The long-range order in the ground state of the linear antiferromagnetic chain is investigated. A chain of $N$ identical spin-\textonehalf{} particles coupled by the anisotropic Hamiltonian $\mathcal{H}=2J\ensuremath{\Sigma}\stackrel{N}{j=1}{{S}_{j}}^{z}{{S}_{j+1}}^{z}+bJ\ensuremath{\Sigma}\stackrel{N}{j=1}({{S}_{j}}^{+}{{S}_{j+1}}^{\ensuremath{-}}+{{S}_{j}}^{\ensuremath{-}}{{S}_{j+1}}^{+})$ is considered, and the periodic boundary condition, ${S}_{i+N}\ensuremath{\equiv}{S}_{i}$, is imposed so that the chain may be considered a loop of particles. The operator $\mathfrak{M}$ , which rotates the particles in the loop so that each of the particles moves to the next highest site number, is introduced. It is proved that simultaneous eigenstates of $\mathcal{H}$ and $\mathfrak{M}$ have no long-range order as measured by ${\ensuremath{\sigma}}^{*}$ of Kasteleijn. The ground state of the Hamiltonian for $b\ensuremath{\ne}0$, $N<\ensuremath{\infty}$ is shown to be nondegenerate. It is therefore necessarily an eigenstate of $\mathfrak{M}$ and as such has no long-range order.The long-range order in the ground state of the linear antiferromagnetic chain is investigated. A chain of $N$ identical spin-\textonehalf{} particles coupled by the anisotropic Hamiltonian $\mathcal{H}=2J\ensuremath{\Sigma}\stackrel{N}{j=1}{{S}_{j}}^{z}{{S}_{j+1}}^{z}+bJ\ensuremath{\Sigma}\stackrel{N}{j=1}({{S}_{j}}^{+}{{S}_{j+1}}^{\ensuremath{-}}+{{S}_{j}}^{\ensuremath{-}}{{S}_{j+1}}^{+})$ is considered, and the periodic boundary condition, ${S}_{i+N}\ensuremath{\equiv}{S}_{i}$, is imposed so that the chain may be considered a loop of particles. The operator $\mathfrak{M}$, which rotates the particles in the loop so that each of the particles moves to the next highest site number, is introduced. It is proved that simultaneous eigenstates of $\mathcal{H}$ and $\mathfrak{M}$ have no long-range order as measured by ${\ensuremath{\sigma}}^{*}$ of Kasteleijn. The ground state of the Hamiltonian for $b\ensuremath{\ne}0$, $N<\ensuremath{\infty}$ is shown to be nondegenerate. It is therefore necessarily an eigenstate of $\mathfrak{M}$ and as such has no long-range order.

Physical ReviewMay 15, 19622 citations

Long range order in the anisotropic quantum ferromagnetic Heisenberg model

Tom Kennedy

Communications in Mathematical PhysicsSep 1, 198544 citations

Néel order in a spatially anisotropic Heisenberg antiferromagnet

Y. Q. Wang, G. S. Tian, Hua Lin

We investigate the existence of N\'eel long-range order in a spatially anisotropic antiferromagnetic Heisenberg model. As the anisotropical coupling ${J}_{\ensuremath{\perp}}$ increases from $0$ to $1,$ the model interpolates between an infinite number of decoupled one-dimensional gapped spin chains and a two-dimensional gapless isotropic Heisenberg antiferromagnet. By applying a method based on reflection positivity [F. J. Dyson, E. H. Lieb, and B. Simon, J. Stat. Phys. $18,$ 335 (1978)], we determine the upper bound of the critical values of ${J}_{\ensuremath{\perp}}^{c}$ such that when ${J}_{\ensuremath{\perp}}>~{J}_{\ensuremath{\perp}}^{c},$ the ground state of this model has an antiferromagnetic long-range order. For the case of spin $S=1,$ our result confirms and strengthens previous results derived by numerical calculations on finite-size samples, while for other values of S, our results are new.We investigate the existence of N\'eel long-range order in a spatially anisotropic antiferromagnetic Heisenberg model. As the anisotropical coupling ${J}_{\ensuremath{\perp}}$ increases from $0$ to $1,$ the model interpolates between an infinite number of decoupled one-dimensional gapped spin chains and a two-dimensional gapless isotropic Heisenberg antiferromagnet. By applying a method based on reflection positivity [F. J. Dyson, E. H. Lieb, and B. Simon, J. Stat. Phys. $18,$ 335 (1978)], we determine the upper bound of the critical values of ${J}_{\ensuremath{\perp}}^{c}$ such that when ${J}_{\ensuremath{\perp}}>~{J}_{\ensuremath{\perp}}^{c},$ the ground state of this model has an antiferromagnetic long-range order. For the case of spin $S=1,$ our result confirms and strengthens previous results derived by numerical calculations on finite-size samples, while for other values of S, our results are new.

Physical review. B, Condensed matterFeb 7, 20037 citations

Néel order in a spatially anisotropic Heisenberg antiferromagnet with single-ion anisotropy

Y. Q. Wang, G. S. Tian, Hua Lin

In the present paper, we investigate the existence of the antiferromagnetic long-range order in a $S=1$ spatially anisotropic antiferromagnetic Heisenberg model with a single-ion energy term. By applying the reflection-positivity method developed by Dyson, Lieb, and Simon [J. Stat. Phys. 53, 1019 (1988)], we are able to determine the parameter regions where either the longitudinal or the transverse antiferromagnetic long-range order exist. We find that the single-ion energy anisotropy has a strong influence on the critical value of the spin coupling constant between the one-dimensional integer-spin chains. Qualitatively, our results agree with the previous ones obtained by the mean-field theory and hence, reestablish the conclusions on a mathematically rigorous basis.In the present paper, we investigate the existence of the antiferromagnetic long-range order in a $S=1$ spatially anisotropic antiferromagnetic Heisenberg model with a single-ion energy term. By applying the reflection-positivity method developed by Dyson, Lieb, and Simon [J. Stat. Phys. 53, 1019 (1988)], we are able to determine the parameter regions where either the longitudinal or the transverse antiferromagnetic long-range order exist. We find that the single-ion energy anisotropy has a strong influence on the critical value of the spin coupling constant between the one-dimensional integer-spin chains. Qualitatively, our results agree with the previous ones obtained by the mean-field theory and hence, reestablish the conclusions on a mathematically rigorous basis.

Physical Review BJul 19, 20054 citations

The long-range order in the ground state of the linear antiferromagnetic chain is investigated. A chain of

N

identical spin-\textonehalf{} particles coupled by the anisotropic Hamiltonian $\mathcal{H}=2J\ensuremath{\Sigma}\stackrel{N}{j=1}{{S}_{j}}^{z}{{S}_{j+1}}^{z}+bJ\ensuremath{\Sigma}\stackrel{N}{j=1}({{S}_{j}}^{+}{{S}_{j+1}}^{\ensuremath{-}}+{{S}_{j}}^{\ensuremath{-}}{{S}_{j+1}}^{+})$ is considered, and the periodic boundary condition, ${S}_{i+N}\ensuremath{\equiv}{S}_{i}$, is imposed so that the chain may be considered a loop of particles. The operator

\mathfrak{M}

, which rotates the particles in the loop so that each of the particles moves to the next highest site number, is introduced. It is proved that simultaneous eigenstates of

\mathcal{H}

and

\mathfrak{M}

have no long-range order as measured by ${\ensuremath{\sigma}}^{*}$ of Kasteleijn. The ground state of the Hamiltonian for $b\ensuremath{\ne}0$, $N<\ensuremath{\infty}$ is shown to be nondegenerate. It is therefore necessarily an eigenstate of

\mathfrak{M}

and as such has no long-range order.

We investigate the existence of N\'eel long-range order in a spatially anisotropic antiferromagnetic Heisenberg model. As the anisotropical coupling ${J}_{\ensuremath{\perp}}$ increases from

0

1,

the model interpolates between an infinite number of decoupled one-dimensional gapped spin chains and a two-dimensional gapless isotropic Heisenberg antiferromagnet. By applying a method based on reflection positivity [F. J. Dyson, E. H. Lieb, and B. Simon, J. Stat. Phys.

18,

335 (1978)], we determine the upper bound of the critical values of ${J}_{\ensuremath{\perp}}^{c}$ such that when ${J}_{\ensuremath{\perp}}>~{J}_{\ensuremath{\perp}}^{c},$ the ground state of this model has an antiferromagnetic long-range order. For the case of spin

S=1,

our result confirms and strengthens previous results derived by numerical calculations on finite-size samples, while for other values of S, our results are new.

In the present paper, we investigate the existence of the antiferromagnetic long-range order in a

S=1

spatially anisotropic antiferromagnetic Heisenberg model with a single-ion energy term. By applying the reflection-positivity method developed by Dyson, Lieb, and Simon [J. Stat. Phys. 53, 1019 (1988)], we are able to determine the parameter regions where either the longitudinal or the transverse antiferromagnetic long-range order exist. We find that the single-ion energy anisotropy has a strong influence on the critical value of the spin coupling constant between the one-dimensional integer-spin chains. Qualitatively, our results agree with the previous ones obtained by the mean-field theory and hence, reestablish the conclusions on a mathematically rigorous basis.