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Physical Review A

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

Edward B. Brown

Published Apr 1, 1991• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physreva.43.4544

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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.
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.
Recent attempts to develop double-time Green's-function theories for the transverse Ising model (TIM) have been subject to certain difficulties; the appearance of zero-frequency poles in the commutator Green's function (CGF) led to the avoidance of CGF and an incomplete treatment based only upon anticommutator Green's functions (AGF), and ambiguities in the determination of some thermal averages. Upon decoupling the equations-of-motion hierarchy, we develop a consistent Green's-function theory of the TIM. The AGF and CGF are shown to provide consistent results, the problem of zero-frequency poles in CGF being treated appropriately. We resolve the ambiguity problem and obtain reasonable values for the Weiss critical field and the ground-state magnetization.
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.