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Physical review. B, Condensed matter

S=1 antiferromagnetic Heisenberg chain with an axial-symmetry-breaking anisotropy

Tiôru Sakai, Minoru Takahashi

Published Nov 1, 1991• 5 citations
Publisher

Abstract

The ground state of the S=1 antiferromagnetic Heisenberg chain with an axial-symmetry-breaking anisotropy ${\mathit{E}\mathit{tsum}}_{\mathit{j}}$ [(${\mathit{S}}_{\mathit{j}}^{\mathit{x}}$${)}^{2}$-(${\mathit{S}}_{\mathit{j}}^{\mathit{y}}$${)}^{2}$] is studied by the finite-size scaling and phenomenological renormalization-group technique based on numerical diagonalizations up to N=14. It is found that two phase transitions exist at ${\mathit{E}}_{\mathit{c}1}$=0.20\ifmmode\pm\else\textpm\fi{}0.01 and ${\mathit{E}}_{\mathit{c}2}$=1.71\ifmmode\pm\else\textpm\fi{}0.01, and both transitions belong to the same universality class as the two-dimensional Ising model.

DOI
10.1103/physrevb.44.10385

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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.

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Spin-correlation functions of theS=1Heisenberg-Ising chain by the large-cluster-decomposition Monte Carlo method

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We study the spin-correlation functions of the one-dimensional S=1 antiferromagnetic Heisenberg-Ising model by the large-cluster-decomposition Monte Carlo method. It is found that this model shows a second-order transition at the anisotropy parameter ${\ensuremath{\lambda}}_{c}$=1.167\ifmmode\pm\else\textpm\fi{}0.007. The obtained critical exponents (\ensuremath{\eta}=0.253\ifmmode\pm\else\textpm\fi{}0.002, \ensuremath{\beta}=0.126\ifmmode\pm\else\textpm\fi{}0.007, \ensuremath{\nu}=0.98\ifmmode\pm\else\textpm\fi{}0.02) strongly suggest that this transition belongs to the same universality class as the two-dimensional Ising model.

Physical review. B, Condensed matter•Nov 1, 1989•39 citations
The ground state of the S=1 antiferromagnetic Heisenberg chain with an axial-symmetry-breaking anisotropy Etsumj{\mathit{E}\mathit{tsum}}_{\mathit{j}}Etsumj​ [({\mathit{S}}_{\mathit{j}}^{\mathit{x}}$${)}^{2}-({\mathit{S}}_{\mathit{j}}^{\mathit{y}}$${)}^{2}] is studied by the finite-size scaling and phenomenological renormalization-group technique based on numerical diagonalizations up to N=14. It is found that two phase transitions exist at Ec1{\mathit{E}}_{\mathit{c}1}Ec1​=0.20\ifmmode\pm\else\textpm\fi{}0.01 and Ec2{\mathit{E}}_{\mathit{c}2}Ec2​=1.71\ifmmode\pm\else\textpm\fi{}0.01, and both transitions belong to the same universality class as the two-dimensional Ising model.
We study the S =1 antiferromagnetic Heisenberg chain with the anisotropies E ∑ j [( S j x ) 2 -( S j y ) 2 ] and D ∑ j ( S j x ) 2 in a magnetic field H along z -axis, using numerical diagonalization up to N =14 and the finite-size scaling and phenomenological renormalization. We give the phase diagrams in the H E and H D planes repectively at T =0. It is found that a magnetic field breaking the axial symmetry of the Hamiltonian brings about canted Néel order for H c1 < H < H c2 , if the system has the Haldane gap. The two transitions at H c1 and H c2 are revealed to belong to the same universality class as the two-dimensional Ising model. In addition it is found that even if the system is in the large- D phase ( D > D c ≈1), the same transitions occur in the region where the value of D is about 1.0∼1.5.
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.
One-dimensional S=1S=1S=1 XXZ\mathrm{XXZ}XXZ chains with uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite-size systems. The numerical data are analyzed using conformal field theory, the level spectroscopy, phenomenological renormalization-group, and finite-size scaling method. We thus present a quantitatively reliable ground-state phase diagram of this model. The ground states of this model contain the Haldane phase, large-D phase, N\'eel phase, two XY\mathrm{XY}XY phases, and the ferromagnetic phase. There are four different types of transitions between these phases: the Brezinskii-Kosterlitz-Thouless-type transitions, the Gaussian-type transitions, the Ising-type transitions, and the first-order transitions. The locations of these critical lines are accurately determined.
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.
Ground-state properties of the spin-1 antiferromagnetic Heisenberg chain with bond alternation and uniaxial single-ion-type anisotropy, which is described by the Hamiltonian \({\cal H}_0 = \sum_{\ell=1}^N\{1-(-1)^\ell\alpha\}{\mib S}_\ell\cdot{\mib S}_{\ell+1} + D\sum_{\ell=1}^N\;(S_\ell^z)^2\) (0.0 ≤α≤1.0, -∞< D < ∞), are studied. An exact diagonalization method by means of the Lanczös technique is used. The ground-state phase diagram on the α versus D plane at the thermodynamic limit is determined by analyzing the Binder parameters associated with Néel order and z -direction string order for finite-size ( N = 6, 8, …, 16) chains. It is found that there exists a multicritical point at which the Haldane, Néel, and singlet-dimer phases meet and that the singlet-dimer phase and the large- D phase belong to the same unique phase. The ground-state magnetization curves at the thermodynamic limit for various values of α and D are obtained for the system described by the Hamiltonian, H 0 plus the Zeeman term - H ∑ ℓ=1 N S ℓ z . The results show that the 1/2-plateau appears in the magnetization curve at least when D ≥0.0 and 0.0≲α≤1.00.0\lesssim\alpha \leq 1.00.0≲α≤1.0.
We study the spin-correlation functions of the one-dimensional S=1 antiferromagnetic Heisenberg-Ising model by the large-cluster-decomposition Monte Carlo method. It is found that this model shows a second-order transition at the anisotropy parameter ${\ensuremath{\lambda}}_{c}$=1.167\ifmmode\pm\else\textpm\fi{}0.007. The obtained critical exponents (\ensuremath{\eta}=0.253\ifmmode\pm\else\textpm\fi{}0.002, \ensuremath{\beta}=0.126\ifmmode\pm\else\textpm\fi{}0.007, \ensuremath{\nu}=0.98\ifmmode\pm\else\textpm\fi{}0.02) strongly suggest that this transition belongs to the same universality class as the two-dimensional Ising model.