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arXiv:cond-mat/9507098

A novel intermediate phase in S=2 antiferromagnetic chains with uniaxial anisotropy

Masaki Oshikawa, Masaru Yamanaka, Seiji Miyashita

Published Jul 24, 1995• 0 citations
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Abstract

We study ground state properties of the $S=2$ quantum antiferromagnetic chain with a uniaxially anisotropic Hamiltonian: $ H = \sum_{j} [S_{j} \cdot S_{j+1} + D (S^{z}_{j})^2 ] $ by a Monte Carlo calculation. While it has been reported that a string order parameter vanishes exponentially at the isotropic Heisenberg point ($D=0$), we found it tends to remain finite in the thermodynamic limit around $D=1.2$. This implies that the model for $S=2$ has a novel intermediate phase between the Haldane phase and the large-$D$ phase, as was anticipated from the earlier argument based on the Valence-Bond-Solid picture.

DOI
10.48550/arxiv.cond-mat/9507098

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We study ground state properties of the S=2S=2S=2 quantum antiferromagnetic chain with a uniaxially anisotropic Hamiltonian: H=∑j[Sj⋅Sj+1+D(Sjz)2]H = \sum_{j} [S_{j} \cdot S_{j+1} + D (S^{z}_{j})^2 ]H=∑j​[Sj​⋅Sj+1​+D(Sjz​)2] by a Monte Carlo calculation. While it has been reported that a string order parameter vanishes exponentially at the isotropic Heisenberg point (D=0D=0D=0), we found it tends to remain finite in the thermodynamic limit around D=1.2D=1.2D=1.2. This implies that the model for S=2S=2S=2 has a novel intermediate phase between the Haldane phase and the large-DDD phase, as was anticipated from the earlier argument based on the Valence-Bond-Solid picture.
The ground state of the ferromagnetic-antiferromagnetic alternating spin-1/2 Heisenberg chain is studied with Ising-type anisotropy on the ferromagnetic bond. This model tends to the spin-1 antiferromagnetic Heisenberg chain with on-site anisotropy in the limit of strong ferromagnetic bond. It is found that the Haldane-gap phase, the large-D phase, and the two types of N\'eel phases appear as the ground state of this model. The duality relations between these four phases are proved for several limiting cases. The string order parameter characterizing the large-D phase is introduced based on these duality relations. The ground-state phase diagram is determined by the numerical diagonalization of the finite-size system.
Ground-state properties of the spin-1 antiferromagnetic Heisenberg chain with bond alternation and uniaxial single-ion-type anisotropy, which is described by the Hamiltonian \({\cal H}_0 = \sum_{\ell=1}^N\{1-(-1)^\ell\alpha\}{\mib S}_\ell\cdot{\mib S}_{\ell+1} + D\sum_{\ell=1}^N\;(S_\ell^z)^2\) (0.0 ≤α≤1.0, -∞< D < ∞), are studied. An exact diagonalization method by means of the Lanczös technique is used. The ground-state phase diagram on the α versus D plane at the thermodynamic limit is determined by analyzing the Binder parameters associated with Néel order and z -direction string order for finite-size ( N = 6, 8, …, 16) chains. It is found that there exists a multicritical point at which the Haldane, Néel, and singlet-dimer phases meet and that the singlet-dimer phase and the large- D phase belong to the same unique phase. The ground-state magnetization curves at the thermodynamic limit for various values of α and D are obtained for the system described by the Hamiltonian, H 0 plus the Zeeman term - H ∑ ℓ=1 N S ℓ z . The results show that the 1/2-plateau appears in the magnetization curve at least when D ≥0.0 and 0.0≲α≤1.00.0\lesssim\alpha \leq 1.00.0≲α≤1.0.
We have investigated Haldane's conjecture for the S=2 antiferromagnetic quantum spin chain with nearest-neighbor exchange J. Using a density matrix renormalization group algorithm for chains up to L=350 spins, we find in the thermodynamic limit a finite gap of \ensuremath{\Delta}=0.085(5)J and a finite spin-spin correlation length \ensuremath{\xi}=49(1) lattice spacings. We have confirmed the gap value by a zero-temperature quantum Monte Carlo study. We show that the ground state has a hidden topological order that is revealed in a nonlocal string correlation function which saturates to a nonzero value in the thermodynamic limit. We investigate the behavior of the spin-2 chain under an easy-plane anisotropy D(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2, D\ensuremath{\gtrsim}0, and find that the Haldane and the large-D phase are separated by an XY phase. The string correlation function vanishes only in the D\ensuremath{\rightarrow}\ensuremath{\infty} limit and does not distinguish between the Haldane phase and the perturbative large-D phase. An analysis of the transition mechanism and of the S=2 phase diagram in the presence of easy-plane and exchange anisotropy, markedly different from the S=1 phase diagram, allow us to conjecture how the classical limit is reached from increasing integer spins. \textcopyright{} 1996 The American Physical Society.
We provide evidence of an intermediate Haldane phase in a spin-2 quantum chain. By combining effective field theory and numerical approaches, we show that the phase diagram of the proposed model includes SO(5) Haldane, intermediate Haldane, and large-DDD phases. We determine the characteristic properties of these phases, including edge states, string order parameters, and degeneracies in the entanglement spectrum. The symmetries responsible for the degeneracy patterns observed in the entanglement spectrum are also discussed.
In this paper we study an antiferromagnetic Heisenberg S=2S=2S=2 quantum spin chain at T=0T=0T=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=1S=1S=1 and the isotropic point for S=2S=2S=2 are by now excellently understood, contradictory scenarios exist for the S=2S=2S=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=1S=1S=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=2S=2S=2, from which the $S=\ensuremath{\infty}$ phase diagram evolves quite trivially; the S=1S=1S=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=2S=2S=2 chain. The question of the existence of hidden topological order in the anisotropic S=2S=2S=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-DDD 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=2S=2S=2 and thus higher spin chains.
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.