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Physical Review A

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

Jingbo Zhang

Published Apr 1, 1991• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physreva.43.4547

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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.
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.
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.