We have investigated Haldane's conjecture for the S = 2 antiferromagnetic quantum spin chain with nearest-neighbour exchange J . Using a density matrix renormalization group algorithm for chains up to L = 350 spins, we show that the ground state has hidden topological order: the physics of the S = 2 chain can be captured by a valence bond solid description. We also observe effective free spin-1 states at the ends of an open S = 2 chain. We find in the thermodynamic limit a gap Δ = 0.085(5) J and a spin correlation length ξ = 49(1) lattice spacings in the isotropic case. We give the phase diagram in the presence of single-ion and exchange anisotropy, showing how the Haldane phase is squeezed out when the spin S increases.
In this paper we study an antiferromagnetic Heisenberg
S=2 quantum spin chain at
T=0 with both interaction and on-site anisotropy, $H={\ensuremath{\sum}}_{i}\frac{1}{2}{(S}_{i}^{+}{S}_{i+1}^{\ensuremath{-}}{+S}_{i}^{\ensuremath{-}}{S}_{i+1}^{+}{)+J}^{z}{S}_{i}^{z}{S}_{i+1}^{z}{+D(S}_{i}^{z}{)}^{2}$. While the phase diagram of this chain for
S=1 and the isotropic point for
S=2 are by now excellently understood, contradictory scenarios exist for the
S=2 anisotropic phase diagram, which imply completely different mechanisms of the emergence of the classical $\stackrel{\ensuremath{\rightarrow}}{S}\ensuremath{\infty}$ 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 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, from which the $S=\ensuremath{\infty}$ phase diagram evolves quite trivially; the
S=1 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 chain. The question of the existence of hidden topological order in the anisotropic
S=2 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 ${S}^{+}{S}^{\ensuremath{-}}$ 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 and thus higher spin chains.
The antiferromagnetic Heisenberg spin chain with integer spin has a short-range magnetic order and an excitation energy gap above the ground state. This so-called Haldane gap is proportional to the exchange coupling constant J of the Heisenberg chain. In this study, we discuss recent results about the spin dependence of the Haldane gap and conjecture an analytical formula valid asymptotically for large spin values. We then discuss the robustness of the edge states of the spin-1 chain by studying a spin-1 ladder using the density matrix renormalization group (DMRG) algorithm. We show that the peculiar hidden topological order of the spin-1 chain disappears smoothly by increasing the ladder rung coupling without any intervening phase transition. This is evidence for the fragile character of the topological order of the spin-1 chain.
Thermodynamic properties of S=1 antiferromagnetic Heisenberg chains with free and periodic boundaries are investigated by a quantum Monte Carlo method. In particular, temperature dependences of the specific heat, the magnetic susceptibility, and the hidden order parameter, which are inherent to the Haldane phase, are investigated. The specific heat turns out to have a peak at a temperature
Tpeak\ensuremath{\sim}2\ensuremath{\Delta}, where \ensuremath{\Delta} is the energy gap of the present model, although the temperature dependence of the specific heat is very similar to the Schottky type. In open chains, due to the fourfold degeneracy of the ground state, the magnetic susceptibility shows a Curie-like divergence regardless of the number of spins in the chain. The amplitude of the Curie-like divergence is consistent with the edge moments (S=1/2). On the other hand, it is confirmed that the long-range hidden order exists only in the ground state. How the hidden order grows as the temperature goes to zero (T\ensuremath{\rightarrow}0) is investigated by introducing an intrinsic correlation length ${\ensuremath{\xi}}_{\mathit{D}}$ of the ordering, which diverges as T\ensuremath{\rightarrow}0. A qualitative difference in fluctuations between the ground state and the finite-temperature states is discussed making use of snapshots of the transformed two-dimensional Ising system on which a Monte Carlo simulation is being performed.
We study a Heisenberg antiferromagnetic chain of spin S=1 with nearest-neighbor exchange interaction J and easy-plane single-ion anisotropy D. We follow the splitting of the Haldane gap into two components as a function of the anisotropy. Chains of length up to 16 sites are studied by a Lancz\"os method and longer chains up to 32 sites have been investigated by a quantum Monte Carlo simulation for D/J=0.10, 0.15, 0.20, and 0.25. A careful finite-size scaling treatment of our data leads to values that are directly relevant to the interpretation of experiments on Ni(
C2H8N2)2NO2ClO4 (NENP). We discuss the relationship between our results and the magnetic-susceptibility measurements as well as the inelastic-neutron-scattering data and on the Haldane conjecture. Values are proposed for the parameters D and J in the case of NENP.
The phase diagram of antiferromagnetic spin-
S chain with
XY-type anisotropy and frustrating next-nearest-neighbor interaction is studied in the limit of large integer S with the help of a field-theoretical approach. It is shown that the existence of gapless and gapped chiral phases found in recent numerical studies [M. Kaburagi et al., J. Phys. Soc. Jpn. 68, 3185 (1999); T. Hikihara et al., J. Phys. Soc. Jpn. 69, 259 (2000)] is not specific for
S=1, but is rather a generic large-
S feature. Estimates for the corresponding transition boundaries are obtained, and a sketch of the typical phase diagram is presented. It is also shown that frustration stabilizes the Haldane phase against the variation of the anisotropy.