Physical review. B, Condensed matter

Green’s function theory of the spin-1 low-dimensional quantumXYferromagnet

M. E. Gouvêa, Antonio S. T. Pires

Published Mar 5, 2001• 9 citations

Abstract

We calculate the spin-spin correlations and the magnetic susceptibility of the quantum one- and two-dimensional $\mathrm{XY}$ models with $S=1,$ using the two-time Green's function method and performing a decoupling proposed by Kondo and Yamaji for cases with no long-range order. A set of self-consistent equations of the correlation functions are derived and solved numerically. We present results for the correlation functions, susceptibilities, and specific heat for the whole range of temperature.

DOI: 10.1103/physrevb.63.134408

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

Jun Kondo, Yamaji K.

Progress of Theoretical PhysicsMar 1, 1972246 citations

Green’s function study of one-dimensional spin-1/2 Ising-Heisenberg ferromagnets

W. J. Zhang, J. L. Shen, Jian Xu +1 more

Physical review. B, Condensed matterFeb 1, 19957 citations

Thermodynamic properties of one-dimensional spin-1 antiferromagnetic Heisenberg chains: Green’s-function approach

Shu-Qing Bao, Hu Zhao, Jue-Lian Shen +1 more

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Magnetic Properties of One-Dimensional Ferromagnetic Mixed-Spin Model within Tyablikov Decoupling Approximation

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Application of the Method of Two-Time Green's Function to the Spin-½ Heisenberg Ferromagnet

Tohru Morita, Tomoyasu Tanaka

Physical ReviewJan 18, 196523 citations

Physical review. B, Condensed matter

Green’s function theory of the spin-1 low-dimensional quantumXYferromagnet

M. E. Gouvêa, Antonio S. T. Pires

Published Mar 5, 2001• 9 citations

Publisher

Abstract

We calculate the spin-spin correlations and the magnetic susceptibility of the quantum one- and two-dimensional

\mathrm{XY}

models with

S=1,

using the two-time Green's function method and performing a decoupling proposed by Kondo and Yamaji for cases with no long-range order. A set of self-consistent equations of the correlation functions are derived and solved numerically. We present results for the correlation functions, susceptibilities, and specific heat for the whole range of temperature.

DOI: 10.1103/physrevb.63.134408

Related papers

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

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W. J. Zhang, J. L. Shen, Jian Xu +1 more

The one-dimensional spin-1/2 Ising-Heisenberg ferromagnets are studied by the Green's function method at finite temperatures. The Green's function is decoupled in terms of the correlation functions of 〈 ${\mathit{S}}_{0}^{\mathit{z}}$ ${\mathit{S}}_{\mathit{n}}^{\mathit{z}}$ 〉 and 〈 ${\mathit{S}}_{0}^{+}$ ${\mathit{S}}_{\mathit{n}}^{\mathrm{\ensuremath{-}}}$〉 under the constraint of 〈 ${\mathit{S}}^{\mathit{z}}$ 〉=0, from which a set of self-consistent equations are derived and solved numerically. The thermodynamic properties such as internal energy, specific heat, and zero-field parallel susceptibility are calculated. Our results predict the correct behavior for the correlation functions and thermodynamic quantities in the entire temperature region, in agreement both with the exact numerical results on the finite chain and the experimental data.The one-dimensional spin-1/2 Ising-Heisenberg ferromagnets are studied by the Green's function method at finite temperatures. The Green's function is decoupled in terms of the correlation functions of 〈${\mathit{S}}_{0}^{\mathit{z}}$${\mathit{S}}_{\mathit{n}}^{\mathit{z}}$〉 and 〈${\mathit{S}}_{0}^{+}$${\mathit{S}}_{\mathit{n}}^{\mathrm{\ensuremath{-}}}$〉 under the constraint of 〈${\mathit{S}}^{\mathit{z}}$〉=0, from which a set of self-consistent equations are derived and solved numerically. The thermodynamic properties such as internal energy, specific heat, and zero-field parallel susceptibility are calculated. Our results predict the correct behavior for the correlation functions and thermodynamic quantities in the entire temperature region, in agreement both with the exact numerical results on the finite chain and the experimental data.

Physical review. B, Condensed matterFeb 1, 19957 citations

Thermodynamic properties of one-dimensional spin-1 antiferromagnetic Heisenberg chains: Green’s-function approach

Shu-Qing Bao, Hu Zhao, Jue-Lian Shen +1 more

The one-dimensional spin-1 Heisenberg antiferromagnetic chains are studied by a generalized two-time Green's-function method at finite temperature. 〈 ${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$ 〉 is set to zero at each site to guarantee the lack of the long-range order in the one-dimensional Heisenberg system. The Green's functions are decoupled in terms of the correlation functions of spin operators. A set of self-consistent equations of the correlation functions are derived and solved numerically. Correlation functions of spin operators, thermodynamic properties such as the internal energy, and specific heat in the entire temperature region are obtained. The Haldane gap in the excitation spectrum appears naturally in the analytical result. When k is zero, there is a gap 2\ensuremath{\Delta}, which is 1.0J and close to the theoretical result of White and Huse. And there is a broad maximum in the temperature dependence of the specific heat. The properties of the thermodynamic quantities are consistent with the numerical results on the finite chains. \textcopyright{} 1996 The American Physical Society.The one-dimensional spin-1 Heisenberg antiferromagnetic chains are studied by a generalized two-time Green's-function method at finite temperature. 〈${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$〉 is set to zero at each site to guarantee the lack of the long-range order in the one-dimensional Heisenberg system. The Green's functions are decoupled in terms of the correlation functions of spin operators. A set of self-consistent equations of the correlation functions are derived and solved numerically. Correlation functions of spin operators, thermodynamic properties such as the internal energy, and specific heat in the entire temperature region are obtained. The Haldane gap in the excitation spectrum appears naturally in the analytical result. When k is zero, there is a gap 2\ensuremath{\Delta}, which is 1.0J and close to the theoretical result of White and Huse. And there is a broad maximum in the temperature dependence of the specific heat. The properties of the thermodynamic quantities are consistent with the numerical results on the finite chains. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matterJan 1, 199624 citations

Magnetic Properties of One-Dimensional Ferromagnetic Mixed-Spin Model within Tyablikov Decoupling Approximation

Yuan Chen, Xiang Ying, Chuang-Chuang Song

In this paper, we apply the two-time Green's function method, and provide a simple way to study the magnetic properties of one-dimensional spin-(S,s) Heisenberg ferromagnets. The magnetic susceptibility and correlation functions are obtained by using the Tyablikov decoupling approximation. Our results show that the magnetic susceptibility and correlation length are a monotonically decreasing function of temperature regardless of the mixed spins. It is found that in the case of S=s, our results of one-dimensional mixed-spin model is reduced to be those of the isotropic ferromagnetic Heisenberg chain in the whole temperature region. Our results for the susceptibility are in agreement with those obtained by other theoretical approaches.In this paper, we apply the two-time Green's function method, and provide a simple way to study the magnetic properties of one-dimensional spin-(S,s) Heisenberg ferromagnets. The magnetic susceptibility and correlation functions are obtained by using the Tyablikov decoupling approximation. Our results show that the magnetic susceptibility and correlation length are a monotonically decreasing function of temperature regardless of the mixed spins. It is found that in the case of S=s, our results of one-dimensional mixed-spin model is reduced to be those of the isotropic ferromagnetic Heisenberg chain in the whole temperature region. Our results for the susceptibility are in agreement with those obtained by other theoretical approaches.

Communications in Theoretical PhysicsOct 1, 20104 citations

Green's Function Approach to the Modified Quantum Heisenberg Model

Tosizumi Aoki

The two-time Green's function method is applied to the modified quantum Heisenberg model with the following exchange interactions: The interactions decay as inverse square of the distance between sites and have alternate positive and negative signs with increasing distance between sites; moreover, they work well only along the coordinate axes. The critical temperature and zero-temperature spin reduction are calculated for the one-, two- and three-dimensional cases. The temperature dependence of spin reductions in two and three dimensions is estimated in the low-temperature region. The critical behavior of long-range order and susceptibility is also investigated. Lastly, it is attempted to estimate the long-distance behavior of static correlation function in one dimension.The two-time Green's function method is applied to the modified quantum Heisenberg model with the following exchange interactions: The interactions decay as inverse square of the distance between sites and have alternate positive and negative signs with increasing distance between sites; moreover, they work well only along the coordinate axes. The critical temperature and zero-temperature spin reduction are calculated for the one-, two- and three-dimensional cases. The temperature dependence of spin reductions in two and three dimensions is estimated in the low-temperature region. The critical behavior of long-range order and susceptibility is also investigated. Lastly, it is attempted to estimate the long-distance behavior of static correlation function in one dimension.

Journal of the Physical Society of JapanNov 1, 19981 citation

Application of the Method of Two-Time Green's Function to the Spin-½ Heisenberg Ferromagnet

Tohru Morita, Tomoyasu Tanaka

In the usual theory of the two-time Green's function, decoupling techniques are introduced in order to obtain a self-closed equation for the lowest order Green's function. These decouplings give neither the correct temperature dependence (the absence of a ${T}^{3}$ term) of magnetization nor the correct correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$, which should be zero by definition. In this paper the equation of motion for the higher order Green's function is solved in the approximation which will lead to correct results up to terms of order ${T}^{3}$ in these quantities. Employing this higher order Green's function, it is found that the correction to the magnetization consists of a term of order ${T}^{4}$ and a divergent term and that the correlation function consists of a term of order ${T}^{3}$ and a divergent term. A method extracting a significant part from this divergent term is found by analyzing a system of two interacting \textonehalf{} spins. By this method, it is shown that the ${T}^{3}$ term in the magnetization is completely eliminated and that the correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$ vanishes identically.In the usual theory of the two-time Green's function, decoupling techniques are introduced in order to obtain a self-closed equation for the lowest order Green's function. These decouplings give neither the correct temperature dependence (the absence of a ${T}^{3}$ term) of magnetization nor the correct correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$, which should be zero by definition. In this paper the equation of motion for the higher order Green's function is solved in the approximation which will lead to correct results up to terms of order ${T}^{3}$ in these quantities. Employing this higher order Green's function, it is found that the correction to the magnetization consists of a term of order ${T}^{4}$ and a divergent term and that the correlation function consists of a term of order ${T}^{3}$ and a divergent term. A method extracting a significant part from this divergent term is found by analyzing a system of two interacting \textonehalf{} spins. By this method, it is shown that the ${T}^{3}$ term in the magnetization is completely eliminated and that the correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$ vanishes identically.

Physical ReviewJan 18, 196523 citations

We calculate the spin-spin correlations and the magnetic susceptibility of the quantum one- and two-dimensional

\mathrm{XY}

models with

S=1,

{\mathit{S}}_{0}^{\mathit{z}}

{\mathit{S}}_{\mathit{n}}^{\mathit{z}}

〉 and 〈

{\mathit{S}}_{0}^{+}

${\mathit{S}}_{\mathit{n}}^{\mathrm{\ensuremath{-}}}$〉 under the constraint of 〈

{\mathit{S}}^{\mathit{z}}

〉=0, from which a set of self-consistent equations are derived and solved numerically. The thermodynamic properties such as internal energy, specific heat, and zero-field parallel susceptibility are calculated. Our results predict the correct behavior for the correlation functions and thermodynamic quantities in the entire temperature region, in agreement both with the exact numerical results on the finite chain and the experimental data.

The one-dimensional spin-1 Heisenberg antiferromagnetic chains are studied by a generalized two-time Green's-function method at finite temperature. 〈

{\mathit{S}}_{\mathit{i}}^{\mathit{z}}

〉 is set to zero at each site to guarantee the lack of the long-range order in the one-dimensional Heisenberg system. The Green's functions are decoupled in terms of the correlation functions of spin operators. A set of self-consistent equations of the correlation functions are derived and solved numerically. Correlation functions of spin operators, thermodynamic properties such as the internal energy, and specific heat in the entire temperature region are obtained. The Haldane gap in the excitation spectrum appears naturally in the analytical result. When k is zero, there is a gap 2\ensuremath{\Delta}, which is 1.0J and close to the theoretical result of White and Huse. And there is a broad maximum in the temperature dependence of the specific heat. The properties of the thermodynamic quantities are consistent with the numerical results on the finite chains. \textcopyright{} 1996 The American Physical Society.

{T}^{3}

term) of magnetization nor the correct correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$, which should be zero by definition. In this paper the equation of motion for the higher order Green's function is solved in the approximation which will lead to correct results up to terms of order

{T}^{3}

in these quantities. Employing this higher order Green's function, it is found that the correction to the magnetization consists of a term of order

{T}^{4}

and a divergent term and that the correlation function consists of a term of order

{T}^{3}

and a divergent term. A method extracting a significant part from this divergent term is found by analyzing a system of two interacting \textonehalf{} spins. By this method, it is shown that the

{T}^{3}

term in the magnetization is completely eliminated and that the correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$ vanishes identically.