Physical review. B, Condensed matter

Critical exponent η in S =1 antiferromagnetic Heisenberg chains with alternating interaction

Shoji Yamamoto

Published Oct 1, 1995• 19 citations

Abstract

Low-temperature properties of S=1 antiferromagnetic Heisenberg chains with alternating interaction, scrH=${\mathit{Jtsum}}_{\mathit{i}}$[1-(-1${)}^{\mathit{i}}$\ensuremath{\delta}]${\mathrm{S}}_{\mathit{i}}$\ensuremath{\cdot}${\mathrm{S}}_{\mathit{i}+1}$, are studied by a quantum Monte Carlo method. Spatial distributions of the spin correlation function 〈${\mathit{S}}_{1}^{\mathit{z}}$${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$〉 and the local magnetic moment 〈${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$〉 are visualized as a function of \ensuremath{\delta}. The continuous phase transition at \ensuremath{\delta}=0.25, which was indicated by several authors, is confirmed by the \ensuremath{\delta} dependence of the low-lying level structure, the divergence of the correlation length, and also the collapse of the edge states. The critical exponent \ensuremath{\eta}, governing the power-law decay of the spin correlation function, is estimated to be 0.99\ifmmode\pm\else\textpm\fi{}0.05. The present result raises a possibility that the variety of quantum spin chains including not only the half-odd-integer-S ones but also the integer-S ones may be described in terms of the generic critical theory.

DOI: 10.1103/physrevb.52.10170

Related papers

Elementary excitations of S =1 antiferromagnetic Heisenberg chains with bond alternation

Shoji Yamamoto

Physical review. B, Condensed matterJun 1, 199526 citations

Ground-State Properties ofS=1 Antiferromagnetic Heisenberg Chains with Bond Alternation

Shoji Yamamoto

Journal of the Physical Society of JapanDec 15, 199454 citations

Thermodynamic properties ofS=1 antiferromagnetic Heisenberg chains as Haldane systems

Shoji Yamamoto, Seiji Miyashita

Physical review. B, Condensed matterOct 1, 199364 citations

S=1 antiferromagnetic Heisenberg chain in a magnetic field

Tôru Sakai, Minoru Takahashi

Physical review. B, Condensed matterJun 1, 1991150 citations

Critical phenomena and quantum phase transition in long range Heisenberg antiferromagnetic chains

Nicolas Laflorencie, Ian K. Affleck, Mona Berciu

arXiv:cond-mat/0509390Journal of Statistical Mechanics Theory and ExperimentDec 1, 200577 citations

Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S =1 Heisenberg chain

Steven R. White, David A. Huse

Physical review. B, Condensed matterAug 1, 1993605 citations

Physical review. B, Condensed matter

Critical exponent η in S =1 antiferromagnetic Heisenberg chains with alternating interaction

Shoji Yamamoto

Published Oct 1, 1995• 19 citations

Publisher

Abstract

Low-temperature properties of S=1 antiferromagnetic Heisenberg chains with alternating interaction, scrH=

{\mathit{Jtsum}}_{\mathit{i}}

[1-(-1

{)}^{\mathit{i}}

\ensuremath{\delta}]

{\mathrm{S}}_{\mathit{i}}

\ensuremath{\cdot}

{\mathrm{S}}_{\mathit{i}+1}

, are studied by a quantum Monte Carlo method. Spatial distributions of the spin correlation function 〈

{\mathit{S}}_{1}^{\mathit{z}}$${\mathit{S}}_{\mathit{i}}^{\mathit{z}}

〉 and the local magnetic moment 〈

{\mathit{S}}_{\mathit{i}}^{\mathit{z}}

〉 are visualized as a function of \ensuremath{\delta}. The continuous phase transition at \ensuremath{\delta}=0.25, which was indicated by several authors, is confirmed by the \ensuremath{\delta} dependence of the low-lying level structure, the divergence of the correlation length, and also the collapse of the edge states. The critical exponent \ensuremath{\eta}, governing the power-law decay of the spin correlation function, is estimated to be 0.99\ifmmode\pm\else\textpm\fi{}0.05. The present result raises a possibility that the variety of quantum spin chains including not only the half-odd-integer-S ones but also the integer-S ones may be described in terms of the generic critical theory.

DOI: 10.1103/physrevb.52.10170

Related papers

Elementary excitations of S =1 antiferromagnetic Heisenberg chains with bond alternation

Shoji Yamamoto

A world-line quantum Monte Carlo method is employed to study the elementary excitations of S=1 antiferromagnetic Heisenberg chains with bond alternation described by the Hamiltonian, scrH=J ${\mathit{tsum}}_{\mathit{i}}$ [1-(-1 ${)}^{\mathit{i}}$ \ensuremath{\delta}] ${\mathrm{S}}_{\mathit{i}}$ \ensuremath{\cdot} ${\mathrm{S}}_{\mathit{i}+1}$ . Imaginary-time correlation functions of this system at low-enough temperatures exhibit almost single-exponential decay and thus the lowest excitation spectrum can be successfully extracted from them. As \ensuremath{\delta} increases from 0 to 1, the system encounters a continuous phase transition at \ensuremath{\delta}\ensuremath{\simeq}0.25, where the energy(E)-momentum(q) relation is fitted well to the form E(q)=( ${\mathit{v}}^{2}$ ${\mathrm{sin}}^{2}$ q+${\mathrm{\ensuremath{\Delta}}}^{2}$ ${)}^{1/2}$ with the effective light velocity v=2.46\ifmmode\pm\else\textpm\fi{}0.08 and the energy gap \ensuremath{\Delta} which vanishes in the long chain limit. The central charge c at the massless critical point is estimated to be 1.02\ifmmode\pm\else\textpm\fi{}0.09, which suggests the universality class of the Gaussian model.A world-line quantum Monte Carlo method is employed to study the elementary excitations of S=1 antiferromagnetic Heisenberg chains with bond alternation described by the Hamiltonian, scrH=J ${\mathit{tsum}}_{\mathit{i}}$ [1-(-1${)}^{\mathit{i}}$\ensuremath{\delta}]${\mathrm{S}}_{\mathit{i}}$\ensuremath{\cdot}${\mathrm{S}}_{\mathit{i}+1}$. Imaginary-time correlation functions of this system at low-enough temperatures exhibit almost single-exponential decay and thus the lowest excitation spectrum can be successfully extracted from them. As \ensuremath{\delta} increases from 0 to 1, the system encounters a continuous phase transition at \ensuremath{\delta}\ensuremath{\simeq}0.25, where the energy(E)-momentum(q) relation is fitted well to the form E(q)=(${\mathit{v}}^{2}$ ${\mathrm{sin}}^{2}$q+${\mathrm{\ensuremath{\Delta}}}^{2}$${)}^{1/2}$ with the effective light velocity v=2.46\ifmmode\pm\else\textpm\fi{}0.08 and the energy gap \ensuremath{\Delta} which vanishes in the long chain limit. The central charge c at the massless critical point is estimated to be 1.02\ifmmode\pm\else\textpm\fi{}0.09, which suggests the universality class of the Gaussian model.

Physical review. B, Condensed matterJun 1, 199526 citations

Ground-State Properties ofS=1 Antiferromagnetic Heisenberg Chains with Bond Alternation

Shoji Yamamoto

Ground-state properties of S =1 antiferromagnetic Heisenberg chains with bond alternation, H = J ∑ i {1-(-1) i δ} S i · S i +1 , are studied by a quantum Monte Carlo method. Spin correlation between chain ends on the open chain appears for δ<0.25, while it vanishes for δ≥0.25. At δ≃0.25, the spin correlation length has a peak and the S =1/2 edge states for the open chain disappear. The fourfold degeneracy of the ground state of the open chain holds for δ<0.25, while it is removed for δ≥0.25. All the results support the continuous phase transition at δ≃0.25 demonstrated in the previous works.Ground-state properties of S =1 antiferromagnetic Heisenberg chains with bond alternation, H = J ∑ i {1-(-1) i δ} S i · S i +1 , are studied by a quantum Monte Carlo method. Spin correlation between chain ends on the open chain appears for δ<0.25, while it vanishes for δ≥0.25. At δ≃0.25, the spin correlation length has a peak and the S =1/2 edge states for the open chain disappear. The fourfold degeneracy of the ground state of the open chain holds for δ<0.25, while it is removed for δ≥0.25. All the results support the continuous phase transition at δ≃0.25 demonstrated in the previous works.

Journal of the Physical Society of JapanDec 15, 199454 citations

Thermodynamic properties ofS=1 antiferromagnetic Heisenberg chains as Haldane systems

Shoji Yamamoto, Seiji Miyashita

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 ${\mathit{T}}_{\mathrm{peak}}$ \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.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 ${\mathit{T}}_{\mathrm{peak}}$\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.

Physical review. B, Condensed matterOct 1, 199364 citations

S=1 antiferromagnetic Heisenberg chain in a magnetic field

Tôru Sakai, Minoru Takahashi

A one-dimensional S=1 Heisenberg antiferromagnet in a magnetic field H (\ensuremath{\parallel}z axis) at T=0 is studied by numerical diagonalizations up to N=16. We give the magnetization curve at the thermodynamic limit, and derive an anomaly at ${\mathit{H}}_{\mathit{c}1}$ (=\ensuremath{\Delta}), where \ensuremath{\Delta} is the Haldane gap. It is also found that the transverse spin correlation has the asymptotic form 〈 ${\mathit{S}}_{0}^{\mathit{x}}$ ${\mathit{S}}_{\mathit{r}}^{\mathit{x}}$ 〉\ensuremath{\sim}(-1 ${)}^{\mathit{r}}$ ${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$, and the transverse staggered susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{st}}^{\mathit{x}\mathit{x}}$ diverges between ${\mathit{H}}_{\mathit{c}1}$ and ${\mathit{H}}_{\mathit{c}2}$ (=4). The exponent \ensuremath{\eta} has a minimum (\ensuremath{\eta}\ensuremath{\simeq}0.3) at magnetization m\ensuremath{\simeq}1/3 and \ensuremath{\eta}\ensuremath{\simeq}0.5 at ${\mathit{H}}_{\mathit{c}1}$ and ${\mathit{H}}_{\mathit{c}2}$ . If the system is quasi-one-dimensional, even small interchain couplings can create a canted N\'eel order, within a mean-field approximation for interchain interactions. This is consistent with a recent NMR measurement for Ni( ${\mathrm{C}}_{2}$ ${\mathrm{H}}_{8}$ ${\mathrm{N}}_{2}$ ${)}_{2}$ ${\mathrm{NO}}_{2}$ ( ${\mathrm{ClO}}_{4}$ ) (NENP) at low temperature.A one-dimensional S=1 Heisenberg antiferromagnet in a magnetic field H (\ensuremath{\parallel}z axis) at T=0 is studied by numerical diagonalizations up to N=16. We give the magnetization curve at the thermodynamic limit, and derive an anomaly at ${\mathit{H}}_{\mathit{c}1}$ (=\ensuremath{\Delta}), where \ensuremath{\Delta} is the Haldane gap. It is also found that the transverse spin correlation has the asymptotic form 〈${\mathit{S}}_{0}^{\mathit{x}}$${\mathit{S}}_{\mathit{r}}^{\mathit{x}}$〉\ensuremath{\sim}(-1${)}^{\mathit{r}}$${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$, and the transverse staggered susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{st}}^{\mathit{x}\mathit{x}}$ diverges between ${\mathit{H}}_{\mathit{c}1}$ and ${\mathit{H}}_{\mathit{c}2}$ (=4). The exponent \ensuremath{\eta} has a minimum (\ensuremath{\eta}\ensuremath{\simeq}0.3) at magnetization m\ensuremath{\simeq}1/3 and \ensuremath{\eta}\ensuremath{\simeq}0.5 at ${\mathit{H}}_{\mathit{c}1}$ and ${\mathit{H}}_{\mathit{c}2}$. If the system is quasi-one-dimensional, even small interchain couplings can create a canted N\'eel order, within a mean-field approximation for interchain interactions. This is consistent with a recent NMR measurement for Ni(${\mathrm{C}}_{2}$${\mathrm{H}}_{8}$${\mathrm{N}}_{2}$${)}_{2}$${\mathrm{NO}}_{2}$(${\mathrm{ClO}}_{4}$) (NENP) at low temperature.

Physical review. B, Condensed matterJun 1, 1991150 citations

Critical phenomena and quantum phase transition in long range Heisenberg antiferromagnetic chains

Nicolas Laflorencie, Ian K. Affleck, Mona Berciu

Antiferromagnetic Hamiltonians with short range, non-frustrating interactions are well known to exhibit long range magnetic order in dimensions d ≥2 but exhibit only quasi-long-range order, with power-law decay of correlations, in d = 1 (for half-integer spin). On the other hand, non-frustrating long range interactions can induce long range order in d = 1. We study Hamiltonians in which the long range interactions have an adjustable amplitude λ, as well as an adjustable power law 1/| x | α , using a combination of quantum Monte Carlo and analytic methods: spin-wave, large N non-linear σ model, and renormalization group methods. We map out the phase diagram in the λ–α plane and study the nature of the critical line separating the phases with long range and quasi-long-range order. We find that this corresponds to a novel line of critical points with continuously varying critical exponents and a dynamical exponent, z <1.Antiferromagnetic Hamiltonians with short range, non-frustrating interactions are well known to exhibit long range magnetic order in dimensions d ≥2 but exhibit only quasi-long-range order, with power-law decay of correlations, in d = 1 (for half-integer spin). On the other hand, non-frustrating long range interactions can induce long range order in d = 1. We study Hamiltonians in which the long range interactions have an adjustable amplitude λ, as well as an adjustable power law 1/| x | α , using a combination of quantum Monte Carlo and analytic methods: spin-wave, large N non-linear σ model, and renormalization group methods. We map out the phase diagram in the λ–α plane and study the nature of the critical line separating the phases with long range and quasi-long-range order. We find that this corresponds to a novel line of critical points with continuously varying critical exponents and a dynamical exponent, z <1.

arXiv:cond-mat/0509390Journal of Statistical Mechanics Theory and ExperimentDec 1, 200577 citations

Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S =1 Heisenberg chain

Steven R. White, David A. Huse

We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be ${\mathit{e}}_{0}$ \ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be ${\mathit{e}}_{0}$\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.

Physical review. B, Condensed matterAug 1, 1993605 citations

Low-temperature properties of S=1 antiferromagnetic Heisenberg chains with alternating interaction, scrH=

{\mathit{Jtsum}}_{\mathit{i}}

[1-(-1

{)}^{\mathit{i}}

\ensuremath{\delta}]

{\mathrm{S}}_{\mathit{i}}

\ensuremath{\cdot}

{\mathrm{S}}_{\mathit{i}+1}

, are studied by a quantum Monte Carlo method. Spatial distributions of the spin correlation function 〈

{\mathit{S}}_{1}^{\mathit{z}}$${\mathit{S}}_{\mathit{i}}^{\mathit{z}}

〉 and the local magnetic moment 〈

{\mathit{S}}_{\mathit{i}}^{\mathit{z}}

A world-line quantum Monte Carlo method is employed to study the elementary excitations of S=1 antiferromagnetic Heisenberg chains with bond alternation described by the Hamiltonian, scrH=J

{\mathit{tsum}}_{\mathit{i}}

[1-(-1

{)}^{\mathit{i}}

\ensuremath{\delta}]

{\mathrm{S}}_{\mathit{i}}

\ensuremath{\cdot}

{\mathrm{S}}_{\mathit{i}+1}

. Imaginary-time correlation functions of this system at low-enough temperatures exhibit almost single-exponential decay and thus the lowest excitation spectrum can be successfully extracted from them. As \ensuremath{\delta} increases from 0 to 1, the system encounters a continuous phase transition at \ensuremath{\delta}\ensuremath{\simeq}0.25, where the energy(E)-momentum(q) relation is fitted well to the form E(q)=(

{\mathit{v}}^{2}

{\mathrm{sin}}^{2}

q+${\mathrm{\ensuremath{\Delta}}}^{2}$

{)}^{1/2}

with the effective light velocity v=2.46\ifmmode\pm\else\textpm\fi{}0.08 and the energy gap \ensuremath{\Delta} which vanishes in the long chain limit. The central charge c at the massless critical point is estimated to be 1.02\ifmmode\pm\else\textpm\fi{}0.09, which suggests the universality class of the Gaussian model.

{\mathit{T}}_{\mathrm{peak}}

\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.

{\mathit{H}}_{\mathit{c}1}

(=\ensuremath{\Delta}), where \ensuremath{\Delta} is the Haldane gap. It is also found that the transverse spin correlation has the asymptotic form 〈

{\mathit{S}}_{0}^{\mathit{x}}

{\mathit{S}}_{\mathit{r}}^{\mathit{x}}

〉\ensuremath{\sim}(-1

{)}^{\mathit{r}}

${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$, and the transverse staggered susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{st}}^{\mathit{x}\mathit{x}}$ diverges between

{\mathit{H}}_{\mathit{c}1}

and

{\mathit{H}}_{\mathit{c}2}

(=4). The exponent \ensuremath{\eta} has a minimum (\ensuremath{\eta}\ensuremath{\simeq}0.3) at magnetization m\ensuremath{\simeq}1/3 and \ensuremath{\eta}\ensuremath{\simeq}0.5 at

{\mathit{H}}_{\mathit{c}1}

and

{\mathit{H}}_{\mathit{c}2}

. If the system is quasi-one-dimensional, even small interchain couplings can create a canted N\'eel order, within a mean-field approximation for interchain interactions. This is consistent with a recent NMR measurement for Ni(

{\mathrm{C}}_{2}

{\mathrm{H}}_{8}

{\mathrm{N}}_{2}

{)}_{2}

{\mathrm{NO}}_{2}

(

{\mathrm{ClO}}_{4}

) (NENP) at low temperature.

{\mathit{e}}_{0}

\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.