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Progress of Theoretical Physics

Renormalization of Two-electron Green-function

Sigenobu Sunakawa, Tsutomu Imamura, Ryôyû Utiyama

Published Nov 1, 1954• 0 citations
Publisher

Abstract

The difference between the ordinary method of renormalization of Dyson and that of the present authors developed in the previous paper is emphasized. In order to clarify this difference, the renormalization of the two-electron Green-function is investigated. The proof of the finiteness of the renormalized Green-function seems to be given in quite a brief way. It is the essential point in this proof that there is no ambiguity in the order of integration appearing in course of calculation of Green-functions.

DOI
10.1143/ptp.12.642

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The difference between the ordinary method of renormalization of Dyson and that of the present authors developed in the previous paper is emphasized. In order to clarify this difference, the renormalization of the two-electron Green-function is investigated. The proof of the finiteness of the renormalized Green-function seems to be given in quite a brief way. It is the essential point in this proof that there is no ambiguity in the order of integration appearing in course of calculation of Green-functions.
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The functional-derivative method of calculating the Green's function developed earlier for the Hubbard model is generalized and used to obtain a fully renormalized solution. Higher-order functional derivatives operating on the basic Green's functions, GGG and $\ensuremath{\Gamma}$, are all evaluated explicitly, thus making the solution applicable to the narrow-band region as well as the wide-band region. Correction terms $\ensuremath{\Phi}$ generated from functional derivatives of equal-time Green's functions of the type $\frac{{\ensuremath{\delta}}^{n}〈N〉}{\ensuremath{\delta}{\ensuremath{\epsilon}}^{n}}$, etc., with $n\ensuremath{\ge}2$. It is found that the $\ensuremath{\Phi}'\mathrm{s}$ are, in fact, renormalization factors involved in the self-energy $\ensuremath{\Sigma}$ and that the structure of the $\ensuremath{\Phi}'\mathrm{s}$ resembles that of $\ensuremath{\Sigma}$ and contains the same renormalization factors $\ensuremath{\Phi}$. The renormalization factors $\ensuremath{\Phi}$ are shown to satisfy a set of equations and can be evaluated self-consistently. In the presence of the $\ensuremath{\Phi}'\mathrm{s}$, all difficulties found in the previous results (papers I and II) are removed, and the energy spectrum $\ensuremath{\omega}$ can now be evaluated for all occupations nnn. The Schwinger relation is the only basic relation used in generating this fully self-consistent Green's function, and the Baym-Kadanoff continuity condition is automatically satisfied.
For systems governed by two kinds of interactions it is shown that these can be built successively into the Green’s functions describing the system’s response. Whereas for the ordinary Green’s function the Dyson equation to solve has the same form in each step, we derive the non-trivial second-step equation for the transfer or coupling function, which on the one hand is closely related to the self-energy and on the other hand of practical relevance in transport calculations. PACS numbers: 02.60.Cb, 02.90.+p, 72.10.Bg 1.
Recently, general point interactions in one dimension has been used to model a large number of different phenomena in quantum mechanics. Such potentials, however, require some sort of regularization to lead to meaningful results. The usual ways to do so rely on technicalities that may hide important physical aspects of the problem. In this work we present a method to calculate the exact Green functions for general point interactions in one dimension. Our approach differs from previous ones because it is based only on physical quantities, namely, the scattering coefficients R and T to construct G. Renormalization or particular mathematical prescriptions are not invoked. The simple formulation of the method makes it easy to extend to more general contexts, such as for lattices of N general point interactions, on a line, on a half-line, under periodic boundary conditions, and confined in a box.