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Zeitschrift für Naturforschung A

Zur Beseitigung von t-Matrix-Singularitäten bei nichtverschwindendem Gesamtimpuls

Stephan Braun

Published Sep 1, 1964• 0 citations
Publisher

Abstract

As has been shown by TOMASINI it is possible to develop a perturbation theory for a many-fermion system by starting with a BOGOLJUBOV-transformed HAMILTONian having an energy gap in the unperturbed HAMILTONian. An investigation of the analogue to the BRUECKNER ladder approximation then shows this to be free of the COOPER-type divergencies of the original BRUECKNER t-matrix provided only zero total momentum of the pairs is taken into account. This paper extends the investigation to non-vanishing total momentum. For a separable potential and weak interaction it is shown that there are no divergencies for any total momentum. The reason for this (as in the case of zero total momentum) is the existence of the energy gap in the unperturbed spectrum.

DOI
10.1515/zna-1964-0903

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Physical Review•Aug 15, 1962•8 citations
As has been shown by TOMASINI it is possible to develop a perturbation theory for a many-fermion system by starting with a BOGOLJUBOV-transformed HAMILTONian having an energy gap in the unperturbed HAMILTONian. An investigation of the analogue to the BRUECKNER ladder approximation then shows this to be free of the COOPER-type divergencies of the original BRUECKNER t-matrix provided only zero total momentum of the pairs is taken into account. This paper extends the investigation to non-vanishing total momentum. For a separable potential and weak interaction it is shown that there are no divergencies for any total momentum. The reason for this (as in the case of zero total momentum) is the existence of the energy gap in the unperturbed spectrum.
The application of Brueckner's perturbation approach to interacting Fermion systems is prevented by the poles of the t-matrix which can be present for finite as well as infinite systems if the interaction is attractive. It is discussed that the singularity is removed if we take account of the contribution to the energy from diagrams involving two hole labels a number of times. These diagrams are not necessarily of ladder type and correspond to the possibility that a scattered pair of particles can go to the levels which they have left in the unperturbed state. Though the removal of poles makes one entertain a hope for the perturbation approach to finite systems, it is discussed that the same is not the case with infinite systems. It is pointed out that the difficulty in dealing with infinite systems is not caused by the pairs corresponding to the poles as is usually considered to be, but is due to pairs lying more closely to the Fermi surface.
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In order to investigate the validity of the Brueckner and the simple sum of the ladder diagrams approximations to the energy of an infinite system of fermions, we have calculated as a function of density all the terms in the perturbation expansion through the fourth order in the strength of the potential. We have done this for a repulsive, square-well, two-body potential spin-\textonehalf{} fermions. We are able to construct rigorous upper and lower bounds from the coefficients for the value of the ladder approximation and find that the standard solution procedures give reasonably accurate results (within a few percent). We find that the error in the solution as obtained in practice to the equations of the Brueckner approximation is large compared to the size of its departure from the ladder approximation. We further find for low-to-moderate densities and for low-to-moderate potential strengths that the Brueckner approximation both as a sum of a certain class of diagrams and as computed in practice lies above the ladder approximation while the complete perturbation theory lies below it. This result arises from the neglect of the ring diagrams by the Brueckner approximation.
This paper contains a critique of the Brueckner-Goldstone perturbation series for the ground-state energy of an interacting gas of fermions. We have calculated this energy by first constructing the grand partition function at finite temperature, and then carefully taking the limit as $T\ensuremath{\rightarrow}0$. In general this leads to a series which differs from that of Brueckner and Goldstone. An exception is the case where both the unperturbed single-particle energy as well as the interaction potential have spherical symmetry. Reasons for the breakdown of the Brueckner-Goldstone formalism are briefly discussed.
The Galitskii-Feynman TTT matrix, which sums the ladder diagrams for both particle-particle and hole-hole scattering in a many-fermion system, is shown to have a singularity at negative values of the c.m. energy of the interacting pair. The singularity appears as a first-order pole in the TTT matrix, the position of which is a function of both the density of the system and the total c.m. momentum of the pair, and corresponds to the formation of bound pairs in the medium. The singularity structure exists in two distinct Fermi systems (liquid He3{\mathrm{He}}^{3}He3 and nuclear matter), and detailed numerical results are presented using a potential appropriate to liquid He3{\mathrm{He}}^{3}He3. The singularity in this case is shown to be present at all densities above a certain critical density which is lower than the physical density.
In the approximation neglecting any but single pair correlations, singularities of the reaction matrix render Brueckner's integral for the average energy per particle a singular integral. Several attempts to overcome this difficulty have been unsuccessful. By considering the infinite Fermi system to be a limit of finite systems, it is shown that the correct result merely involves replacing Brueckner's ordinary integral over diagonal reaction matrix elements by a principal value integral. In a finite system the level shift of a Bethe-Goldstone state differs from the diagonal reaction matrix element by a normalization factor which does not approach unity uniformly in the integration variable as the volume becomes infinite. In the neighborhood of a singularity the expression for the two-particle energy shift takes the form $\frac{\mathrm{cy}}{({y}^{2}+c{\mathcal{U}}^{\ensuremath{-}1})}$, where yyy is the unperturbed energy measured from the singularity, ccc is the square of a matrix element, and U\mathcal{U}U is the quantization volume. Hence as $\mathcal{U}\ensuremath{\rightarrow}\ensuremath{\infty}$ the sum over the energy yyy indeed approaches a principal value integral. An alternative derivation, employing a modified reaction matrix for which there is no difference between level shift and matrix element, leads to the same result. The general derivations are preceded by a soluble example.The connection of the Brueckner approximation with a phase-shift approximation for low-density systems is discussed. Some corrections to the higher order terms in existing derivations of the "separation method" expansion of the Brueckner reaction matrix are given.