Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Singularities in the Galitskii-Feynman T matrix

R. F. Bishop, Michael R. Strayer, J M Irvine

Published Dec 1, 1974• 26 citations
Publisher

Abstract

The Galitskii-Feynman $T$ 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 $T$ 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 ${\mathrm{He}}^{3}$ and nuclear matter), and detailed numerical results are presented using a potential appropriate to liquid ${\mathrm{He}}^{3}$. 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.

DOI
10.1103/physreva.10.2423

Related papers

On the Singularity oft-Matrix of Interacting Fermion Systems

Shuichiro Yamasaki

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.

Progress of Theoretical Physics•Sep 1, 1962•1 citation

Effect of singular interaction terms on two-dimensional Fermi liquids

Philip C. E. Stamp

The structure of perturbation theory for an interacting two-dimensional Fermi system is analyzed, in the case where singular terms exist in the quasiparticle interactions. These give a hierarchy of singular terms in the one-particle self-energy. The dominant terms are self-consistently summed, to give a quasiparticle pole vanishing as Z(\ensuremath{\omega})\ensuremath{\sim}(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$${)}_{0}^{\mathrm{\ensuremath{\rho}},\mathrm{\ensuremath{\delta}}2}$/${\mathrm{\ensuremath{\Elzxh}}}^{2}$N(0)${\mathrm{\ensuremath{\omega}}}_{0}$ near the Fermi energy (signaling a breakdown of Fermi-liquid theory), and a form for the scattering amplitude which is consistent with the assumed form for the quasiparticle interaction.

Physical Review Letters•Apr 1, 1992•31 citations

Some aspects of singular interactions in condensed Fermi systems

Philip C. E. Stamp

This article gives a fairly detailed survey of some of the problems raised when the interaction energy fσσ'kk' between 2 fermionic quasiparticles (in 2 dimensions) is singular when $|\underline{k}-\underline{k}'|\to 0$. Before dealing with singular interactions, it is shown how a non-singular fσσ'kk' leads to a 2-dimensional Fermi liquid theory, which is internally consistent, at least as far as its infrared properties are concerned. The quasiparticle properties are calculated in detail. The question of whether singular interactions arise for the dilute Fermi gas, with short-range repulsive interactions, is investigated perturbatively. One finds a weak singularity in fσσ'kk', when the dimensionality D = 2, but it does not destabilize the Fermi liquid. A more sophisticated analysis is then given, to all orders in the interaction, using the Lippman-Schwinger equation as well as a phase shift analysis for a finite box. The conclusion is that any breakdown of Fermi liquid theory must come from non-perturbative effects. An examination is then made of some of the consequences arising if a singular interaction is introduced — the form proposed by Anderson is used as an example. A hierarchy of singular terms arise in all quantities — this is shown for the self-energy, and also the 3–point and 4–point scattering functions. These may be summed in a perfectly consistent manner. Most attention is given to the particle-hole channel, since it appears to lead to results different from those of Anderson. Nevertheless it appears that it is possible to derive a sensible theory starting from a singular effective Hamiltonian — although Fermi Liquid theory breaks down, all fermionic quantities may be calculated consistently. Finally, the effect of a magnetic field (which cuts off the infrared divergences) is investigated, and the de Haas-van Alphen amplitude calculated, for such a singular Fermionic system.

Journal de Physique I•Feb 1, 1993•16 citations

Excitation Spectrum of a Fermion System

Sadaaki Yanagawa

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.

Journal of Mathematical Physics•May 1, 1962•0 citations

Self-Consistent Pair Interaction for Many-Fermion System

K. Sawada, Toshio Soda

The equations which determine the one-particle energy and effective two-body interaction in an interacting Fermi gas are constructed within the approximation which sums up all pair creation-annihilation processes. The equation corresponds to the familiar equation for the $K$ matrix which represents the interaction between particles (or holes) and sums up the particle-particle (or hole-hole) scattering processes. The method of the equation of motion is used in this paper. Our result for the one-particle energy is shown to lead to the result previously obtained by Quinn and Ferrell, and by Rockmore for the case of the electron gas with Coulomb interactions, when we replace screened potentials by bare potential in the self-consistent energy equation. For nuclear matter, it is shown that the presence of an attractive interaction in the equation of motion for number density causes an "enhancement" of exchange forces, whereas in the electron gas repulsive Coulomb interactions lead to "screening" of the exchange force. The strength of the isospin density interaction pseudopotential is enhanced by a factor of two when one solves the self-consistent equation; and a simple estimate shows that the Goldhaber-Teller mode lies about 15% higher than the value $\frac{{p}_{F}q}{m}$ previously estimated by Glassgold et al. ($q$: momentum of the oscillation, ${p}_{F}$: Fermi momentum).

Physical Review•Aug 15, 1961•1 citation

VERTEX FUNCTION OF A STRONGLY INTERACTING ELECTRON-HOLE PAIR NEAR THE FERMI SURFACE

Sérgio Luiz Garavelli, A. FERRAZ, Jon Henderson +1 more

In this work we calculate and compare all the second- and third-order diagrams of the irreducible part K of the electron-hole scattering function for strongly interacting particles near the Fermi surface. We show that the exchange ladder diagrams for K are indeed the most singular diagrams up to this order of perturbation theory in the proximity of the Fermi surface. From this result emerges a new low energy scale parameter and a new effective coupling constant associated with the strongly interacting regime. The summation of exchange ladder diagrams to infinite order produces a purely imaginary pole in K. As a result the vertex function Γ diverges and the Fermi liquid becomes unstable in this strongly interacting regime.

Modern Physics Letters B•Jul 20, 1995•0 citations
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.
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.
The structure of perturbation theory for an interacting two-dimensional Fermi system is analyzed, in the case where singular terms exist in the quasiparticle interactions. These give a hierarchy of singular terms in the one-particle self-energy. The dominant terms are self-consistently summed, to give a quasiparticle pole vanishing as Z(\ensuremath{\omega})\ensuremath{\sim}(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$${)}_{0}^{\mathrm{\ensuremath{\rho}},\mathrm{\ensuremath{\delta}}2}$/${\mathrm{\ensuremath{\Elzxh}}}^{2}$N(0)${\mathrm{\ensuremath{\omega}}}_{0}$ near the Fermi energy (signaling a breakdown of Fermi-liquid theory), and a form for the scattering amplitude which is consistent with the assumed form for the quasiparticle interaction.
This article gives a fairly detailed survey of some of the problems raised when the interaction energy fσσ'kk' between 2 fermionic quasiparticles (in 2 dimensions) is singular when ∣k‾−k‾′∣→0|\underline{k}-\underline{k}'|\to 0∣k​−k​′∣→0. Before dealing with singular interactions, it is shown how a non-singular fσσ'kk' leads to a 2-dimensional Fermi liquid theory, which is internally consistent, at least as far as its infrared properties are concerned. The quasiparticle properties are calculated in detail. The question of whether singular interactions arise for the dilute Fermi gas, with short-range repulsive interactions, is investigated perturbatively. One finds a weak singularity in fσσ'kk', when the dimensionality D = 2, but it does not destabilize the Fermi liquid. A more sophisticated analysis is then given, to all orders in the interaction, using the Lippman-Schwinger equation as well as a phase shift analysis for a finite box. The conclusion is that any breakdown of Fermi liquid theory must come from non-perturbative effects. An examination is then made of some of the consequences arising if a singular interaction is introduced — the form proposed by Anderson is used as an example. A hierarchy of singular terms arise in all quantities — this is shown for the self-energy, and also the 3–point and 4–point scattering functions. These may be summed in a perfectly consistent manner. Most attention is given to the particle-hole channel, since it appears to lead to results different from those of Anderson. Nevertheless it appears that it is possible to derive a sensible theory starting from a singular effective Hamiltonian — although Fermi Liquid theory breaks down, all fermionic quantities may be calculated consistently. Finally, the effect of a magnetic field (which cuts off the infrared divergences) is investigated, and the de Haas-van Alphen amplitude calculated, for such a singular Fermionic system.
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 equations which determine the one-particle energy and effective two-body interaction in an interacting Fermi gas are constructed within the approximation which sums up all pair creation-annihilation processes. The equation corresponds to the familiar equation for the KKK matrix which represents the interaction between particles (or holes) and sums up the particle-particle (or hole-hole) scattering processes. The method of the equation of motion is used in this paper. Our result for the one-particle energy is shown to lead to the result previously obtained by Quinn and Ferrell, and by Rockmore for the case of the electron gas with Coulomb interactions, when we replace screened potentials by bare potential in the self-consistent energy equation. For nuclear matter, it is shown that the presence of an attractive interaction in the equation of motion for number density causes an "enhancement" of exchange forces, whereas in the electron gas repulsive Coulomb interactions lead to "screening" of the exchange force. The strength of the isospin density interaction pseudopotential is enhanced by a factor of two when one solves the self-consistent equation; and a simple estimate shows that the Goldhaber-Teller mode lies about 15% higher than the value pFqm\frac{{p}_{F}q}{m}mpF​q​ previously estimated by Glassgold et al. (qqq: momentum of the oscillation, pF{p}_{F}pF​: Fermi momentum).
In this work we calculate and compare all the second- and third-order diagrams of the irreducible part K of the electron-hole scattering function for strongly interacting particles near the Fermi surface. We show that the exchange ladder diagrams for K are indeed the most singular diagrams up to this order of perturbation theory in the proximity of the Fermi surface. From this result emerges a new low energy scale parameter and a new effective coupling constant associated with the strongly interacting regime. The summation of exchange ladder diagrams to infinite order produces a purely imaginary pole in K. As a result the vertex function Γ diverges and the Fermi liquid becomes unstable in this strongly interacting regime.