Journal of Mathematical Physics

Linear Heisenberg Model of Ferro- and Antiferromagnetism

Shigetoshi Katsura, Sakari Inawashiro

Published Aug 1, 1964• 27 citations

Abstract

The partition function of the one-dimensional Heisenberg model is considered. Hamiltonian of the system H=−12 ∑ [J⊥(σlxσl+1x+σlyσl+1y)+J∥σlzσl+1z]−mH ∑ σlzis expressed in terms of Fermi operators. The term which contains J∥, the quartic term and a part of quadratic term in Fermi operators, have been regarded as perturbation, keeping the symmetry with respect to the magnetic field. Linked-cluster expansion in an appropriate form for this case has been developed and the partition function has been obtained up to the third order in J∥. Numerical values of energy, specific heat, and susceptibility up to second order in J∥ are shown. The ground-state energy is EN|J⊥|=−2π−2π2(J∥|J⊥|)−16π3(16−π2144)(J∥J⊥)2+O[(J∥J⊥)3].E/N|J∥| for the antiferromagnetic case J∥ = −|J∥| = J is −0.8899. Agreement with the exact value, −0.8863, is quite satisfactory.

DOI: 10.1063/1.1704213

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Journal of Mathematical Physics

Linear Heisenberg Model of Ferro- and Antiferromagnetism

Shigetoshi Katsura, Sakari Inawashiro

Published Aug 1, 1964• 27 citations

Publisher

Abstract

DOI: 10.1063/1.1704213

Related papers

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

Classical Heisenberg Model

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Exact expressions for the partition function, spin pair correlation function, and susceptibility of the onedimensional isotropic classical Heisenberg model are obtained in zero external field with cyclic boundary conditions. It is shown that the methods used to derive these results enable the partition functions and susceptibilities of finite clusters of interacting classical spins to be evaluated in terms of the $3n\ensuremath{-}j$ symbols of Wigner. Exact results in one dimension are also obtained for the partition function and susceptibility of a "planar" classical Heisenberg model. In this model the spin vectors interact via a Heisenberg coupling but each spin vector is restricted to lie in a plane.The anisotropic classical Heisenberg model described by the Hamiltonian $\mathcal{H}=\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{(\mathrm{ij})}2({{J}_{\mathrm{ij}}}^{x}{{s}_{i}}^{x}{{s}_{j}}^{x}+{{J}_{\mathrm{ij}}}^{y}{{s}_{i}}^{y}{{s}_{j}}^{y}+{{J}_{\mathrm{ij}}}^{z}{{s}_{i}}^{z}{{s}_{j}}^{z})\ensuremath{-}mH\ensuremath{\Sigma}\stackrel{N}{j=1}{{s}_{i}}^{z},$ where ${{s}_{i}}^{x}$ , ${{s}_{i}}^{y}$ , and ${{s}_{i}}^{z}$ are components of the unit vector ${\mathbf{s}}_{i}$ , is also considered. A perturbation series for the zero-field free energy of the anisotropic model in one dimension with nearest-neighbor interactions ${{J}_{\mathrm{ij}}}^{x}={{J}_{\mathrm{ij}}}^{y}=J$ and ${{J}_{\mathrm{ij}}}^{z}=\ensuremath{\gamma}J$ is developed in powers of $\ensuremath{\gamma}\ensuremath{-}1$ using the isotropic model as the unperturbed system. Detailed calculations are performed to third order in $\ensuremath{\gamma}\ensuremath{-}1$. It is found that the perturbation series for the energy per spin breaks down as $T\ensuremath{\rightarrow}0$. A high-temperature series expansion for the anisotropic model, which is valid for a general interaction potential and lattice, is derived by generalizing the methods developed by Horwitz and Callen for the Ising model. This series is rearranged to give a simplified diagram expansion. Finally, a practical technique for calculating the high-temperature series expansions of the zero-field free energy and susceptibility of the isotropic classical Heisenberg model is presented.Exact expressions for the partition function, spin pair correlation function, and susceptibility of the onedimensional isotropic classical Heisenberg model are obtained in zero external field with cyclic boundary conditions. It is shown that the methods used to derive these results enable the partition functions and susceptibilities of finite clusters of interacting classical spins to be evaluated in terms of the $3n\ensuremath{-}j$ symbols of Wigner. Exact results in one dimension are also obtained for the partition function and susceptibility of a "planar" classical Heisenberg model. In this model the spin vectors interact via a Heisenberg coupling but each spin vector is restricted to lie in a plane.The anisotropic classical Heisenberg model described by the Hamiltonian $\mathcal{H}=\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{(\mathrm{ij})}2({{J}_{\mathrm{ij}}}^{x}{{s}_{i}}^{x}{{s}_{j}}^{x}+{{J}_{\mathrm{ij}}}^{y}{{s}_{i}}^{y}{{s}_{j}}^{y}+{{J}_{\mathrm{ij}}}^{z}{{s}_{i}}^{z}{{s}_{j}}^{z})\ensuremath{-}mH\ensuremath{\Sigma}\stackrel{N}{j=1}{{s}_{i}}^{z},$ where ${{s}_{i}}^{x}$, ${{s}_{i}}^{y}$, and ${{s}_{i}}^{z}$ are components of the unit vector ${\mathbf{s}}_{i}$, is also considered. A perturbation series for the zero-field free energy of the anisotropic model in one dimension with nearest-neighbor interactions ${{J}_{\mathrm{ij}}}^{x}={{J}_{\mathrm{ij}}}^{y}=J$ and ${{J}_{\mathrm{ij}}}^{z}=\ensuremath{\gamma}J$ is developed in powers of $\ensuremath{\gamma}\ensuremath{-}1$ using the isotropic model as the unperturbed system. Detailed calculations are performed to third order in $\ensuremath{\gamma}\ensuremath{-}1$. It is found that the perturbation series for the energy per spin breaks down as $T\ensuremath{\rightarrow}0$. A high-temperature series expansion for the anisotropic model, which is valid for a general interaction potential and lattice, is derived by generalizing the methods developed by Horwitz and Callen for the Ising model. This series is rearranged to give a simplified diagram expansion. Finally, a practical technique for calculating the high-temperature series expansions of the zero-field free energy and susceptibility of the isotropic classical Heisenberg model is presented.

Physical ReviewMar 10, 1967196 citations

Statistical Mechanics of the Finite Heisenberg Model. I

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The eigen-functions and eigen-values of finite three-dimensional Heisenberg models with anisotropy constant as a parameter (spin=1/2) were obtained exactly with the aid of group theoretical technique and by using a high speed computer. The calculated eigenvalues are identical with those obtained by Serber and Dresselhaus for the isotropic simple cubic 2×2×2 lattice. However, it was found that some terms in their full expressions were to be corrected. The zeros of the partition function of the above 2×2×2 lattice have been investigated as functions of the anisotropy parameter both in the complex temperature plane and in the complex magnetic field plane. Thermodynamic functions such as energy, specific heat, magnetization, and susceptibility have been numerically computed for these finite system.The eigen-functions and eigen-values of finite three-dimensional Heisenberg models with anisotropy constant as a parameter (spin=1/2) were obtained exactly with the aid of group theoretical technique and by using a high speed computer. The calculated eigenvalues are identical with those obtained by Serber and Dresselhaus for the isotropic simple cubic 2×2×2 lattice. However, it was found that some terms in their full expressions were to be corrected. The zeros of the partition function of the above 2×2×2 lattice have been investigated as functions of the anisotropy parameter both in the complex temperature plane and in the complex magnetic field plane. Thermodynamic functions such as energy, specific heat, magnetization, and susceptibility have been numerically computed for these finite system.

Journal of the Physical Society of JapanJan 1, 197014 citations

Statistical Mechanics of the Finite Heisenberg Model. II

Chikao Kawabata

The eigenfunctions of a two-dimensional finite 3×3 lattice of Heisenberg model ( s =1/2) are obtained exactly with the aid of group theoretical technique and by using a computer. The eigenvalues of the anisotropic exchange Hamiltonian have been computed. The zeros of the partition function for the two-dimensional finite 3×3 lattice lie on the same unit circle in the complex magnetic field plane as those obtained in the three-dimensional finite 2×2×2 lattice. The zeros of the partition function obtained by coarse-graining of eigenvalues distribute on a line in the complex temperature plane. The thermodynamic quantities–specific heat, entropy, magnetization–has been numerically computed for ferro- and antiferromagnetic coupling. The critical index of the magnetization versus field is discussed on the basis of these numerical results.The eigenfunctions of a two-dimensional finite 3×3 lattice of Heisenberg model ( s =1/2) are obtained exactly with the aid of group theoretical technique and by using a computer. The eigenvalues of the anisotropic exchange Hamiltonian have been computed. The zeros of the partition function for the two-dimensional finite 3×3 lattice lie on the same unit circle in the complex magnetic field plane as those obtained in the three-dimensional finite 2×2×2 lattice. The zeros of the partition function obtained by coarse-graining of eigenvalues distribute on a line in the complex temperature plane. The thermodynamic quantities–specific heat, entropy, magnetization–has been numerically computed for ferro- and antiferromagnetic coupling. The critical index of the magnetization versus field is discussed on the basis of these numerical results.

Journal of the Physical Society of JapanApr 1, 197011 citations

An Exactly Soluble Model Showing Ferromagnetism

Shigetoshi Katsura, Sakari Inawashiro

A Hamiltonian of a one-dimensional Heisenberg model with ferromagnetic interaction is expressed in terms of fermion operators, and renormalized linked cluster expansion has been carried out. It is shown that the approximation leads to ferromagnetic behaviors similar to the molecular field approximation. A model for which the second- and higher-order terms vanish is presented and regarded as an example of quantum-mechanical systems that shows ferromagnetism rigorously. It is also noted that the model is very much like the Husimi-Temperley model or the van der Waals gas with respect to a long-range character of interaction and the vanishing of higher-order terms.A Hamiltonian of a one-dimensional Heisenberg model with ferromagnetic interaction is expressed in terms of fermion operators, and renormalized linked cluster expansion has been carried out. It is shown that the approximation leads to ferromagnetic behaviors similar to the molecular field approximation. A model for which the second- and higher-order terms vanish is presented and regarded as an example of quantum-mechanical systems that shows ferromagnetism rigorously. It is also noted that the model is very much like the Husimi-Temperley model or the van der Waals gas with respect to a long-range character of interaction and the vanishing of higher-order terms.

Journal of Mathematical PhysicsDec 1, 19652 citations

Green's-Function Formalism of the One-Dimensional Heisenberg Spin System

Jun Kondo, Yamaji K.

The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.

Progress of Theoretical PhysicsMar 1, 1972246 citations

{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.

{{s}_{i}}^{x}

{{s}_{i}}^{y}

, and

{{s}_{i}}^{z}

are components of the unit vector

{\mathbf{s}}_{i}

, is also considered. A perturbation series for the zero-field free energy of the anisotropic model in one dimension with nearest-neighbor interactions

{{J}_{\mathrm{ij}}}^{x}={{J}_{\mathrm{ij}}}^{y}=J

and ${{J}_{\mathrm{ij}}}^{z}=\ensuremath{\gamma}J$ is developed in powers of $\ensuremath{\gamma}\ensuremath{-}1$ using the isotropic model as the unperturbed system. Detailed calculations are performed to third order in $\ensuremath{\gamma}\ensuremath{-}1$. It is found that the perturbation series for the energy per spin breaks down as $T\ensuremath{\rightarrow}0$. A high-temperature series expansion for the anisotropic model, which is valid for a general interaction potential and lattice, is derived by generalizing the methods developed by Horwitz and Callen for the Ising model. This series is rearranged to give a simplified diagram expansion. Finally, a practical technique for calculating the high-temperature series expansions of the zero-field free energy and susceptibility of the isotropic classical Heisenberg model is presented.