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Progress of Theoretical Physics

On the Singularity oft-Matrix of Interacting Fermion Systems

Shuichiro Yamasaki

Published Sep 1, 1962• 1 citation
Publisher

Abstract

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.

DOI
10.1143/ptp.28.407

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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.
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.
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.
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 binding of two fermions in a degenerate Fermi gas is examined in terms of the T-matrix. We deal with effective T-matrix of two fermions in the degenerate Fermi gas in terms of the scattering amplitude. This makes approach valid for the retarded as well as non-retarded interactions. We show that for the interaction of a pair of fermions in a quiescent Fermi sea there are two different series of the ladder diagrams-Exchange and Hartree diagrams, describing a bound state and producing two different T-scattering matrices. The exchange series of the ladder diagrams corresponds to the T-matrix in the triplet-state. This T-matrix contains the pole solution corresponding to the bound state for the attractive interaction only. In contrast, the Hartree diagrams lead to the T-matrix of a pair of fermions in the singlet-state, which have a pole solution corresponding to the bound state of a pair of fermions for the repulsive interaction. Thus, we are led to assert that the bound state of a pair of fermions in the singlet-state in the quiescent Fermi sea can form for the repulsive interaction only and there is no bound state in the singlet-state from the attractive interaction.
By the method of ``modes'' introduced by Sawada, the excitation spectra of a fermion system with singular interactions have been obtained in three cases; (1) with one additional particle above and and one hole below the Fermi surface, (2) with one additional particle above the Fermi surface, and (3) with two particles above the Fermi surface. The argument holds not only for the system of low density (nuclear matter with a hard-core repulsion), but also for the case of the interacting electron gas in the high-density region. The results have been compared with the Brueckner-Goldstone perturbation-expansion formulas, by using diagrams in the second- and third-order of the expansion in interaction strength. We show that in the approximation of our treatment, the occupation probability function at temperature T has the same form as the Fermi distribution function.
The method of superposition of configurations, which provides a general solution of the quantum-mechanical many-particle problem (for fermions), is reformulated so that equations may be compared with those used in the Brueckner theory. Important differences occur in application to finite or nonuniform systems for which the Hartree-Fock or Brueckner self-consistent orbitals are not plane waves. In such cases nonvanishing single-particle matrix elements occur which cannot be described by the Brueckner formalism based on a two-particle operator. Equations for an effective two-particle operator, equivalent to the variational equations of the method of superposition of configurations, are derived for a basis of Hartree-Fock orbitals. At the expense of making the orbital basis dependent upon the effective two-particle operator, the orbital basis can be determined by a condition which is essentially that of the Brueckner method. This condition removes a class of matrix elements which do not necessarily vanish in the Hartree-Fock basis although they would be neglected to second order in a perturbation calculation starting from the Hartree-Fock wave function. The equations for the effective two-particle operator are formally the same in both cases but lead to different operators because of the different choice of basis. In neither case can the equations be written in terms of products of such operators, a formalism assumed in the Brueckner theory. It is shown that the Brueckner condition is not equivalent to the condition which would determine the best possible orbital basis for the form of wave function implied by the use of an effective operator dependent on two particles only. The argument of the present paper is limited to systems with a finite number of particles, since the variational principle used here is not applicable without modification to systems with an infinite number of particles.