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Physical review. B, Condensed matter

Anisotropy effects on the magnetic properties of anS=1 antiferromagnetic Heisenberg chain

Shoji Yamamoto, Seiji Miyashita

Published Sep 1, 1994• 42 citations
Publisher

Abstract

Effects of an uniaxial single-ion anisotropy ${\mathit{tsum}}_{\mathit{i}}$${\mathit{D}}_{\mathit{z}}$(${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${)}^{2}$ on the magnetic properties of an S=1 antiferromagnetic Heisenberg chain with open boundaries are studied by a quantum Monte Carlo method. In the ground stae, how the anisotropy affects the staggered magnetization 〈${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$〉 and the spin correlation functions 〈${\mathit{S}}_{1}^{\mathit{z}}$${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$〉 and 〈${\mathit{S}}_{1}^{\mathit{x}}$${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$〉 is investigated. Amplitudes of 〈${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$〉 decrease with with increase of ${\mathit{D}}_{\mathit{z}}$. The correlation length ${\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{z}}$ of 〈${\mathit{S}}_{1}^{\mathit{z}}$${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$〉 monotonously decreases with increase of ${\mathit{D}}_{\mathit{z}}$, while the correlation length ${\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{x}}$ of 〈${\mathit{S}}_{1}^{\mathit{x}}$${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$〉 has the opposite but weak ${\mathit{D}}_{\mathit{z}}$ dependence. The fourfold degeneracy of the ground state also holds with nonzero ${\mathit{D}}_{\mathit{z}}$'s in the thermodynamic limit although it is lifted up in finite chains. At finite temperatures, the magnetic susceptibilities along the z axis (${\mathrm{\ensuremath{\chi}}}^{\mathit{z}}$) and the x axis (${\mathrm{\ensuremath{\chi}}}^{\mathit{x}}$) are calculated. Temperature dependences of the bulk susceptibilites, ${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}^{\mathit{x}}$, are consistent with the previous experimental result on Ni(${\mathrm{C}}_{2}$${\mathrm{H}}_{8}$${\mathrm{N}}_{2}$${)}_{2}$${\mathrm{NO}}_{2}$${\mathrm{ClO}}_{4}$ (NENP). Finite-size corrections, ${\mathrm{\ensuremath{\chi}}}_{0}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{0}^{\mathit{x}}$, and the Curie susceptibilities resulting from chain ends, ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}^{\mathit{x}}$, are also investigated in detail. Both ${\mathrm{\ensuremath{\chi}}}_{0}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$ diverge as temperature goes to zero and they are combined through a relation ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$=${\mathrm{\ensuremath{\chi}}}_{0}$+${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}$. Temperature dependences of ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$ are dominated by the edge effect at low temperatures, while at high temperatures they are attributed to an S=1 free spin.

DOI
10.1103/physrevb.50.6277

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Effects of an uniaxial single-ion anisotropy {\mathit{tsum}}_{\mathit{i}}$${\mathit{D}}_{\mathit{z}}({\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${)}^{2} on the magnetic properties of an S=1 antiferromagnetic Heisenberg chain with open boundaries are studied by a quantum Monte Carlo method. In the ground stae, how the anisotropy affects the staggered magnetization 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and the spin correlation functions 〈{\mathit{S}}_{1}^{\mathit{z}}$${\mathit{S}}_{\mathit{i}}^{\mathit{z}}〉 and 〈{\mathit{S}}_{1}^{\mathit{x}}$${\mathit{S}}_{\mathit{i}}^{\mathit{x}}〉 is investigated. Amplitudes of 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 decrease with with increase of Dz{\mathit{D}}_{\mathit{z}}Dz​. The correlation length \ensuremathξSSz{\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{z}}\ensuremathξSSz​ of 〈{\mathit{S}}_{1}^{\mathit{z}}$${\mathit{S}}_{\mathit{i}}^{\mathit{z}}〉 monotonously decreases with increase of Dz{\mathit{D}}_{\mathit{z}}Dz​, while the correlation length \ensuremathξSSx{\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{x}}\ensuremathξSSx​ of 〈{\mathit{S}}_{1}^{\mathit{x}}$${\mathit{S}}_{\mathit{i}}^{\mathit{x}}〉 has the opposite but weak Dz{\mathit{D}}_{\mathit{z}}Dz​ dependence. The fourfold degeneracy of the ground state also holds with nonzero Dz{\mathit{D}}_{\mathit{z}}Dz​'s in the thermodynamic limit although it is lifted up in finite chains. At finite temperatures, the magnetic susceptibilities along the z axis (\ensuremathχz{\mathrm{\ensuremath{\chi}}}^{\mathit{z}}\ensuremathχz) and the x axis (\ensuremathχx{\mathrm{\ensuremath{\chi}}}^{\mathit{x}}\ensuremathχx) are calculated. Temperature dependences of the bulk susceptibilites, \ensuremathχ\ensuremath∞z{\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}^{\mathit{z}}\ensuremathχ\ensuremath∞z​ and \ensuremathχ\ensuremath∞x{\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}^{\mathit{x}}\ensuremathχ\ensuremath∞x​, are consistent with the previous experimental result on Ni({\mathrm{C}}_{2}$${\mathrm{H}}_{8}$${\mathrm{N}}_{2}$${)}_{2}$${\mathrm{NO}}_{2}$${\mathrm{ClO}}_{4} (NENP). Finite-size corrections, \ensuremathχ0z{\mathrm{\ensuremath{\chi}}}_{0}^{\mathit{z}}\ensuremathχ0z​ and \ensuremathχ0x{\mathrm{\ensuremath{\chi}}}_{0}^{\mathit{x}}\ensuremathχ0x​, and the Curie susceptibilities resulting from chain ends, \ensuremathχCz{\mathrm{\ensuremath{\chi}}}_{\mathit{C}}^{\mathit{z}}\ensuremathχCz​ and \ensuremathχCx{\mathrm{\ensuremath{\chi}}}_{\mathit{C}}^{\mathit{x}}\ensuremathχCx​, are also investigated in detail. Both \ensuremathχ0{\mathrm{\ensuremath{\chi}}}_{0}\ensuremathχ0​ and \ensuremathχC{\mathrm{\ensuremath{\chi}}}_{\mathit{C}}\ensuremathχC​ diverge as temperature goes to zero and they are combined through a relation \ensuremathχC{\mathrm{\ensuremath{\chi}}}_{\mathit{C}}\ensuremathχC​=\ensuremathχ0{\mathrm{\ensuremath{\chi}}}_{0}\ensuremathχ0​+\ensuremathχ\ensuremath∞{\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}\ensuremathχ\ensuremath∞​. Temperature dependences of \ensuremathχC{\mathrm{\ensuremath{\chi}}}_{\mathit{C}}\ensuremathχC​ are dominated by the edge effect at low temperatures, while at high temperatures they are attributed to an S=1 free spin.
Properties of the ground state of open chains of the S=1 Heisenberg antiferromagnet are studied by a quantum Monte Carlo method. Size (length) dependences of the ground-state energy, the energy gap, the staggered magnetization 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and the spin-correlation function 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 are investigated under free, fixed, and periodic boundary conditions. The size dependences are significant in chains with less than 25 spins, where the magnitude of the staggered magnetization is rather large. However, the fourfold degeneracy of the ground state and the exponential decay of correlations that are inherent to the Haldane system are recovered when the chain length becomes long enough. Both the staggered magnetization and the spin-correlation function decay exponentially. The correlation length is about six in lattice spacing, which agrees with the value obtained in the bulk.
A one-dimensional S=1 Heisenberg antiferromagnet in a magnetic field H (\ensuremath{\parallel}z axis) at T=0 is studied by numerical diagonalizations up to N=16. We give the magnetization curve at the thermodynamic limit, and derive an anomaly at Hc1{\mathit{H}}_{\mathit{c}1}Hc1​ (=\ensuremath{\Delta}), where \ensuremath{\Delta} is the Haldane gap. It is also found that the transverse spin correlation has the asymptotic form 〈S0x{\mathit{S}}_{0}^{\mathit{x}}S0x​Srx{\mathit{S}}_{\mathit{r}}^{\mathit{x}}Srx​〉\ensuremath{\sim}(-1)r{)}^{\mathit{r}})r${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$, and the transverse staggered susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{st}}^{\mathit{x}\mathit{x}}$ diverges between Hc1{\mathit{H}}_{\mathit{c}1}Hc1​ and Hc2{\mathit{H}}_{\mathit{c}2}Hc2​ (=4). The exponent \ensuremath{\eta} has a minimum (\ensuremath{\eta}\ensuremath{\simeq}0.3) at magnetization m\ensuremath{\simeq}1/3 and \ensuremath{\eta}\ensuremath{\simeq}0.5 at Hc1{\mathit{H}}_{\mathit{c}1}Hc1​ and Hc2{\mathit{H}}_{\mathit{c}2}Hc2​. If the system is quasi-one-dimensional, even small interchain couplings can create a canted N\'eel order, within a mean-field approximation for interchain interactions. This is consistent with a recent NMR measurement for Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​(ClO4{\mathrm{ClO}}_{4}ClO4​) (NENP) at low temperature.
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.
We study the dispersion of magnetic excitations in a Heisenberg antiferromagnetic chain of spin S=1 with exchange J and single-ion anisotropy D. Chains of length up to 18 sites are studied by a Lanczo\ifmmode\ddot\else\textasciidieresis\fi{}s method. For the Haldane phase DDc{\mathit{D}}_{\mathit{c}}Dc​\ensuremath{\approxeq}J we show that the gap at zero momentum and Sz{\mathit{S}}^{\mathit{z}}Sz=1 is given by the sum of the gaps in the sectors Sz{\mathit{S}}^{\mathit{z}}Sz=1 and Sz{\mathit{S}}^{\mathit{z}}Sz=0 at momentum \ensuremath{\pi}. The transition region near D=J is likely to be reduced to an isolated point. Large-D perturbation theory is smoothly recovered as soon as D>2J. We discuss the implications for experiments on Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​ClO4{\mathrm{ClO}}_{4}ClO4​ (NENP) as well as CsFeBr3{\mathrm{CsFeBr}}_{3}CsFeBr3​.
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The neutron-scattering intensity and energy eigenvalues of the elementary excitations of the S=1 antiferromagnetic Heisenberg chain with single-ion anisotropy [scrH=J tsumi=1N{\mathit{tsum}}_{\mathit{i}=1}^{\mathit{N}}tsumi=1N​ Si{\mathbf{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Si+1{\mathbf{S}}_{\mathit{i}+1}Si+1​+D tsum(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2] are calculated with a numerical diagonalization method and a projector Monte Carlo method. We set D/J=0.2,0.0,-0.2 and N=14,16,18,32. For D\ensuremath{\ne}0, the elementary excitations in the subspace of Sz{\mathit{S}}^{\mathit{z}}Sz=\ifmmode\pm\else\textpm\fi{}1 and in the subspace of Sz{\mathit{S}}^{\mathit{z}}Sz=0 are different. The Sz{\mathit{S}}^{\mathit{z}}Sz=\ifmmode\pm\else\textpm\fi{}1 branch is lower than the Sz{\mathit{S}}^{\mathit{z}}Sz=0 branch near Q=\ensuremath{\pi}; however, it becomes slightly higher in other regions. The crossing of levels occurs at Q\ensuremath{\simeq}\ifmmode\pm\else\textpm\fi{}0.7\ensuremath{\pi}. We find that the low-lying excited state has a weight of more than 90% in the dynamical structure factor, except for Q\ensuremath{\simeq}0. Our results explain recent experimental results on Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​ClO4{\mathrm{ClO}}_{4}ClO4​.