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Physical review. B, Condensed matter

Dispersion of magnetic excitations in a spin-1 chain with easy-plane anisotropy

O. Golinelli, Th. Jolicœur, R. Lacaze

Published Nov 1, 1992• 69 citations
Publisher

Abstract

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 D${\mathit{D}}_{\mathit{c}}$\ensuremath{\approxeq}J we show that the gap at zero momentum and ${\mathit{S}}^{\mathit{z}}$=1 is given by the sum of the gaps in the sectors ${\mathit{S}}^{\mathit{z}}$=1 and ${\mathit{S}}^{\mathit{z}}$=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(${\mathrm{C}}_{2}$${\mathrm{H}}_{8}$${\mathrm{N}}_{2}$${)}_{2}$${\mathrm{NO}}_{2}$${\mathrm{ClO}}_{4}$ (NENP) as well as ${\mathrm{CsFeBr}}_{3}$.

DOI
10.1103/physrevb.46.10854

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Elementary excitations of an anisotropic spin-1 chain

Minoru Takahashi

The neutron-scattering intensity and energy eigenvalues of the elementary excitations of the S=1 antiferromagnetic Heisenberg chain with single-ion anisotropy [scrH=J ${\mathit{tsum}}_{\mathit{i}=1}^{\mathit{N}}$ ${\mathbf{S}}_{\mathit{i}}$\ensuremath{\cdot}${\mathbf{S}}_{\mathit{i}+1}$+D tsum(${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${)}^{2}$] are calculated with a numerical diagonalization method and a projector Monte Carlo method. We set D/J=0.2,0.0,-0.2 and N=14,16,18,32. For D\ensuremath{\ne}0, the elementary excitations in the subspace of ${\mathit{S}}^{\mathit{z}}$=\ifmmode\pm\else\textpm\fi{}1 and in the subspace of ${\mathit{S}}^{\mathit{z}}$=0 are different. The ${\mathit{S}}^{\mathit{z}}$=\ifmmode\pm\else\textpm\fi{}1 branch is lower than the ${\mathit{S}}^{\mathit{z}}$=0 branch near Q=\ensuremath{\pi}; however, it becomes slightly higher in other regions. The crossing of levels occurs at Q\ensuremath{\simeq}\ifmmode\pm\else\textpm\fi{}0.7\ensuremath{\pi}. We find that the low-lying excited state has a weight of more than 90% in the dynamical structure factor, except for Q\ensuremath{\simeq}0. Our results explain recent experimental results on Ni(${\mathrm{C}}_{2}$${\mathrm{H}}_{8}$${\mathrm{N}}_{2}$${)}_{2}$${\mathrm{NO}}_{2}$${\mathrm{ClO}}_{4}$.

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

Shoji Yamamoto, Seiji Miyashita

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.

Physical review. B, Condensed matter•Sep 1, 1994•42 citations

S=1 antiferromagnetic Heisenberg chain in a magnetic field

Tôru Sakai, Minoru Takahashi

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 ${\mathit{H}}_{\mathit{c}1}$ (=\ensuremath{\Delta}), where \ensuremath{\Delta} is the Haldane gap. It is also found that the transverse spin correlation has the asymptotic form 〈${\mathit{S}}_{0}^{\mathit{x}}$${\mathit{S}}_{\mathit{r}}^{\mathit{x}}$〉\ensuremath{\sim}(-1${)}^{\mathit{r}}$${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$, and the transverse staggered susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{st}}^{\mathit{x}\mathit{x}}$ diverges between ${\mathit{H}}_{\mathit{c}1}$ and ${\mathit{H}}_{\mathit{c}2}$ (=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 ${\mathit{H}}_{\mathit{c}1}$ and ${\mathit{H}}_{\mathit{c}2}$. 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(${\mathrm{C}}_{2}$${\mathrm{H}}_{8}$${\mathrm{N}}_{2}$${)}_{2}$${\mathrm{NO}}_{2}$(${\mathrm{ClO}}_{4}$) (NENP) at low temperature.

Physical review. B, Condensed matter•Jun 1, 1991•150 citations
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({\mathrm{C}}_{2}$${\mathrm{H}}_{8}$${\mathrm{N}}_{2}$${)}_{2}$${\mathrm{NO}}_{2}$${\mathrm{ClO}}_{4} (NENP) as well as CsFeBr3{\mathrm{CsFeBr}}_{3}CsFeBr3​.
We study a Heisenberg antiferromagnetic chain of spin S=1 with nearest-neighbor exchange interaction J and easy-plane single-ion anisotropy D. We follow the splitting of the Haldane gap into two components as a function of the anisotropy. Chains of length up to 16 sites are studied by a Lancz\"os method and longer chains up to 32 sites have been investigated by a quantum Monte Carlo simulation for D/J=0.10, 0.15, 0.20, and 0.25. A careful finite-size scaling treatment of our data leads to values that are directly relevant to the interpretation of 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). We discuss the relationship between our results and the magnetic-susceptibility measurements as well as the inelastic-neutron-scattering data and on the Haldane conjecture. Values are proposed for the parameters D and J in the case of NENP.
We investigate the magnetic field behaviour of an antiferromagnetic Heisenberg spin-1 chain with the most general single-ion anisotropy. We discuss the regime in which the magnetic field is below the transition value. The splitting of the Haldane triplet is obtained as a function of a field applied in an arbitrary orientation by means of a Lanczos exact diagonalization of chains of up to 16 spins. Our results are well summarized in terms of a first-order perturbation theory. We explain various level crossings that occur by the existence of discrete symmetries. A discussion is given of the electron-spin-resonance and neutron-scattering experiments on the compound Ni(C 2 H 8 N 2 ) 2 NO 2 ClO 4 (NENP).
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​.
Effects of an uniaxial single-ion anisotropy tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​Dz{\mathit{D}}_{\mathit{z}}Dz​(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})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 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and 〈S1x{\mathit{S}}_{1}^{\mathit{x}}S1x​Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​〉 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 ${\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{z}}$ of 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 monotonously decreases with increase of Dz{\mathit{D}}_{\mathit{z}}Dz​, while the correlation length ${\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{x}}$ of 〈S1x{\mathit{S}}_{1}^{\mathit{x}}S1x​Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​〉 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 (${\mathrm{\ensuremath{\chi}}}^{\mathit{z}}$) 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(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). 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.
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.