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Physical review. B, Condensed matter

Comment on "Ground-state properties of a spin-1 antiferromagnetic chain"

Daniel Charles Mattis

Published Apr 1, 1985• 4 citations
Publisher

Abstract

Botet and Jullien [Phys. Rev. B 27, 613 (1983)] and Botet, Jullien, and Kolb [Phys. Rev. B 28, 3914 (1983)] performed a finite-size scaling analysis of the spin-1 antiferromagnetic Heisenberg-Ising chain. Their work was criticized by Bonner and M\"uller [Phys. Rev. B 29, 5216 (1984)] on the grounds that a similar analysis for spin \textonehalf{} yields misleading results. In the present work, we show that the plane-rotator version of spin-1 chain has some features similar to that obtained by Botet et al., although, unfortunately, our model can be solved analytically only over a restricted set of the parameters, and thus does not unambiguously confirm their results. The need for further studies is emphasized.

DOI
10.1103/physrevb.31.4698

Related papers

Comment on "Ground-state properties of a spin-1 antiferromagnetic chain"

Jill C. Bonner, Gerhard H. Muller

Botet and Jullien [Phys. Rev. B 27, 613 (1983)] have conducted a finite-chain scaling analysis on the spin-1 antiferromagnetic Heisenberg-Ising chain. They claim that the ground-state properties are completely different from the case of spin $\frac{1}{2}$. Using finite-size scaling techniques, they conclude that between the $\mathrm{XY}$ gapless phase and the doublet-ground-state N\'eel phase, a new phase appears in an extended anisotropy range. This phase, which encompasses the isotropic Heisenberg model, is characterized by a nonmagnetic singlet ground state and a nonzero excitation-energy gap. We perform a similar finite-chain scaling analysis on the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg-Ising chain and obtain strikingly similar results. Since the spin-$\frac{1}{4}$ system is known rigorously not to exhibit the type of behavior proposed by Botet and Jullien, their analysis is open to doubt. The possibility arises that a strong singularity at the Heisenberg point for spin 1 (similar to the essential singularity for spin $\frac{1}{2}$) may be giving rise to misleading results. The $T=0$ phase behavior of the spin-1 Heisenberg-Ising antiferromagnetic chain, therefore, remains in question and specific additional numerical studies are proposed to clarify the matter.

Physical review. B, Condensed matter•May 1, 1984•80 citations

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 $D$.

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Kiyomi Okamoto, Dai Nishino

The spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain has been studied by Hida [Phys. Rev. B 46, 8268 (1992)]. We point out an error in his bosonized Hamiltonian representation of this model and also show that his numerical results can be well explained by use of the corrected form of the bosonized Hamiltonian.

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Ground-state properties of a spin-1 antiferromagnetic chain

Robert S. Botet, Rémi Jullien

A finite-cell-scaling analysis of the spin-1 antiferromagnetic Heisenberg-Ising chain, shows that the ground-state properties are completely different than in the spin-\textonehalf{} case. Between the $\mathrm{XY}$ gapless phase and the doublet ground-state N\'eel phase, it appears, in an extended range of the anisotropy ($0\ensuremath{\lesssim}\ensuremath{\lambda}\ensuremath{\lesssim}1.18$) a characteristic Heisenberg-phase with a nonmagnetic singlet ground state, nonzero gap, and exponential decay of the correlation functions.

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Reply to "Comment on `Ground-state properties of a spin-1 antiferromagnetic chain' "

Robert S. Botet, Rémi Jullien, M. Kolb

We contrast, on the same figure, our phenomenological renormalization-group results on the $S=1$ Heisenberg-Ising antiferromagnetic chain with results that we have recently obtained for $S=\frac{1}{2}$ and $S=\frac{3}{2}$. The different behaviors of integer- and half-integer-spin chains are revealed very clearly. This confirms the conclusion of our Communication.

Physical review. B, Condensed matter•May 1, 1984•14 citations

Comparison of ground state properties for odd half-integer and integer spin antiferromagnetic Heisenberg chains

M. Kolb, Robert S. Botet, Rémi Jullien

Ground state properties of finite anisotropic antiferromagnetic Heisenberg chains are studied for odd half-integer spins S= 1 / 2 , 3 / 2 ... as well as integer spins S=1, 2 .... Finite size scaling analysis of the results clearly distinguishes the half-integer (S= 1 / 2 , 3 / 2 ) from the integer (S=1) spin situation. It gives strong support to a recent conjecture which postulates that the T=0 phase structure is very different in the two cases. According to this idea there exists a new phase between the planar and the antiferromagnetic region, for integer spins only. This phase, which includes the isotropic point, has a finite energy gap and no long range order.

Journal of Physics A Mathematical and General•Dec 1, 1983•75 citations
Botet and Jullien [Phys. Rev. B 27, 613 (1983)] and Botet, Jullien, and Kolb [Phys. Rev. B 28, 3914 (1983)] performed a finite-size scaling analysis of the spin-1 antiferromagnetic Heisenberg-Ising chain. Their work was criticized by Bonner and M"uller [Phys. Rev. B 29, 5216 (1984)] on the grounds that a similar analysis for spin \textonehalf{} yields misleading results. In the present work, we show that the plane-rotator version of spin-1 chain has some features similar to that obtained by Botet et al., although, unfortunately, our model can be solved analytically only over a restricted set of the parameters, and thus does not unambiguously confirm their results. The need for further studies is emphasized.
Botet and Jullien [Phys. Rev. B 27, 613 (1983)] have conducted a finite-chain scaling analysis on the spin-1 antiferromagnetic Heisenberg-Ising chain. They claim that the ground-state properties are completely different from the case of spin 12\frac{1}{2}21​. Using finite-size scaling techniques, they conclude that between the XY\mathrm{XY}XY gapless phase and the doublet-ground-state N\'eel phase, a new phase appears in an extended anisotropy range. This phase, which encompasses the isotropic Heisenberg model, is characterized by a nonmagnetic singlet ground state and a nonzero excitation-energy gap. We perform a similar finite-chain scaling analysis on the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg-Ising chain and obtain strikingly similar results. Since the spin-14\frac{1}{4}41​ system is known rigorously not to exhibit the type of behavior proposed by Botet and Jullien, their analysis is open to doubt. The possibility arises that a strong singularity at the Heisenberg point for spin 1 (similar to the essential singularity for spin 12\frac{1}{2}21​) may be giving rise to misleading results. The T=0T=0T=0 phase behavior of the spin-1 Heisenberg-Ising antiferromagnetic chain, therefore, remains in question and specific additional numerical studies are proposed to clarify the matter.
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.
A finite-cell-scaling analysis of the spin-1 antiferromagnetic Heisenberg-Ising chain, shows that the ground-state properties are completely different than in the spin-\textonehalf{} case. Between the XY\mathrm{XY}XY gapless phase and the doublet ground-state N\'eel phase, it appears, in an extended range of the anisotropy ($0\ensuremath{\lesssim}\ensuremath{\lambda}\ensuremath{\lesssim}1.18$) a characteristic Heisenberg-phase with a nonmagnetic singlet ground state, nonzero gap, and exponential decay of the correlation functions.
We contrast, on the same figure, our phenomenological renormalization-group results on the S=1S=1S=1 Heisenberg-Ising antiferromagnetic chain with results that we have recently obtained for S=12S=\frac{1}{2}S=21​ and S=32S=\frac{3}{2}S=23​. The different behaviors of integer- and half-integer-spin chains are revealed very clearly. This confirms the conclusion of our Communication.
Ground state properties of finite anisotropic antiferromagnetic Heisenberg chains are studied for odd half-integer spins S= 1 / 2 , 3 / 2 ... as well as integer spins S=1, 2 .... Finite size scaling analysis of the results clearly distinguishes the half-integer (S= 1 / 2 , 3 / 2 ) from the integer (S=1) spin situation. It gives strong support to a recent conjecture which postulates that the T=0 phase structure is very different in the two cases. According to this idea there exists a new phase between the planar and the antiferromagnetic region, for integer spins only. This phase, which includes the isotropic point, has a finite energy gap and no long range order.