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

Linearized Many-Body Problem

Nobuyuki Fukuda, Fumiaki Iwamoto, K. Sawada

Published Aug 17, 1964• 75 citations
Publisher

Abstract

The theory of an electron gas at high density is generalized so as to include the exchange scattering of a particle and hole in the singlet as well as in the triplet spin state. This approximation is the maximum possible linearization of a quantum-mechanical many-body problem, and corresponds to the theory of small-amplitude oscillations of a classical many-body system. The linearized Hamiltonian is that of Wentzel's meson pair theory, in which now two kinds of mesons are involved, one with spin zero corresponding to the singlet state of a particle and hole and the other with spin one corresponding to the triplet state. The correlation energy is shown to be the sum of the change in zero-point energies of the two-meson fields with correction only in the second-order term. The theory is then formulated in terms of Green's function, and it is proved explicitly that the linearized theory (or the random-phase approximation with exchange effects taken into account) is equivalent to the "ladder approximation" for the particle-hole scattering. The particle-particle scattering by means of the screened interaction is also discussed. The case of the $\ensuremath{\delta}$-function potential is explicitly solved, and it is shown that the correlation energy becomes complex in the region where the instability of the paramagnetic state occurs.

DOI
10.1103/physrev.135.a932

Related papers

Correlations in two-dimensional electron gas: Random-phase approximation with exchange and ladder results

Francesco Pederiva, E. Lipparini, K Takayanagi

We have evaluated the density-density response of the two-dimensional electron gas at zero temperature by solving the Dyson equation for the particle-hole Green's function, including exchange Coulomb matrix elements and short-range contributions in the ladder approximation. We study the effect of these correlations on the total energy, compressibility per particle, local field factor G ( q ), static structure factor and pair-correlation function. Results are compared with the normal random-phase approximation, local field theories and quantum Monte Carlo calculations.

Europhysics Letters (EPL)•Dec 15, 1997•6 citations

Density-functional Green function theory: dynamical exchange-correlation field in lieu of self-energy

F. Aryasetiawan

The one-particle Green function of a many-electron system is traditionally formulated within the self-energy picture. A different formalism was recently proposed, in which the self-energy is replaced by a dynamical exchange-correlation field, which acts on the Green function locally in both space and time. It was found that there exists a fundamental quantity, referred to as the dynamical exchange-correlation hole, which can be interpreted as effective density fluctuations induced in a many-electron system when a hole or an electron is introduced into the system, as in photoemission and inverse photoemission experiments. The dynamical exchange-correlation potential is simply the Coulomb potential of this exchange-correlation hole, which fulfils a sum rule and an exact constraint, identical to those satisfied by the static exchange-correlation hole in density-functional theory. The proposed formalism has been applied to a number of model systems such as the half-filled one-dimensional Hubbard model, the one-dimensional antiferromagnetic Heisenberg model, and the single-impurity Anderson model. The dynamical exchange-correlation hole and field of the homogeneous electron gas have also been studied with the view of constructing a density-functional approximation such as the local-density approximation. The availability of simple but accurate approximations for the exchange-correlation potential would circumvent costly computations of the traditional self-energy. The formalism may also provide new perspectives and insights into the many-body problem.

arXiv:2505.19852•Journal of Physics Condensed Matter•Jul 15, 2025•0 citations

Many-body effects in the one-dimensional electron gas with short-range interaction

Alfred Gold

We discuss interaction effects for the one-dimensional electron gas with a repulsive \ensuremath{\delta}-function interaction potential by using the random-phase approximation and a local-field correction. Analytical results for the local-field correction of charge-density fluctuations and spin-density fluctuations are obtained. The ground-state energy is found to be in better agreement with the exact result than the ground-state energy calculated within the random-phase approximation. We calculate the pair-correlation functions for electrons with parallel and antiparallel spins and the paramagnetic susceptibility. The energies of the collective density modes ${\mathrm{\ensuremath{\omega}}}_{\mathrm{d}}$ (q) and the collective spin modes ${\mathrm{\ensuremath{\omega}}}_{\mathrm{s}}$ (q) are calculated and compared with the energy of the electron-hole excitations ${\mathrm{\ensuremath{\omega}}}_{\mathrm{eh}\ifmmode\pm\else\textpm\fi{}}$ (q): ${\mathrm{\ensuremath{\omega}}}_{\mathrm{d}}$ (q\ensuremath{\rightarrow}0)>${\mathrm{\ensuremath{\omega}}}_{\mathrm{eh}\ifmmode\pm\else\textpm\fi{}}$ (q\ensuremath{\rightarrow}0)>${\mathrm{\ensuremath{\omega}}}_{\mathrm{s}}$ (q\ensuremath{\rightarrow}0). The critical exponents for the long-distance behavior of correlation functions are found to be described by local-field corrections. We compare our results for the critical exponents with recent results obtained by bosonization techniques and the conformal field theory.

Physical review. B, Condensed matter•Apr 15, 1997•17 citations

Correlation beyond the random phase approximation: A consistent many-body perturbation theory approach

F. Bechstedt

The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function $L$. It obeys a Bethe-Salpeter equation (BSE), whose kernel is the functional derivative of the particle self-energy. It is solved on equal footing for two-particle and single-particle Green's functions within Hedin's GW approximation. According to the Hartree and exchange-correlation contributions to the kernel, the correlation expression decays into random-phase-approximation (RPA) -like and screened-exchange-like parts. The actual treatment of the screened potential $W$ in the BSE for $L$ leads to different second-order screened exchange corrections (SOSEX). Including correlation in the kernel even yields approximations beyond SOSEX. The progress beyond RPA is illustrated for spinless and spin-polarized homogeneous electron gases, indicating that the screening treatment is substantial.

Physical review. B./Physical review. B•Jun 15, 2018•5 citations

Many-body perturbation theory: The GW approximation

Christoph Friedrich, Arno Schindlmayr

In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.

•Jan 1, 2006•19 citations

Particle-hole excitations within a self-consistent random-phase approximation

D. Gambacurta, F. Catara

We present a method for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The main differences with respect to previous approaches are underlined and discussed. In particular, no use of renormalized operators is made. By means of the method of linearization of equations of motion, we derive a set of RPA-like equations depending only on the one-body density matrix. The latter is no more assumed to be diagonal and it is expressed in terms of the $X$ and $Y$ amplitudes of the particle-hole phonon operators. This set of nonlinear equations is solved via an iterative procedure, which allows us to calculate the energies and the wave functions of the excited states. After presenting our approach in its general formulation, we test its quality by using metal clusters as a good test laboratory for a generic many-body system. Comparison to RPA shows significant improvements. We discuss also how the present approach can be further extended beyond the particle-hole configuration space, getting an approximation scheme in which energy weighted sum rules are exactly preserved, thus solving a problem common to all extension of RPA proposed until now. The implementation of such approach will be afforded in a future work.

Physical Review B•May 23, 2008•14 citations
The theory of an electron gas at high density is generalized so as to include the exchange scattering of a particle and hole in the singlet as well as in the triplet spin state. This approximation is the maximum possible linearization of a quantum-mechanical many-body problem, and corresponds to the theory of small-amplitude oscillations of a classical many-body system. The linearized Hamiltonian is that of Wentzel's meson pair theory, in which now two kinds of mesons are involved, one with spin zero corresponding to the singlet state of a particle and hole and the other with spin one corresponding to the triplet state. The correlation energy is shown to be the sum of the change in zero-point energies of the two-meson fields with correction only in the second-order term. The theory is then formulated in terms of Green's function, and it is proved explicitly that the linearized theory (or the random-phase approximation with exchange effects taken into account) is equivalent to the "ladder approximation" for the particle-hole scattering. The particle-particle scattering by means of the screened interaction is also discussed. The case of the \ensuremathδ\ensuremath{\delta}\ensuremathδ-function potential is explicitly solved, and it is shown that the correlation energy becomes complex in the region where the instability of the paramagnetic state occurs.
We have evaluated the density-density response of the two-dimensional electron gas at zero temperature by solving the Dyson equation for the particle-hole Green's function, including exchange Coulomb matrix elements and short-range contributions in the ladder approximation. We study the effect of these correlations on the total energy, compressibility per particle, local field factor G ( q ), static structure factor and pair-correlation function. Results are compared with the normal random-phase approximation, local field theories and quantum Monte Carlo calculations.
The one-particle Green function of a many-electron system is traditionally formulated within the self-energy picture. A different formalism was recently proposed, in which the self-energy is replaced by a dynamical exchange-correlation field, which acts on the Green function locally in both space and time. It was found that there exists a fundamental quantity, referred to as the dynamical exchange-correlation hole, which can be interpreted as effective density fluctuations induced in a many-electron system when a hole or an electron is introduced into the system, as in photoemission and inverse photoemission experiments. The dynamical exchange-correlation potential is simply the Coulomb potential of this exchange-correlation hole, which fulfils a sum rule and an exact constraint, identical to those satisfied by the static exchange-correlation hole in density-functional theory. The proposed formalism has been applied to a number of model systems such as the half-filled one-dimensional Hubbard model, the one-dimensional antiferromagnetic Heisenberg model, and the single-impurity Anderson model. The dynamical exchange-correlation hole and field of the homogeneous electron gas have also been studied with the view of constructing a density-functional approximation such as the local-density approximation. The availability of simple but accurate approximations for the exchange-correlation potential would circumvent costly computations of the traditional self-energy. The formalism may also provide new perspectives and insights into the many-body problem.
We discuss interaction effects for the one-dimensional electron gas with a repulsive \ensuremath{\delta}-function interaction potential by using the random-phase approximation and a local-field correction. Analytical results for the local-field correction of charge-density fluctuations and spin-density fluctuations are obtained. The ground-state energy is found to be in better agreement with the exact result than the ground-state energy calculated within the random-phase approximation. We calculate the pair-correlation functions for electrons with parallel and antiparallel spins and the paramagnetic susceptibility. The energies of the collective density modes ${\mathrm{\ensuremath{\omega}}}_{\mathrm{d}}$ (q) and the collective spin modes ${\mathrm{\ensuremath{\omega}}}_{\mathrm{s}}$ (q) are calculated and compared with the energy of the electron-hole excitations ${\mathrm{\ensuremath{\omega}}}_{\mathrm{eh}\ifmmode\pm\else\textpm\fi{}}$ (q): ${\mathrm{\ensuremath{\omega}}}_{\mathrm{d}}$ (q\ensuremath{\rightarrow}0)>${\mathrm{\ensuremath{\omega}}}_{\mathrm{eh}\ifmmode\pm\else\textpm\fi{}}$ (q\ensuremath{\rightarrow}0)>${\mathrm{\ensuremath{\omega}}}_{\mathrm{s}}$ (q\ensuremath{\rightarrow}0). The critical exponents for the long-distance behavior of correlation functions are found to be described by local-field corrections. We compare our results for the critical exponents with recent results obtained by bosonization techniques and the conformal field theory.
The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function LLL. It obeys a Bethe-Salpeter equation (BSE), whose kernel is the functional derivative of the particle self-energy. It is solved on equal footing for two-particle and single-particle Green's functions within Hedin's GW approximation. According to the Hartree and exchange-correlation contributions to the kernel, the correlation expression decays into random-phase-approximation (RPA) -like and screened-exchange-like parts. The actual treatment of the screened potential WWW in the BSE for LLL leads to different second-order screened exchange corrections (SOSEX). Including correlation in the kernel even yields approximations beyond SOSEX. The progress beyond RPA is illustrated for spinless and spin-polarized homogeneous electron gases, indicating that the screening treatment is substantial.
In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.
We present a method for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The main differences with respect to previous approaches are underlined and discussed. In particular, no use of renormalized operators is made. By means of the method of linearization of equations of motion, we derive a set of RPA-like equations depending only on the one-body density matrix. The latter is no more assumed to be diagonal and it is expressed in terms of the XXX and YYY amplitudes of the particle-hole phonon operators. This set of nonlinear equations is solved via an iterative procedure, which allows us to calculate the energies and the wave functions of the excited states. After presenting our approach in its general formulation, we test its quality by using metal clusters as a good test laboratory for a generic many-body system. Comparison to RPA shows significant improvements. We discuss also how the present approach can be further extended beyond the particle-hole configuration space, getting an approximation scheme in which energy weighted sum rules are exactly preserved, thus solving a problem common to all extension of RPA proposed until now. The implementation of such approach will be afforded in a future work.