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

International Journal of Modern Physics B

LONG-RANGE ORDER IN TWO-DIMENSIONAL XXZ MODEL

Wen‐Long You

Published Apr 10, 2009• 2 citations
Publisher

Abstract

In the present paper, by applying the reflection positivity method due to Dyson, Lieb, and Simon, we rigorously establish the sufficient condition for existence of the Néel long-range order (NLRO) in two-dimensional spin-1/2 XXZ. Our result shows that if the anisotropic coupling Δ satisfies 0 ≤ Δ ≤ 0.30 or Δ ≥ 1.52, the existence of the NLRO along easy axis is proved.

DOI
10.1142/s0217979209052145

Related papers

Extended proof of long-range order in the two-dimensional quantum spin-1/2 XXZ-model at T=0

Hans-Aloys Wischmann, E. Müller‐Hartmann

We prove the existence of long-range order (LRO) in the ground state of the 2D quantum spin-1/2 XXZ-model for all anisotropies $0 \leq \Delta\ leq 0.22$ and Δ≥1.47. We achieve this extension of previous proofs by developing an improved variant of the Dyson-Lieb-Simon proof of LRO in quantum spin systems and by deriving improved upper and lower bounds for the ground state energy of the XXZ-model.

Journal de Physique I•May 1, 1991•11 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.

Physical review. B, Condensed matter•Feb 7, 2003•7 citations

Ground-State Long-Range Order in the Two-DimensionalXXZMode

Hidetoshi Nishimori, Yukiyasu Ozeki

We prove the existence of Néel-type long-range order in the ground state of the spin-1/2 XXZ model with Δ (exchange anisotropy)>1.67 on the square lattice. We further show the existence of long-range order for Δ >1.10 and 0≦ Δ <0.59 by assuming monotonicity of nearest neighbor correlations as functions of the system size. The assumption of monotonicity is supported by numerical calculations.

Journal of the Physical Society of Japan•Mar 15, 1989•42 citations

Long-Range Order in Antiferromagnetic Quantum Spin Systems

Yukiyasu Ozeki, Hidetoshi Nishimori, Yasuhiro Tomita

The existence of long-range order is proved under certain conditions for the antiferromagnetic XYZ model on the simple cubic or the square lattice. In particular, the spin-1/2 XXZ model on the square lattice is shown to have ground-state long-range order if the exchange anisotropy Δ satisfies 0≦ Δ <0.20 or Δ >1.72, which improves the result of Kubo and Kishi. The existence of long-range order of the z -component of the spin operator is proved for the XXZ model with XY -like anisotropy (0≦ Δ ≦1) under certain conditions. A similar result is shown to hold for the long-range order in the x -direction for the Ising-like model ( Δ ≧1). The XXZ model on the two-dimensional hexagonal lattice is proved to have finite ground-state long-range order for any value of Δ (≧0) if S ≧1 and for Δ >2.55 when S =1/2.

Journal of the Physical Society of Japan•Jan 15, 1989•27 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.

Physical Review B•Jul 19, 2005•4 citations

Existence of Long-Range Order in the XXZ Model

Kenn Kubo, Tatsuya Kishi

The method of Kennedy, Lieb, and Shastry is applied to the antiferromagnetic $\mathrm{XXZ}$ model to examine the existence of long-range order (LRO). On the 3D simple-cubic lattice LRO is proved to exist for any $S\ensuremath{\geqq}\frac{1}{2}$ and $\ensuremath{\Delta}\ensuremath{\geqq}0$ at low temperature. On the 2D square lattice LRO exists in the ground state for any $\ensuremath{\Delta}\ensuremath{\leqq}0$ if $S\ensuremath{\geqq}1$ and for $0\ensuremath{\leqq}\ensuremath{\Delta}<0.13$ and $\ensuremath{\Delta}=0$ if $S=\frac{1}{2}$. The existence of LRO in the ground state of the $S=\frac{1}{2} \mathrm{XY}$ model ($\ensuremath{\Delta}=0$), therefore, has been confirmed, in agreement with the conjecture by Nishimori et al. The ground state of the $S=\frac{1}{2}$ antiferromagnetic Heisenberg model in two dimensions still remains to be clarified.

Physical Review Letters•Nov 28, 1988•106 citations
In the present paper, by applying the reflection positivity method due to Dyson, Lieb, and Simon, we rigorously establish the sufficient condition for existence of the Néel long-range order (NLRO) in two-dimensional spin-1/2 XXZ. Our result shows that if the anisotropic coupling Δ satisfies 0 ≤ Δ ≤ 0.30 or Δ ≥ 1.52, the existence of the NLRO along easy axis is proved.
We prove the existence of long-range order (LRO) in the ground state of the 2D quantum spin-1/2 XXZ-model for all anisotropies 0≤Δ leq0.220 \leq \Delta\ leq 0.220≤Δ leq0.22 and Δ≥1.47. We achieve this extension of previous proofs by developing an improved variant of the Dyson-Lieb-Simon proof of LRO in quantum spin systems and by deriving improved upper and lower bounds for the ground state energy of the XXZ-model.
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 000 to 1,1,1, 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. 18,18,18, 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,S=1,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.
The existence of long-range order is proved under certain conditions for the antiferromagnetic XYZ model on the simple cubic or the square lattice. In particular, the spin-1/2 XXZ model on the square lattice is shown to have ground-state long-range order if the exchange anisotropy Δ satisfies 0≦ Δ <0.20 or Δ >1.72, which improves the result of Kubo and Kishi. The existence of long-range order of the z -component of the spin operator is proved for the XXZ model with XY -like anisotropy (0≦ Δ ≦1) under certain conditions. A similar result is shown to hold for the long-range order in the x -direction for the Ising-like model ( Δ ≧1). The XXZ model on the two-dimensional hexagonal lattice is proved to have finite ground-state long-range order for any value of Δ (≧0) if S ≧1 and for Δ >2.55 when S =1/2.
In the present paper, we investigate the existence of the antiferromagnetic long-range order in a S=1S=1S=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.
The method of Kennedy, Lieb, and Shastry is applied to the antiferromagnetic XXZ\mathrm{XXZ}XXZ model to examine the existence of long-range order (LRO). On the 3D simple-cubic lattice LRO is proved to exist for any $S\ensuremath{\geqq}\frac{1}{2}$ and $\ensuremath{\Delta}\ensuremath{\geqq}0$ at low temperature. On the 2D square lattice LRO exists in the ground state for any $\ensuremath{\Delta}\ensuremath{\leqq}0$ if $S\ensuremath{\geqq}1$ and for $0\ensuremath{\leqq}\ensuremath{\Delta}<0.13$ and $\ensuremath{\Delta}=0$ if S=12S=\frac{1}{2}S=21​. The existence of LRO in the ground state of the S=12XYS=\frac{1}{2} \mathrm{XY}S=21​XY model ($\ensuremath{\Delta}=0$), therefore, has been confirmed, in agreement with the conjecture by Nishimori et al. The ground state of the S=12S=\frac{1}{2}S=21​ antiferromagnetic Heisenberg model in two dimensions still remains to be clarified.