A dyadic Tamm-Dancoff method by using a matrix-valued generator coordinate

Behram N. Kurșunoǧlu

Seiya Nishiyama, Hiroyuki Morita, Hiromasa Ohnishi

The traditional Tamm–Dancoff (TD) method is one of the standard procedures for solving the Schrödinger equation of fermion many-body systems. However, it meets a serious difficulty when an instability occurs in the symmetry-adapted ground state of the independent particle approximation (IPA) and when the stable IPA ground state becomes of broken symmetry. If one uses the stable but broken symmetry IPA ground state as the starting approximation, TD wave functions also become of broken symmetry. On the contrary, if we start from a symmetry-adapted but unstable wave function, the convergence of the TD expansion becomes bad. Thus, the requirements of symmetry and rapid convergence are not in general compatible in the conventional TD expansion of the systems with strong collective correlations. Along the same line as Fukutome's, we give a group-theoretical deduction of a U ( n ) dyadic TD equation by using a matrix-valued generator coordinate.

Günter Ludyk

Hideo Fukutome

A new Tamm-Dancoff method is developed on the basis of the SO(2N + 1) regular representation formalism of Fermion many-body systems. The Schrodinger equation in the SO(2N + 1) generator coordinate representation is decomposed into symmetry subspaces by means of a Peierls-Yoccoz, projection procedure. The decomposed Schrödinger equation is transformed to a quasi-particle frame by a transformation of the generator coordinate and expanded by the symmetry projected Tamm-Dancoff bases. The Tamm-Danco[f equation and wave function thus obtained satisfy all symmetry requirements at every order of the expansion. The first order Tamm-Dancoff wave function is neither an independent particle nor quasi-particle approximation but represents Bose condensed Fermion pairs (and an unpaired particle) performing a coherent motion. The higher order terms represent fluctuations of Fermion pairs (and the unpaired particle) from the Bose condensate.

Francisco Soto, Braulio Misael Villegas-Martínez, H. M. Moya-Cessa

A. Friederich, W. Gerling

Abstract Instead of emphasizing the ground state as is done in Green's function method, we take a finite-dimensional subspace of the Hilbert space: the space of the "intermediate states". A systematic introduction of intermediate states is effected by an extension of the method of generating functionals: we combine the generating functionals of the n-point Green's functions to a "matrix functional" T, and form new matrix functionals, which are matrix functions of T. The aim of this paper is to develop the functional calculus in such a way that the transition from scalar functionals to matrix functionals is straightforward, and the method of obtaining further results becomes clear. Following the lines of Dürr and Wagner we get u η-and ζ-rules with intermediate states". Using them we define a truncation procedure for the equations of motion of the n-point Green's functions, the "New Tamm-Dancoff method with intermediate states". This extension makes it possible to treat the effect of nearby levels in many body systems with Green's functions. In ad-dition to well-known approximations, such as the Hartree-Fock and the Hartree-Bogoliubov theory, the RPA and the quasiparticle RPA, we obtain a series of new approximations. Among these are the "Hartree-Fock theory with intermediate states" and the "random-phase approximation with intermediate states", which we already applied with great success to some exactly soluble models.