Physical Review A

Undetermined-constant method in the boson Green’s-function theory

J. B. Zhang, Z. K. Qin

Published Jul 1, 1987• 10 citations

Abstract

It is shown in the present paper that if operators of a boson Green's function involve arbitrary c numbers, then the corresponding correlation function obtained from the spectral theorem is independent of the c numbers, which is in contradiction with the definition of the said correlation function. In order to solve this problem we have introduced an undetermined-constant method and applied it to ferromagnetic and ferroelectric systems, respectively. It is pointed out that the introduction of the undetermined constant is an intrinsic characteristic of the boson Green's-function method and that contradictory results can often be obtained if it is not introduced.

DOI: 10.1103/physreva.36.915

Related papers

Comment on ‘‘Undetermined-constant method in the boson Green’s-function theory’’

Edward B. Brown

Physical Review AApr 1, 19911 citation

Reply to ‘‘Comment on ‘Undetermined-constant method in the boson Green’s-function theory’ ’’

Jingbo Zhang

In this Reply to the preceding Comment by Brown [Phys. Rev. A 43, XXXX (1991)], the following two points are discussed. (1) The undetermined-constant method solves problems by using only boson Green's functions, but the method given by Brown in the Comment must use both boson and fermion Green's functions. (2) When the undetermined-constant method is applied to the example given by Ramos and Gomes [Nuovo Cimento 3A, 441 (1971)], exactly the same results can be obtained.

Physical Review AApr 1, 19911 citation

Spectral theorem of many-body Green’s function theory when there are zero eigenvalues of the matrix governing the equations of motion

Peter Fröbrich, P. J. Kuntz

arXiv:cond-mat/0305444Physical review. B, Condensed matterJul 9, 200315 citations

Theory of the Ising Ferromagnet Using the Green Function Method

Takehiko Oguchi, Ikuo Ono

Progress of Theoretical PhysicsJun 1, 196615 citations

The treatment of zero eigenvalues of the matrix governing the equations of motion in many-body Green function theory

Peter Fröbrich, P. J. Kuntz

arXiv:cond-mat/0407084Journal of Physics Condensed MatterFeb 5, 20058 citations

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 A

Undetermined-constant method in the boson Green’s-function theory

J. B. Zhang, Z. K. Qin

Published Jul 1, 1987• 10 citations

Publisher

Abstract

DOI: 10.1103/physreva.36.915

Related papers

Comment on ‘‘Undetermined-constant method in the boson Green’s-function theory’’

Edward B. Brown

The apparent fundamental inconsistency in the behavior of double-time Green's functions and the correlations related to them by the spectral theorem claimed by Zhang and Qin [Phys. Rev. A 36, 915 (1987)] is shown to arise from the use of an incorrect version of the spectral theorem. The ``method of undetermined constants'' constructed by Zhang and Qin to remove this apparent inconsistency is not an intrinsic characteristic of the boson Green's-function method. Rather, it is unnecessary and leads to incorrect results in both systems considered by Zhang and Qin, i.e., the mean-field transverse Ising model and the symmetrically decoupled zero-field Heisenberg model. Use of the correct spectral theorem is shown to remove the inconsistency. Correct treatments based on the correct spectral theorem of the models considered by Zhang and Qin are also presented.The apparent fundamental inconsistency in the behavior of double-time Green's functions and the correlations related to them by the spectral theorem claimed by Zhang and Qin [Phys. Rev. A 36, 915 (1987)] is shown to arise from the use of an incorrect version of the spectral theorem. The ``method of undetermined constants'' constructed by Zhang and Qin to remove this apparent inconsistency is not an intrinsic characteristic of the boson Green's-function method. Rather, it is unnecessary and leads to incorrect results in both systems considered by Zhang and Qin, i.e., the mean-field transverse Ising model and the symmetrically decoupled zero-field Heisenberg model. Use of the correct spectral theorem is shown to remove the inconsistency. Correct treatments based on the correct spectral theorem of the models considered by Zhang and Qin are also presented.

Physical Review AApr 1, 19911 citation

Reply to ‘‘Comment on ‘Undetermined-constant method in the boson Green’s-function theory’ ’’

Jingbo Zhang

Physical Review AApr 1, 19911 citation

Spectral theorem of many-body Green’s function theory when there are zero eigenvalues of the matrix governing the equations of motion

Peter Fröbrich, P. J. Kuntz

In using the spectral theorem of many-body Green's function theory in order to relate correlations to commutator Green's functions, it is necessary in the standard procedure to consider the anti-commutator Green's functions as well whenever the matrix governing the equations of motion for the commutator Green's functions has zero eigenvalues. We show that a singular-value decomposition of this matrix allows one to reformulate the problem in terms of a smaller set of Green's functions with an associated matrix having no zero eigenvalues, thus eliminating the need for the anti-commutator functions. The procedure is quite general and easy to apply. It is illustrated for the field-induced reorientation of the magnetization of a ferromagnetic Heisenberg monolayer and it is expected to work for more complicated cases as well.In using the spectral theorem of many-body Green's function theory in order to relate correlations to commutator Green's functions, it is necessary in the standard procedure to consider the anti-commutator Green's functions as well whenever the matrix governing the equations of motion for the commutator Green's functions has zero eigenvalues. We show that a singular-value decomposition of this matrix allows one to reformulate the problem in terms of a smaller set of Green's functions with an associated matrix having no zero eigenvalues, thus eliminating the need for the anti-commutator functions. The procedure is quite general and easy to apply. It is illustrated for the field-induced reorientation of the magnetization of a ferromagnetic Heisenberg monolayer and it is expected to work for more complicated cases as well.

arXiv:cond-mat/0305444Physical review. B, Condensed matterJul 9, 200315 citations

Theory of the Ising Ferromagnet Using the Green Function Method

Takehiko Oguchi, Ikuo Ono

The Green function method is applied to the Ising model of ferromagnetism. It is essential in this paper that we have solved the equations for the Green functions in terms of the correlation functions without using the approximation of decoupling the Green functions as is usually done. Although we get several rigorous relations between the many-spin correlation functions, we do not succeed in solving for the correlation functions, except in the case of a linear chain. We introduce the approximation of eliminating some of the correlation functions using the high temperature expansion, so as to reduce the number of the independent correlation functions. The Curie points of a square lattice and a simple cubic lattice are calculated.The Green function method is applied to the Ising model of ferromagnetism. It is essential in this paper that we have solved the equations for the Green functions in terms of the correlation functions without using the approximation of decoupling the Green functions as is usually done. Although we get several rigorous relations between the many-spin correlation functions, we do not succeed in solving for the correlation functions, except in the case of a linear chain. We introduce the approximation of eliminating some of the correlation functions using the high temperature expansion, so as to reduce the number of the independent correlation functions. The Curie points of a square lattice and a simple cubic lattice are calculated.

Progress of Theoretical PhysicsJun 1, 196615 citations

The treatment of zero eigenvalues of the matrix governing the equations of motion in many-body Green function theory

Peter Fröbrich, P. J. Kuntz

The spectral theorem of many-body Green function theory relates thermodynamic correlations to Green functions. More often than not, the matrix governing the equations of motion has zero eigenvalues. In this case, the standard textbook approach requires both commutator and anti-commutator Green functions to obtain equations for that part of the correlation which does not lie in the null space of the matrix. In this paper, we show that this procedure fails if the projector onto the null space is dependent on the momentum vector, k . We propose an alternative formulation of the theory in terms of the non-null space alone and we show that a solution is possible if one can find a momentum-independent projector onto some subspace of the non-null space. To do this, we enlist the aid of the singular value decomposition of in order to project out the null space, thus reducing the size of the matrix and eliminating the need for the anti-commutator Green function. We extend our previous work (Fröbrich and Kuntz 2003 Phys. Rev. B 68 014410), dealing with a ferromagnetic Heisenberg monolayer and a momentum-independent projector onto the null space, to models where both multilayer films and a momentum-dependent projector are considered. We develop the numerical methods capable of handling these cases and offer a computational algorithm that should be applicable to any similar problem arising in Green function theory.The spectral theorem of many-body Green function theory relates thermodynamic correlations to Green functions. More often than not, the matrix governing the equations of motion has zero eigenvalues. In this case, the standard textbook approach requires both commutator and anti-commutator Green functions to obtain equations for that part of the correlation which does not lie in the null space of the matrix. In this paper, we show that this procedure fails if the projector onto the null space is dependent on the momentum vector, k . We propose an alternative formulation of the theory in terms of the non-null space alone and we show that a solution is possible if one can find a momentum-independent projector onto some subspace of the non-null space. To do this, we enlist the aid of the singular value decomposition of in order to project out the null space, thus reducing the size of the matrix and eliminating the need for the anti-commutator Green function. We extend our previous work (Fröbrich and Kuntz 2003 Phys. Rev. B 68 014410), dealing with a ferromagnetic Heisenberg monolayer and a momentum-independent projector onto the null space, to models where both multilayer films and a momentum-dependent projector are considered. We develop the numerical methods capable of handling these cases and offer a computational algorithm that should be applicable to any similar problem arising in Green function theory.

arXiv:cond-mat/0407084Journal of Physics Condensed MatterFeb 5, 20058 citations

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

{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.