Physical Review

Linear Magnetic Chains with Anisotropic Coupling

Jill C. Bonner, Michael Ellis Fisher

Published Aug 3, 1964• 2,497 citations

Abstract

Linear chains (and rings) of $S=\frac{1}{2}$ spins with the anisotropic (Ising-Heisenberg) Hamiltonian $\mathcal{H}=\ensuremath{-}2J\ensuremath{\Sigma}\stackrel{N}{i=1}{{{S}_{i}}^{z}{{S}_{i+1}}^{z}+\ensuremath{\gamma}({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})}\ensuremath{-}g\ensuremath{\beta}\ensuremath{\Sigma}\stackrel{N}{i=1}\mathrm{H}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{i}$ have been studied by exact machine calculations for $N=2 \mathrm{to} 11$, $\ensuremath{\gamma}=0 \mathrm{to} 1$ and for ferro- and antiferro-magnetic coupling. The results reveal the dependence on finite size and anisotropy of the spectrum and dispersion laws, of the energy, entropy, and specific heat, of the magnetization and susceptibilities, and of the pair correlations. The limiting $N\ensuremath{\rightarrow}\ensuremath{\infty}$ behavior is accurately indicated, for all $\ensuremath{\gamma}$, in the region $\frac{\mathrm{kT}}{|J|}>~0.5$ which includes the maxima in the specific heat and susceptibility. The behavior of thermal and magnetic properties of infinite chains at lower temperatures is estimated by extrapolation. For infinite antiferromagnetic chains the ground-state degeneracy, the anisotropy gap, and the magnetization, perpendicular susceptibility, and pair correlations at $T=0$ are similarly studied. Estimates of the long-range order suggest that it vanishes only at the Heisenberg limit $\ensuremath{\gamma}=1$ and confirm the accuracy of Walker's perturbation series in $\ensuremath{\gamma}$.

DOI: 10.1103/physrev.135.a640

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Theory of Alternating Antiferromagnetic Heisenberg Linear Chains

William Duffy, Kevin P. Barr

Physical ReviewJan 10, 1968227 citations

Ferrimagnetic Heisenberg chains [(1/2-S] (S=1to (5/2): Thermal and magnetic properties

Marc Drillon, Eugenio Coronado, Roland Georges +2 more

Physical review. B, Condensed matterDec 1, 1989101 citations

Low-temperature thermodynamic properties of a random anisotropic antiferromagnetic chain

J. E. Hirsch

Physical review. B, Condensed matterDec 1, 198048 citations

Numerical study of antiferromagnetic Heisenberg-Ising chains for spin s=1/2

Djalrna Medeiros, G. G. Cabrera

Jan 1, 19910 citations

Statistical Mechanics of the Anisotropic Linear Heisenberg Model

Shigetoshi Katsura

Physical ReviewSep 1, 1962787 citations

Linear Antiferromagnetic Chain with Anisotropic Coupling

R. Orbach

Physical ReviewOct 15, 1958532 citations

Physical Review

Linear Magnetic Chains with Anisotropic Coupling

Jill C. Bonner, Michael Ellis Fisher

Published Aug 3, 1964• 2,497 citations

Publisher

Abstract

Linear chains (and rings) of

S=\frac{1}{2}

spins with the anisotropic (Ising-Heisenberg) Hamiltonian

\mathcal{H}=\ensuremath{-}2J\ensuremath{\Sigma}\stackrel{N}{i=1}{{{S}_{i}}^{z}{{S}_{i+1}}^{z}+\ensuremath{\gamma}({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})}\ensuremath{-}g\ensuremath{\beta}\ensuremath{\Sigma}\stackrel{N}{i=1}\mathrm{H}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{i}

have been studied by exact machine calculations for

N=2 \mathrm{to} 11

\ensuremath{\gamma}=0 \mathrm{to} 1

and for ferro- and antiferro-magnetic coupling. The results reveal the dependence on finite size and anisotropy of the spectrum and dispersion laws, of the energy, entropy, and specific heat, of the magnetization and susceptibilities, and of the pair correlations. The limiting

N\ensuremath{\rightarrow}\ensuremath{\infty}

behavior is accurately indicated, for all

\ensuremath{\gamma}

, in the region

\frac{\mathrm{kT}}{|J|}>~0.5

which includes the maxima in the specific heat and susceptibility. The behavior of thermal and magnetic properties of infinite chains at lower temperatures is estimated by extrapolation. For infinite antiferromagnetic chains the ground-state degeneracy, the anisotropy gap, and the magnetization, perpendicular susceptibility, and pair correlations at

T=0

are similarly studied. Estimates of the long-range order suggest that it vanishes only at the Heisenberg limit

\ensuremath{\gamma}=1

and confirm the accuracy of Walker's perturbation series in

\ensuremath{\gamma}

DOI: 10.1103/physrev.135.a640

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Theory of Alternating Antiferromagnetic Heisenberg Linear Chains

William Duffy, Kevin P. Barr

The exact eigenvalue spectrum and thermodynamic properties of the spin Hamiltonian $\mathcal{H}=2|J|\ensuremath{\Sigma}\stackrel{\frac{N}{2}}{i=1}({\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i\ensuremath{-}1}+a{\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i+1})$ are calculated for short chains of $N=4,6,8, \mathrm{and} 10$ spins, each of spin \textonehalf{}, for alternation parameter $a=0.2,0.4,0.6, \mathrm{and} 0.8$ , with ${\mathrm{S}}_{N+1}\ensuremath{\equiv}{\mathrm{S}}_{1}$. The behavior for $N=\ensuremath{\infty}$ is estimated by extrapolation. Comparison is made with the known results for $a=0$ and 1. The ground-state energy, ground-state short-range order, energy gap, and triplet excitation spectrum are compared with various approximate theories. Zero-temperature infinite-chain magnetization curves are inferred from the finite-chain results. The energy, entropy, specific heat, and magnetic susceptibility for $N=10$ are shown to approximate well the behavior for $N=\ensuremath{\infty}$ when $\frac{\mathrm{kT}}{|J|}>a$ . The magnetic-susceptibility data on the free radical 2,2-bis ( $p$ -nitrophenyl)-1-picrylhydrazyl are shown to agree well with the theoretical results for an alternating chain with $a\ensuremath{\approx}0.6$.The exact eigenvalue spectrum and thermodynamic properties of the spin Hamiltonian $\mathcal{H}=2|J|\ensuremath{\Sigma}\stackrel{\frac{N}{2}}{i=1}({\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i\ensuremath{-}1}+a{\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i+1})$ are calculated for short chains of $N=4,6,8, \mathrm{and} 10$ spins, each of spin \textonehalf{}, for alternation parameter $a=0.2,0.4,0.6, \mathrm{and} 0.8$, with ${\mathrm{S}}_{N+1}\ensuremath{\equiv}{\mathrm{S}}_{1}$. The behavior for $N=\ensuremath{\infty}$ is estimated by extrapolation. Comparison is made with the known results for $a=0$ and 1. The ground-state energy, ground-state short-range order, energy gap, and triplet excitation spectrum are compared with various approximate theories. Zero-temperature infinite-chain magnetization curves are inferred from the finite-chain results. The energy, entropy, specific heat, and magnetic susceptibility for $N=10$ are shown to approximate well the behavior for $N=\ensuremath{\infty}$ when $\frac{\mathrm{kT}}{|J|}>a$. The magnetic-susceptibility data on the free radical 2,2-bis ($p$-nitrophenyl)-1-picrylhydrazyl are shown to agree well with the theoretical results for an alternating chain with $a\ensuremath{\approx}0.6$.

Physical ReviewJan 10, 1968227 citations

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Energy spectra and thermodynamic quantities (partition function, entropy, specific heat, susceptibility) of ferrimagnetic Heisenberg chains, made up of two spin sublattices [(1/2-S], are discussed from the exact results computed for finite rings. Extrapolation of the theoretical data versus the length of finite rings N is discussed for S ranging from 1 to (5/2 and compared with previous reported findings for the [(1/2-\ensuremath{\infty}] spin chain. In the low-temperature region, similarities between the ferrimagnetic chains [(1/2-S] and the regular ferromagnetic ones (with spin S-(1/2) are used for the determination of the limiting curves.Energy spectra and thermodynamic quantities (partition function, entropy, specific heat, susceptibility) of ferrimagnetic Heisenberg chains, made up of two spin sublattices [(1/2-S], are discussed from the exact results computed for finite rings. Extrapolation of the theoretical data versus the length of finite rings N is discussed for S ranging from 1 to (5/2 and compared with previous reported findings for the [(1/2-\ensuremath{\infty}] spin chain. In the low-temperature region, similarities between the ferrimagnetic chains [(1/2-S] and the regular ferromagnetic ones (with spin S-(1/2) are used for the determination of the limiting curves.

Physical review. B, Condensed matterDec 1, 1989101 citations

Low-temperature thermodynamic properties of a random anisotropic antiferromagnetic chain

J. E. Hirsch

The low-temperature thermodynamic properties of a spin-\textonehalf{} one-dimensional random anisotropic (Heisenberg-Ising) antiferromagnet described by the Hamiltonian $H=\ensuremath{\Sigma}{i}^{}{J}_{i}({\ensuremath{\sigma}}_{x}^{i}{\ensuremath{\sigma}}_{x}^{i+1}+{\ensuremath{\sigma}}_{y}^{i}{\ensuremath{\sigma}}_{y}^{i+1}+\ensuremath{\gamma}{\ensuremath{\sigma}}_{z}^{i}{\ensuremath{\sigma}}_{z}^{i+1})$ are studied as a function of disorder and anisotropy. The ${J}_{i}>~0$ are independent random variables obeying a probability distribution $P(J)$ , and $0<~\ensuremath{\gamma}<~\ensuremath{\infty}$. The approach used is a numerical implementation of a real-space renormalization-group (RG) method previously introduced. The isotropic Heisenberg case ($\ensuremath{\gamma}=1$), the $\mathrm{XY}$ case ($\ensuremath{\gamma}=0$). and the Ising case ($\ensuremath{\gamma}=\ensuremath{\infty}$) are fixed points of the RG transformation. It is found that in the $\mathrm{XY}$ region $\ensuremath{\gamma}<~1$, including the Heisenberg point, the system exhibits singular behavior in the thermodynamic properties for arbitrary probability distributions for the couplings. For the $\mathrm{XY}$ limit ($\ensuremath{\gamma}=0$) this is in agreement with known exact results. The functional form for the low-temperature susceptibility is found to be $\ensuremath{\chi}\ensuremath{\sim}\frac{1}{(T{\mathrm{ln}}^{m}(\frac{T}{{T}_{0}}))}$ in the entire region $0<~\ensuremath{\gamma}<~1$ for arbitrary probability distributions. In the Ising region ($\ensuremath{\gamma}>1$) the susceptibility shows an approximate power-law divergence for small anisotropy but goes eventually to zero as $T\ensuremath{\rightarrow}0$. Possible relevance of these results to recent experiments on Qn ${(\mathrm{TCNQ})}_{2}$ is discussed.The low-temperature thermodynamic properties of a spin-\textonehalf{} one-dimensional random anisotropic (Heisenberg-Ising) antiferromagnet described by the Hamiltonian $H=\ensuremath{\Sigma}{i}^{}{J}_{i}({\ensuremath{\sigma}}_{x}^{i}{\ensuremath{\sigma}}_{x}^{i+1}+{\ensuremath{\sigma}}_{y}^{i}{\ensuremath{\sigma}}_{y}^{i+1}+\ensuremath{\gamma}{\ensuremath{\sigma}}_{z}^{i}{\ensuremath{\sigma}}_{z}^{i+1})$ are studied as a function of disorder and anisotropy. The ${J}_{i}>~0$ are independent random variables obeying a probability distribution $P(J)$, and $0<~\ensuremath{\gamma}<~\ensuremath{\infty}$. The approach used is a numerical implementation of a real-space renormalization-group (RG) method previously introduced. The isotropic Heisenberg case ($\ensuremath{\gamma}=1$), the $\mathrm{XY}$ case ($\ensuremath{\gamma}=0$). and the Ising case ($\ensuremath{\gamma}=\ensuremath{\infty}$) are fixed points of the RG transformation. It is found that in the $\mathrm{XY}$ region $\ensuremath{\gamma}<~1$, including the Heisenberg point, the system exhibits singular behavior in the thermodynamic properties for arbitrary probability distributions for the couplings. For the $\mathrm{XY}$ limit ($\ensuremath{\gamma}=0$) this is in agreement with known exact results. The functional form for the low-temperature susceptibility is found to be $\ensuremath{\chi}\ensuremath{\sim}\frac{1}{(T{\mathrm{ln}}^{m}(\frac{T}{{T}_{0}}))}$ in the entire region $0<~\ensuremath{\gamma}<~1$ for arbitrary probability distributions. In the Ising region ($\ensuremath{\gamma}>1$) the susceptibility shows an approximate power-law divergence for small anisotropy but goes eventually to zero as $T\ensuremath{\rightarrow}0$. Possible relevance of these results to recent experiments on Qn${(\mathrm{TCNQ})}_{2}$ is discussed.

Physical review. B, Condensed matterDec 1, 198048 citations

Numerical study of antiferromagnetic Heisenberg-Ising chains for spin s=1/2

Djalrna Medeiros, G. G. Cabrera

The Heisenberg-Ising Hamiltonian has been exactly solved us- ing the Lanczos method, for chain of up to N = 28 spins. In the whole anisotropic region, the mass gap displays a dominant exponential behavior with size. The extrapolated ground state is doubly degenerate, and presents long-range order with dominant components of the Neel type. Quantum fluctuations become more important as long as the anisotropy is reduced, leading to a disordered singlet ground state for the infinite system at the isotropic point.The Heisenberg-Ising Hamiltonian has been exactly solved us- ing the Lanczos method, for chain of up to N = 28 spins. In the whole anisotropic region, the mass gap displays a dominant exponential behavior with size. The extrapolated ground state is doubly degenerate, and presents long-range order with dominant components of the Neel type. Quantum fluctuations become more important as long as the anisotropy is reduced, leading to a disordered singlet ground state for the infinite system at the isotropic point.

Jan 1, 19910 citations

Statistical Mechanics of the Anisotropic Linear Heisenberg Model

Shigetoshi Katsura

The anisotropic Hamiltonian, $H=\ensuremath{-}\frac{1}{2}\ensuremath{\Sigma}({J}_{x}{{\ensuremath{\sigma}}_{l}}^{x}{{\ensuremath{\sigma}}_{l+1}}^{x}+{J}_{y}{{\ensuremath{\sigma}}_{l}}^{y}{{\ensuremath{\sigma}}_{l+1}}^{y}+{J}_{z}{{\ensuremath{\sigma}}_{l}}^{z}{{\ensuremath{\sigma}}_{l+1}}^{z})\ensuremath{-}m\mathcal{H}\ensuremath{\Sigma}{{\ensuremath{\sigma}}_{l}}^{z},$ of the linear spin array in the Heisenberg model of magnetism is examined. The eigenstate and the partition function for the case ${J}_{z}=0$ are obtained exactly for a finite system and for an infinite system with the aid of annihilation and creation operators, and the free energy $F$ of the latter is given by $\ensuremath{-}\frac{F}{\mathrm{NkT}}=(\frac{1}{\ensuremath{\pi}})\ensuremath{\int}{0}^{\ensuremath{\pi}}\mathrm{ln}{2cosh{[{{K}_{x}}^{2}+{{K}_{y}}^{2}+2{K}_{x}{K}_{y}cos2\ensuremath{\omega}\ensuremath{-}2C({K}_{x}+{K}_{y})cos\ensuremath{\omega}+{C}^{2}]}^{\frac{1}{2}}}d\ensuremath{\omega},$ where ${K}_{x}=\frac{{J}_{x}}{2kT}$ , ${K}_{y}=\frac{{J}_{y}}{2kT}$ , $C=\frac{m\mathcal{H}}{\mathrm{kT}}$ . The case ${J}_{x}={J}_{y}={J}_{z}=J$ is discussed with the aid of a high-temperature expansion and of analysis of small systems. Specific heats and susceptibilities in special cases: (i) ${J}_{x}={J}_{y}=J$ , ${J}_{z}=0$ , (ii) ${J}_{x}=J$ , ${J}_{y}={J}_{z}=0$ , ( ${\mathrm{iii}}_{\mathrm{f}}$ ) ${J}_{x}={J}_{y}=0$ , ${J}_{z}=J>0$ , ( ${\mathrm{iii}}_{\mathrm{a}}$ ) ${J}_{x}={J}_{y}=0$ , ${J}_{z}=J<0$ , ( ${\mathrm{iv}}_{\mathrm{f}}$ ) ${J}_{x}={J}_{y}={J}_{z}=J>0$ , ( ${\mathrm{iv}}_{\mathrm{a}}$ ) ${J}_{x}={J}_{y}={J}_{z}=J<0$ are compared and it is shown that (i), ( ${\mathrm{iii}}_{\mathrm{a}}$ ), and ( ${\mathrm{iv}}_{\mathrm{a}}$ ) have the characteristic features of the observed parallel susceptibility of an antiferromagnetic substance, (ii) those of perpendicular susceptibility, and ( ${\mathrm{iii}}_{\mathrm{f}}$ ) and ( ${\mathrm{iv}}_{\mathrm{f}}$ ) those of paramagnetic susceptibility, even though they have no singularities. The distribution of the zeros of the partition function is also discussed.The anisotropic Hamiltonian, $H=\ensuremath{-}\frac{1}{2}\ensuremath{\Sigma}({J}_{x}{{\ensuremath{\sigma}}_{l}}^{x}{{\ensuremath{\sigma}}_{l+1}}^{x}+{J}_{y}{{\ensuremath{\sigma}}_{l}}^{y}{{\ensuremath{\sigma}}_{l+1}}^{y}+{J}_{z}{{\ensuremath{\sigma}}_{l}}^{z}{{\ensuremath{\sigma}}_{l+1}}^{z})\ensuremath{-}m\mathcal{H}\ensuremath{\Sigma}{{\ensuremath{\sigma}}_{l}}^{z},$ of the linear spin array in the Heisenberg model of magnetism is examined. The eigenstate and the partition function for the case ${J}_{z}=0$ are obtained exactly for a finite system and for an infinite system with the aid of annihilation and creation operators, and the free energy $F$ of the latter is given by $\ensuremath{-}\frac{F}{\mathrm{NkT}}=(\frac{1}{\ensuremath{\pi}})\ensuremath{\int}{0}^{\ensuremath{\pi}}\mathrm{ln}{2cosh{[{{K}_{x}}^{2}+{{K}_{y}}^{2}+2{K}_{x}{K}_{y}cos2\ensuremath{\omega}\ensuremath{-}2C({K}_{x}+{K}_{y})cos\ensuremath{\omega}+{C}^{2}]}^{\frac{1}{2}}}d\ensuremath{\omega},$ where ${K}_{x}=\frac{{J}_{x}}{2kT}$, ${K}_{y}=\frac{{J}_{y}}{2kT}$, $C=\frac{m\mathcal{H}}{\mathrm{kT}}$. The case ${J}_{x}={J}_{y}={J}_{z}=J$ is discussed with the aid of a high-temperature expansion and of analysis of small systems. Specific heats and susceptibilities in special cases: (i) ${J}_{x}={J}_{y}=J$, ${J}_{z}=0$, (ii) ${J}_{x}=J$, ${J}_{y}={J}_{z}=0$, (${\mathrm{iii}}_{\mathrm{f}}$) ${J}_{x}={J}_{y}=0$, ${J}_{z}=J>0$, (${\mathrm{iii}}_{\mathrm{a}}$) ${J}_{x}={J}_{y}=0$, ${J}_{z}=J<0$, (${\mathrm{iv}}_{\mathrm{f}}$) ${J}_{x}={J}_{y}={J}_{z}=J>0$, (${\mathrm{iv}}_{\mathrm{a}}$) ${J}_{x}={J}_{y}={J}_{z}=J<0$ are compared and it is shown that (i), (${\mathrm{iii}}_{\mathrm{a}}$), and (${\mathrm{iv}}_{\mathrm{a}}$) have the characteristic features of the observed parallel susceptibility of an antiferromagnetic substance, (ii) those of perpendicular susceptibility, and (${\mathrm{iii}}_{\mathrm{f}}$) and (${\mathrm{iv}}_{\mathrm{f}}$) those of paramagnetic susceptibility, even though they have no singularities. The distribution of the zeros of the partition function is also discussed.

Physical ReviewSep 1, 1962787 citations

Linear Antiferromagnetic Chain with Anisotropic Coupling

R. Orbach

The exact solution is given for a linear chain of $N$ atoms of spin \textonehalf{} coupled together by the anisotropic Hamiltonian $\mathcal{H}=2J\ensuremath{\Sigma}\stackrel{N}{i=1}[{{S}_{i}}^{z}{{S}_{i+1}}^{z}+(1\ensuremath{-}\ensuremath{\alpha})({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})].$ The energy of the antiferromagnetic ground state is computed and comparison is made with a variational method. The parameter $\ensuremath{\alpha}$ is allowed to vary between 0 and 1, regulating the relative amount of Ising anisotropy. The short-range order, $\ensuremath{\Sigma}{i}^{}{{S}_{i}}^{z}{{S}_{i+1}}^{z}$, is calculated exactly from the variation of the ground-state energy with $\ensuremath{\alpha}$. It is shown that a kink in the short-range order curve calculated using the variational method is fictitious, and the associated discontinuity in $\frac{{\ensuremath{\partial}}^{2}E}{\ensuremath{\partial}{\ensuremath{\alpha}}^{2}}$ is nonexistent. A discussion is given of long-range order and criticisms are presented regarding the predictions of the variational method.The exact solution is given for a linear chain of $N$ atoms of spin \textonehalf{} coupled together by the anisotropic Hamiltonian $\mathcal{H}=2J\ensuremath{\Sigma}\stackrel{N}{i=1}[{{S}_{i}}^{z}{{S}_{i+1}}^{z}+(1\ensuremath{-}\ensuremath{\alpha})({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})].$ The energy of the antiferromagnetic ground state is computed and comparison is made with a variational method. The parameter $\ensuremath{\alpha}$ is allowed to vary between 0 and 1, regulating the relative amount of Ising anisotropy. The short-range order, $\ensuremath{\Sigma}{i}^{}{{S}_{i}}^{z}{{S}_{i+1}}^{z}$, is calculated exactly from the variation of the ground-state energy with $\ensuremath{\alpha}$. It is shown that a kink in the short-range order curve calculated using the variational method is fictitious, and the associated discontinuity in $\frac{{\ensuremath{\partial}}^{2}E}{\ensuremath{\partial}{\ensuremath{\alpha}}^{2}}$ is nonexistent. A discussion is given of long-range order and criticisms are presented regarding the predictions of the variational method.

Physical ReviewOct 15, 1958532 citations

Linear chains (and rings) of

S=\frac{1}{2}

spins with the anisotropic (Ising-Heisenberg) Hamiltonian

\mathcal{H}=\ensuremath{-}2J\ensuremath{\Sigma}\stackrel{N}{i=1}{{{S}_{i}}^{z}{{S}_{i+1}}^{z}+\ensuremath{\gamma}({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})}\ensuremath{-}g\ensuremath{\beta}\ensuremath{\Sigma}\stackrel{N}{i=1}\mathrm{H}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{i}

have been studied by exact machine calculations for

N=2 \mathrm{to} 11

\ensuremath{\gamma}=0 \mathrm{to} 1

N\ensuremath{\rightarrow}\ensuremath{\infty}

behavior is accurately indicated, for all

\ensuremath{\gamma}

, in the region

\frac{\mathrm{kT}}{|J|}>~0.5

T=0

are similarly studied. Estimates of the long-range order suggest that it vanishes only at the Heisenberg limit

\ensuremath{\gamma}=1

and confirm the accuracy of Walker's perturbation series in

\ensuremath{\gamma}

N=4,6,8, \mathrm{and} 10

spins, each of spin \textonehalf{}, for alternation parameter

a=0.2,0.4,0.6, \mathrm{and} 0.8

, with ${\mathrm{S}}_{N+1}\ensuremath{\equiv}{\mathrm{S}}_{1}$. The behavior for $N=\ensuremath{\infty}$ is estimated by extrapolation. Comparison is made with the known results for

a=0

and 1. The ground-state energy, ground-state short-range order, energy gap, and triplet excitation spectrum are compared with various approximate theories. Zero-temperature infinite-chain magnetization curves are inferred from the finite-chain results. The energy, entropy, specific heat, and magnetic susceptibility for

N=10

are shown to approximate well the behavior for $N=\ensuremath{\infty}$ when

\frac{\mathrm{kT}}{|J|}>a

. The magnetic-susceptibility data on the free radical 2,2-bis (

p

-nitrophenyl)-1-picrylhydrazyl are shown to agree well with the theoretical results for an alternating chain with $a\ensuremath{\approx}0.6$.

{J}_{i}>~0

are independent random variables obeying a probability distribution

P(J)

, and $0<~\ensuremath{\gamma}<~\ensuremath{\infty}$. The approach used is a numerical implementation of a real-space renormalization-group (RG) method previously introduced. The isotropic Heisenberg case ($\ensuremath{\gamma}=1$), the

\mathrm{XY}

case ($\ensuremath{\gamma}=0$). and the Ising case ($\ensuremath{\gamma}=\ensuremath{\infty}$) are fixed points of the RG transformation. It is found that in the

\mathrm{XY}

region $\ensuremath{\gamma}<~1$, including the Heisenberg point, the system exhibits singular behavior in the thermodynamic properties for arbitrary probability distributions for the couplings. For the

\mathrm{XY}

limit ($\ensuremath{\gamma}=0$) this is in agreement with known exact results. The functional form for the low-temperature susceptibility is found to be $\ensuremath{\chi}\ensuremath{\sim}\frac{1}{(T{\mathrm{ln}}^{m}(\frac{T}{{T}_{0}}))}$ in the entire region $0<~\ensuremath{\gamma}<~1$ for arbitrary probability distributions. In the Ising region ($\ensuremath{\gamma}>1$) the susceptibility shows an approximate power-law divergence for small anisotropy but goes eventually to zero as $T\ensuremath{\rightarrow}0$. Possible relevance of these results to recent experiments on Qn

{(\mathrm{TCNQ})}_{2}

is discussed.

{J}_{z}=0

are obtained exactly for a finite system and for an infinite system with the aid of annihilation and creation operators, and the free energy

F

of the latter is given by $\ensuremath{-}\frac{F}{\mathrm{NkT}}=(\frac{1}{\ensuremath{\pi}})\ensuremath{\int}{0}^{\ensuremath{\pi}}\mathrm{ln}{2cosh{[{{K}_{x}}^{2}+{{K}_{y}}^{2}+2{K}_{x}{K}_{y}cos2\ensuremath{\omega}\ensuremath{-}2C({K}_{x}+{K}_{y})cos\ensuremath{\omega}+{C}^{2}]}^{\frac{1}{2}}}d\ensuremath{\omega},$ where

{K}_{x}=\frac{{J}_{x}}{2kT}

{K}_{y}=\frac{{J}_{y}}{2kT}

C=\frac{m\mathcal{H}}{\mathrm{kT}}

. The case

{J}_{x}={J}_{y}={J}_{z}=J

is discussed with the aid of a high-temperature expansion and of analysis of small systems. Specific heats and susceptibilities in special cases: (i)

{J}_{x}={J}_{y}=J

{J}_{z}=0

, (ii)

{J}_{x}=J

{J}_{y}={J}_{z}=0

, (

{\mathrm{iii}}_{\mathrm{f}}

)

{J}_{x}={J}_{y}=0

{J}_{z}=J>0

, (

{\mathrm{iii}}_{\mathrm{a}}

)

{J}_{x}={J}_{y}=0

{J}_{z}=J<0

, (

{\mathrm{iv}}_{\mathrm{f}}

)

{J}_{x}={J}_{y}={J}_{z}=J>0

, (

{\mathrm{iv}}_{\mathrm{a}}

)

{J}_{x}={J}_{y}={J}_{z}=J<0

are compared and it is shown that (i), (

{\mathrm{iii}}_{\mathrm{a}}

), and (

{\mathrm{iv}}_{\mathrm{a}}

) have the characteristic features of the observed parallel susceptibility of an antiferromagnetic substance, (ii) those of perpendicular susceptibility, and (

{\mathrm{iii}}_{\mathrm{f}}

) and (

{\mathrm{iv}}_{\mathrm{f}}

) those of paramagnetic susceptibility, even though they have no singularities. The distribution of the zeros of the partition function is also discussed.

The exact solution is given for a linear chain of

N

atoms of spin \textonehalf{} coupled together by the anisotropic Hamiltonian $\mathcal{H}=2J\ensuremath{\Sigma}\stackrel{N}{i=1}[{{S}_{i}}^{z}{{S}_{i+1}}^{z}+(1\ensuremath{-}\ensuremath{\alpha})({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})].$ The energy of the antiferromagnetic ground state is computed and comparison is made with a variational method. The parameter $\ensuremath{\alpha}$ is allowed to vary between 0 and 1, regulating the relative amount of Ising anisotropy. The short-range order, $\ensuremath{\Sigma}{i}^{}{{S}_{i}}^{z}{{S}_{i+1}}^{z}$, is calculated exactly from the variation of the ground-state energy with $\ensuremath{\alpha}$. It is shown that a kink in the short-range order curve calculated using the variational method is fictitious, and the associated discontinuity in $\frac{{\ensuremath{\partial}}^{2}E}{\ensuremath{\partial}{\ensuremath{\alpha}}^{2}}$ is nonexistent. A discussion is given of long-range order and criticisms are presented regarding the predictions of the variational method.