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Journal of Physics A Mathematical and General

Self-consistent field method from a τ-functional viewpoint

Takao Komatsu, Seiya Nishiyama

Published Aug 10, 2000• 7 citations
Publisher

Abstract

A unified aspect of the self-consistent field (SCF) method and the τ-functional method is presented. SCF theory in the τ-functional space F ∞ manifestly results in a gauge theory of fermions and then a collective motion appears as a motion of fermion gauges with a common factor. This provides a new algebraic tool for the microscopic understanding of fermion many-body systems.

DOI
10.1088/0305-4470/33/33/307

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A unified aspect of the self-consistent field (SCF) method and the τ-functional method is presented. SCF theory in the τ-functional space F ∞ manifestly results in a gauge theory of fermions and then a collective motion appears as a motion of fermion gauges with a common factor. This provides a new algebraic tool for the microscopic understanding of fermion many-body systems.
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A generalized self-consistent field (SCF) theory of many-particle systems is developed by modifying the usual functional relationship between energy and wave function. A parameter $\ensuremath{\xi}$, $0\ensuremath{\le}\ensuremath{\xi}\ensuremath{\le}1$, is thus introduced into the energy so that for $\ensuremath{\xi}=0$, the particles interact only with a self-consistent field, for $\ensuremath{\xi}=1$ the particles interact dynamically among themselves with the correct field, and for intermediate values of $\ensuremath{\xi}$ both SCF and dynamical interactions occur. A systematic development of the theory is given for time-independent and time-dependent probelms, for finite temperatures, and for both uniform and nonuniform systems. The derivative with respect to $\ensuremath{\xi}$ of the total energy, of free energy, is expressed in terms of the dielectric function and an improved version of the dielectric formulation of the many-body problem thereby obtained. A brief discussion of the advantages of the method, possible applications, and further generalizations or extensions is included.
Toward a unified algebraic understanding of physical concepts of the so-called particle and collective motions, a new theory for unified description of both the motions is proposed with the use of time-dependent Hartree-Fock theory on a circle S 1 . The theory simply and clearly elucidates a collective motion induced by a TD mean-field potential. It also describes symmetry breaking of fermion systems and successive occurrence of the collective motion due to recovery of the symmetry. The theoretical frame asserts that the Fock space of finite-dimensional fermions has an algebraic structure to be embedded into that of infinite-dimensional fermions .
The independent-particle model in the theory of many-particle systems is studied by means of the self-consistent-field (SCF) idea. After a review of the characteristic features of the Hartree and Hartree-Fock schemes, the extension of the SCF method developed by Brueckner is further refined by introducing the exact reaction operator containing all correlation effects. This operator is here simply defined by means of the partitioning technique, and, if the SCF potentials are derived from this operator, one obtains a formalism which is completely analogous to the Hartree scheme but which still renders the exact energy and the exact wave function. An elementary derivation of the linked-cluster theorem is given, and finally the inclusion of various symmetry properties is discussed.