Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

Self-consistent mean-field theory of a spin-1 antiferromagnetic chain

Hanting Wang, Jue-Lian Shen, Kui-Ru Li, Zhao-Bin Su

Published May 1, 1994• 7 citations
Publisher

Abstract

We develop an extended self-consistent mean-field approach to study a spin-1 antiferromagnetic (AF) chain with the Hamiltonian H=${\mathit{tsum}}_{\mathit{i}}$ [${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$${\mathit{S}}_{\mathit{i}+1}^{\mathit{x}}$+${\mathit{S}}_{\mathit{i}}^{\mathit{y}}$${\mathit{S}}_{\mathit{i}+1}^{\mathit{y}}$ +\ensuremath{\lambda}${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${\mathit{S}}_{\mathit{i}+1}^{\mathit{z}}$+D(${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${)}^{2}$] based on Anderson's spin-wave theory. The ground-state energy changes little compared with the traditional spin-wave theory and the excitation spectrum shows a gap which lends support to Haldane's conjecture. At the Heisenberg point (\ensuremath{\lambda}=1,D=0), the ground-state energy and the gap are -1.3505 and 0.3914, respectively. In the Heisenberg-Ising case (\ensuremath{\lambda}>0,D=0), the gap decreases when \ensuremath{\lambda} goes down from 1 and at a critical ${\ensuremath{\lambda}}_{\mathit{c}}$>0 the gap goes to 0, while for the single-ion anisotropy (\ensuremath{\lambda}=1,D>0) the gap decreases with the increasing of D and at a critical value ${\mathit{D}}_{\mathit{c}}$ the gap goes to 0. These results agree well with those of other theoretical methods and experiments. We study the one-magnon excitation spectrum carefully and suggest an explanation for its crossing with the two-magnon excitation spectrum at low momentum. We estimate the crossing occurring at k\ensuremath{\simeq}0.3\ensuremath{\pi}. The results agree well with the very recent theoretical and experimental results. The picture of the Haldane gap is discussed from our extended self-consistent mean-field ground state and many interesting properties of the spin-1 AF chain are discussed.

DOI
10.1103/physrevb.49.12805

Related papers

Properties of anS=1 antiferromagnetic Heisenberg chain with single-ion anisotropy

Hong Chen, Lu Yu, Zhao-Bin Su

A spin-1 antiferromagnetic Heisenberg chain with a single-ion anisotropy ${\mathit{Dtsum}}_{\mathit{j}}$(${\mathit{S}}_{\mathit{j}}^{\mathit{z}}$${)}^{2}$ is investigated on the basis of the continuum representation of the model Hamiltonian. Analytical results for the ground state, the excitation spectrum, and the correlation functions are derived in a self-consistent-field approach. We find that the ground state is made of boson pairs and the correlation functions in the ground state decay exponentially. Two Haldane gaps due to the single-ion anisotropy are obtained and a transition at the critical value ${\mathit{D}}_{\mathit{c}}$=1 is shown in the lower Haldane gap and the correlation functions.

Physical review. B, Condensed matter•Nov 1, 1993•12 citations

Low-temperature properties of a spin-1 antiferromagnetic chain

Han-ting Wang, Jue-Lian Shen, Kui-Ru Li +1 more

We study the low-temperature properties of a spin-1 Heisenberg antiferromagnetic chain in an external magnetic field by an extended self-consistent mean-field approach developed earlier based on Anderson's spin-wave theory. At fixed magnetic field, the excitation energies increase, while the correlation length decreases with the increasing temperatures. The magnetic field leads to the Zeeman splitting of the excitations with the upper branch going up and the lower branch going down with the increasing fields. At zero temperature the correlation length is a constant and the gap closes at a critical magnetic field, while at finite temperature the correlation length decreases and a small gap seems to survive at the critical magnetic field. These results are consistent with those of other theoretical studies and agree with the experiments quite well. Also, we show that the system shares very similar properties with the valence-bond-solid state even at finite temperatures. The magnetization is studied and the coexistence of the hidden long-range order and the field-induced ferromagnetic order is discussed.

Physical review. B, Condensed matter•Jun 1, 1995•1 citation

Finite-size-scaling study of the spin-1 Heisenberg-Ising chain with uniaxial anisotropy

Robert S. Botet, Rémi Jullien, M. Kolb

Finite-cell calculations (up to $N=12$ spins) have been performed on the spin-1 Heisenberg-Ising chain with an uniaxial anisotropy, $\mathcal{H}=\ensuremath{\Sigma}{i}^{}[{S}_{i}^{x}{S}_{i+1}^{x}+{S}_{i}^{y}{S}_{i+1}^{y}+\ensuremath{\lambda}{S}_{i}^{z}{S}_{i+1}^{z}+D{({S}_{i}^{z})}^{2}]$. From a scaling analysis of the gap between the ground state and the first excited state, a phase diagram has been drawn in the ($\ensuremath{\lambda},D$) plane and the transition lines between the "ferromagnetic," $X\ensuremath{-}Y$," "singlet-ground-state," and "antiferromagnetic" phases have been estimated for the infinite-$N$ system. One of the most important results is that a singlet-ground-state phase with a nonzero gap exists in an extended range of $\ensuremath{\lambda}$ and $D$ values including the Heisenberg point $\ensuremath{\lambda}=1$, $D=0$, in contrast with the spin-$\frac{1}{2}$ case. Moreover, for $\ensuremath{\lambda}\ensuremath{\simeq}1$, the gap decreases with increasing positive anisotropy $D$, goes through a minimum, estimated to be zero, and then increases with $D$.

Physical review. B, Condensed matter•Oct 1, 1983•250 citations

Critical behavior of the S=1 antiferromagnetic Heisenberg chain (abstract)

Jill C. Bonner

The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.

Journal of Applied Physics•Apr 15, 1991•1 citation

Bond operator theory of isotropic spin-1 antiferromagnetic chains

Hanting Wang, Jue-Lian Shen, Zhao-Bin Su

The bond operator representation of quantum $S=\frac{1}{2}$ spins with the singlet state being projected out is applied to investigate the isotropic spin-1 Heisenberg antiferromagnetic chain $H={\ensuremath{\sum}}_{i}[{\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}\ensuremath{-}\ensuremath{\beta}({\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}{)}^{2}]$. In the approximation of Bose condensation, the ground-state properties of the system and their $\ensuremath{\beta}$ dependences are studied. The results at Heisenberg point $\ensuremath{\beta}=0$ and at valence-bond-solid (VBS) point $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3}$ agree quite well with the analytical, numerical, and experimental studies. The excitation spectrum is nonsymmetric with respect to $k=\ensuremath{\pi}/2$ and shows a minimum at $k=\ensuremath{\pi}$. The $\ensuremath{\beta}$ dependences of the ground-state energy and the excitation gap show the similarities between the Heisenberg point and VBS point. The related properties of the spin ladders are also discussed.

Physical review. B, Condensed matter•Dec 1, 1997•29 citations

Generalized Heisenberg quantum spin chains (invited)

Jill C. Bonner

Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.

Journal of Applied Physics•Apr 15, 1987•40 citations
We develop an extended self-consistent mean-field approach to study a spin-1 antiferromagnetic (AF) chain with the Hamiltonian H=tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​ [{\mathit{S}}_{\mathit{i}}^{\mathit{x}}$${\mathit{S}}_{\mathit{i}+1}^{\mathit{x}}+{\mathit{S}}_{\mathit{i}}^{\mathit{y}}$${\mathit{S}}_{\mathit{i}+1}^{\mathit{y}} +\ensuremath{\lambda}{\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${\mathit{S}}_{\mathit{i}+1}^{\mathit{z}}+D({\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${)}^{2}] based on Anderson's spin-wave theory. The ground-state energy changes little compared with the traditional spin-wave theory and the excitation spectrum shows a gap which lends support to Haldane's conjecture. At the Heisenberg point (\ensuremath{\lambda}=1,D=0), the ground-state energy and the gap are -1.3505 and 0.3914, respectively. In the Heisenberg-Ising case (\ensuremath{\lambda}>0,D=0), the gap decreases when \ensuremath{\lambda} goes down from 1 and at a critical \ensuremathλc{\ensuremath{\lambda}}_{\mathit{c}}\ensuremathλc​>0 the gap goes to 0, while for the single-ion anisotropy (\ensuremath{\lambda}=1,D>0) the gap decreases with the increasing of D and at a critical value Dc{\mathit{D}}_{\mathit{c}}Dc​ the gap goes to 0. These results agree well with those of other theoretical methods and experiments. We study the one-magnon excitation spectrum carefully and suggest an explanation for its crossing with the two-magnon excitation spectrum at low momentum. We estimate the crossing occurring at k\ensuremath{\simeq}0.3\ensuremath{\pi}. The results agree well with the very recent theoretical and experimental results. The picture of the Haldane gap is discussed from our extended self-consistent mean-field ground state and many interesting properties of the spin-1 AF chain are discussed.
A spin-1 antiferromagnetic Heisenberg chain with a single-ion anisotropy Dtsumj{\mathit{Dtsum}}_{\mathit{j}}Dtsumj​(Sjz{\mathit{S}}_{\mathit{j}}^{\mathit{z}}Sjz​)2{)}^{2})2 is investigated on the basis of the continuum representation of the model Hamiltonian. Analytical results for the ground state, the excitation spectrum, and the correlation functions are derived in a self-consistent-field approach. We find that the ground state is made of boson pairs and the correlation functions in the ground state decay exponentially. Two Haldane gaps due to the single-ion anisotropy are obtained and a transition at the critical value Dc{\mathit{D}}_{\mathit{c}}Dc​=1 is shown in the lower Haldane gap and the correlation functions.
We study the low-temperature properties of a spin-1 Heisenberg antiferromagnetic chain in an external magnetic field by an extended self-consistent mean-field approach developed earlier based on Anderson's spin-wave theory. At fixed magnetic field, the excitation energies increase, while the correlation length decreases with the increasing temperatures. The magnetic field leads to the Zeeman splitting of the excitations with the upper branch going up and the lower branch going down with the increasing fields. At zero temperature the correlation length is a constant and the gap closes at a critical magnetic field, while at finite temperature the correlation length decreases and a small gap seems to survive at the critical magnetic field. These results are consistent with those of other theoretical studies and agree with the experiments quite well. Also, we show that the system shares very similar properties with the valence-bond-solid state even at finite temperatures. The magnetization is studied and the coexistence of the hidden long-range order and the field-induced ferromagnetic order is discussed.
Finite-cell calculations (up to N=12N=12N=12 spins) have been performed on the spin-1 Heisenberg-Ising chain with an uniaxial anisotropy, $\mathcal{H}=\ensuremath{\Sigma}{i}^{}[{S}_{i}^{x}{S}_{i+1}^{x}+{S}_{i}^{y}{S}_{i+1}^{y}+\ensuremath{\lambda}{S}_{i}^{z}{S}_{i+1}^{z}+D{({S}_{i}^{z})}^{2}]$. From a scaling analysis of the gap between the ground state and the first excited state, a phase diagram has been drawn in the ($\ensuremath{\lambda},D$) plane and the transition lines between the "ferromagnetic," $X\ensuremath{-}Y$," "singlet-ground-state," and "antiferromagnetic" phases have been estimated for the infinite-NNN system. One of the most important results is that a singlet-ground-state phase with a nonzero gap exists in an extended range of $\ensuremath{\lambda}$ and DDD values including the Heisenberg point $\ensuremath{\lambda}=1$, D=0D=0D=0, in contrast with the spin-12\frac{1}{2}21​ case. Moreover, for $\ensuremath{\lambda}\ensuremath{\simeq}1$, the gap decreases with increasing positive anisotropy DDD, goes through a minimum, estimated to be zero, and then increases with DDD.
The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.
The bond operator representation of quantum S=12S=\frac{1}{2}S=21​ spins with the singlet state being projected out is applied to investigate the isotropic spin-1 Heisenberg antiferromagnetic chain $H={\ensuremath{\sum}}_{i}[{\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}\ensuremath{-}\ensuremath{\beta}({\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}{)}^{2}]$. In the approximation of Bose condensation, the ground-state properties of the system and their $\ensuremath{\beta}$ dependences are studied. The results at Heisenberg point $\ensuremath{\beta}=0$ and at valence-bond-solid (VBS) point $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3}$ agree quite well with the analytical, numerical, and experimental studies. The excitation spectrum is nonsymmetric with respect to $k=\ensuremath{\pi}/2$ and shows a minimum at $k=\ensuremath{\pi}$. The $\ensuremath{\beta}$ dependences of the ground-state energy and the excitation gap show the similarities between the Heisenberg point and VBS point. The related properties of the spin ladders are also discussed.
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.