnetic Heisenberg chains, revealing an odd-even effect influenced by boundary conditions and quantum fluctuations, which challenges existing theoretical models.

Contribution

It uncovers the limitations of traditional effective Hamiltonians and quantum edge-spin models in explaining the odd-even effect in short chains, emphasizing the role of spatial fluctuations.

Findings

Odd-membered chains follow the Landé pattern E(S) ∝ S(S+1).

Even-membered chains show deviations from this pattern.

Short chains exhibit behavior dominated by spatial fluctuations, not explained by long-chain theories.

Abstract

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) \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 < exp(\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.