In this paper we study an antiferromagnetic Heisenberg $S=2\mathrm{}$ quantum spin chain at $T=0\mathrm{}$ with both interaction and on-site anisotropy, $H={\sum }_i\frac{1}{2}{(S}_i^{+}{S}_{i+1\mathrm{}}^{-}{+S}_i^{-}{S}_{i+1\mathrm{}}^{+}{)+J}^z{S}_i^z{S}_{i+1\mathrm{}}^z{+D(S}_i^z{)}^2$. While the phase diagram of this chain for $S=1\mathrm{}$ and the isotropic point for $S=2\mathrm{}$ are by now excellently understood, contradictory scenarios exist for the $S=2\mathrm{}$ anisotropic phase diagram, which imply completely different mechanisms of the emergence of the classical $\overrightarrow{S}\infty$ limit, hence the importance of the question. One main scenario predicts the emergence of a cascade of phase transitions not seen in the $S=1\mathrm{}$ case, based on an analysis using the Affleck-Kennedy-Lieb-Tasaki (AKLT) model. Another scenario is in favor of an almost classical phase diagram for $S=2\mathrm{}$, from which the $S=\infty$ phase diagram evolves quite trivially; the $S=1\mathrm{}$ case then emerges as very special with its dominant quantum effects. Numerical studies have so far not been conclusive. Using the density matrix renormalization group, we study gaps and correlation functions in the anisotropic $S=2\mathrm{}$ chain. The question of the existence of hidden topological order in the anisotropic $S=2\mathrm{}$ chain is emphasized, as it distinguishes between the proposed scenarios. Careful extrapolation allows us to show that the topological order is zero in the thermodynamical limit in all disordered phases, in particular in the new phase interposed between the Haldane and large-$D$ phases. The decay of the hidden order is substantially faster than that of $S^{+}{S}^{-}$ correlations and excellently fitted by power laws typical of massless phases. This excludes the AKLT-model-based scenario in favor of an almost classical phase diagram for the $S=2\mathrm{}$ and thus higher spin chains.

• Received 17 February 1998

DOI:https://doi.org/10.1103/PhysRevB.58.359