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

arXiv:cond-mat/9804011 · Physical review. B, Condensed matter

Dyson equation approach to many-body Green’s functions and self-consistent RPA: Application to the Hubbard model

Steffen Schäfer, Peter Schuck

Published Jan 15, 1999• 23 citations
arXivHTMLPDF

Abstract

An approach for particle-hole correlation functions, based on the so-called self-consistent random-phase approximation is developed. This leads to a fully self-consistent RPA-like theory that satisfies the $f$-sum rule and several other theorems. As a first step, a simpler self-consistent approach, the renormalized RPA, is solved numerically in the one-dimensional Hubbard model. The charge and the longitudinal spin susceptibility, the momentum distribution, and several ground-state properties are calculated and compared with the exact results. Especially at half-filling, our approach provides quite promising results and matches the exact behavior apart from a general prefactor. The strong-coupling limit of our approach can be described analytically.

DOI
10.1103/physrevb.59.1712

Related papers

New self-consistent many-body perturbation theory: Application to the Hubbard model

A. J. Fedro, R. S. Wilson

A new many-body theory perturbationally consistent to all orders of the perturbation is presented and applied to the Hubbard problem as an example. Use is made of the recent Kim and Wilson commutator projection operator to derive the Dyson equation for the one-particle Green's function and an exact set of coupled differential equations for the damping or self-energy term which are solved self-consistently. The Hubbard $I$ result follows in our first approximation and the Esterling-Lange (EL)-type result in the second. Since we keep the hopping finite throughout the calculation, no degeneracy problem is encountered as in EL.

Physical review. B, Solid state•Mar 15, 1975•13 citations

Random phase approximation for the 1D anti-ferromagnetic Heisenberg model.

Aziz Rabhi, Peter Schuck, João da Providência

The Hartree-Fock-RPA approach is applied to the 1D anti-ferromagnetic Heisenberg model in the Jordan-Wigner representation. Somewhat contrary to expectation, this leads to reasonable results for spectral functions and sum rules in the symmetry unbroken phase. In a preliminary application of Self-Consistent RPA to finite size chains strongly improved results are obtained. pacs: 75.40.Gb, 75.50.Ee, 75.10.Pq, 75.10.Jm The 1D Anti-Ferromagnetic Heisenberg Model (AFHM) [1] belongs to one of the most classic many body research fields. It was the first many-body model to be solved exactly by Bethe with his famous Bethe ansatz solution [2] and it has been a playground for testing and developing many body approaches ever since. In spite of tremendous progress in the understanding of many facets of the model, it continues to be a very active field of interest and research [3, 4]. In an attempt to apply extended RPA-like theories, as e.g. the recently developed Self-Consistent RPA (SCRPA) approach [5, 6], we found out, somewhat to our surprise, that even the standard RPA theory has so far not fully been developed in its application to the 1D-AFHM. This probably stems from the fact that the HF-RPA scheme is usually considered as unreliable in low dimensions [7]. However, it will be found that RPA shows interesting features even in this 1D-version of the model. On the other hand SCRPA shows very promising results in a number of cases [5] and in a first application for a small numbers of sites we here get very close agreement with results from exact diagonalization. We mostly consider the isotropic Heisenberg model with anti-ferromagnetic coupling, however some consideration will also be given to the anisotropic case, for instance to the

•Jan 1, 2007•2 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

Extension of Self Consistent RPA method to off-site Hubbard Model

Said Harir, Mohamed Bennai, Y. Boughaleb

We are interested in the possibility that a generalization of Self Consistent Random Phase Approximation (SCRPA) to the Extended Hubbard Model can lead us to accurate estimations of the ground state energy, for closed chains in one dimension with periodic boundary conditions: N=2 (two states problem). We have considered an extended Hubbard model including on-site and off-site interactions, we have compared the SCRPA with a Direct Analytical(DA) results for the ground state energy, and have shown that SCRPA solves the two states problem exactly for any value of U (on-site interaction energy), V2 (off-site interaction energy with opposite spins) and (off-site interaction energy with same spins).

PRSM•Mar 17, 2011•0 citations

Inhomogeneous Gutzwiller approximation with random phase fluctuations for the Hubbard model

G. Seibold, Federico Becca, J. Lorenzana

We present a detailed study of the time-dependent Gutzwiller approximation for the Hubbard model. The formalism, labeled GA+RPA, allows us to compute random-phase-approximation-like (RPA) fluctuations on top of the Gutzwiller approximation (GA). No restrictions are imposed on the charge and spin configurations which makes the method suitable for the calculation of linear excitations around symmetry-broken solutions. Well-behaved sum rules are obeyed as in the Hartree-Fock (HF) plus RPA approach. Analytical results for a two-site model and numerical results for charge-charge and current-current dynamical correlation functions in one and two dimensions are compared with exact and HF+RPA results, supporting the much better performance of GA+RPA with respect to conventional HF+RPA theory.

arXiv:cond-mat/0209351•Physical review. B, Condensed matter•Feb 26, 2003•32 citations

Approximation Des Phases Aleatoires Self-consistante Dans Le Modele De Hubbard

Steffen Schäfer

La RPA Self-Consistante (SCRPA) est appliquée aux fonctions de corrélation particule-trou dans le Modèle de Hubbard. Pour une fonction de Green générale à $n$ corps cette méthode se dérive à partir de l'Equation de Dyson où seules sont retenues les contributions instantanées de l'opérateur de masse. La fonction de Green est alors donnée par un système d'équations intégrales non-linéaires que l'on cherchera à résoudre de façon self-consistante. Elle satisfait, parmi d'autres théorèmes, la règle de somme pondérée par l'énergie. Pour les fonctions de Green à une et à deux particules, la SCRPA obéit à un principe variationnel. Dans le Modèle de Hubbard les fonctions de corrélation de charge et de spin sont calculées en SCRPA. En négligeant les densités connectées à deux corps, nous obtenons une théorie self-consistante plus simple, la RPA renormalisée. Les deux méthodes sont étudiées et comparées à la RPA standard. Nous établissons et résolvons numériquement les équations de la RPA renormalisée pour les fonctions de corrélations de densité de charge dans le Modèle de Hubbard à une dimension. Les susceptibilités de charge et de spin longitudinal, la distribution des impulsions et plusieurs propriétés du fondamental sont évaluées et comparées aux résultats exacts. Dans la limite du couplage fort de la bande à moitié remplie, la RPA renormalisée possède une solution analytique qui est, à un facteur près, en accord avec le développement pour fortes interactions de l'ansatz de Bethe. Comme prévu, des particularités liées à la dimension spatiale $1$, par exemple un comportement de liquide de Luttinger, n'ont pas pu être retrouvées. Or, la description fournie par notre méthode pourrait être assez réaliste en dimensions plus élevées. Une partie de ces travaux a été publié dans "Dyson Equation Approach to Many-Body Greens Functions and Self-Consistent RPA, First Application to the Hubbard Model" Steffen Schäfer, Peter Schuck, Phys. Rev. B 59, 1712-1733 (1999).

•Jan 1, 1998•0 citations
An approach for particle-hole correlation functions, based on the so-called self-consistent random-phase approximation is developed. This leads to a fully self-consistent RPA-like theory that satisfies the fff-sum rule and several other theorems. As a first step, a simpler self-consistent approach, the renormalized RPA, is solved numerically in the one-dimensional Hubbard model. The charge and the longitudinal spin susceptibility, the momentum distribution, and several ground-state properties are calculated and compared with the exact results. Especially at half-filling, our approach provides quite promising results and matches the exact behavior apart from a general prefactor. The strong-coupling limit of our approach can be described analytically.
A new many-body theory perturbationally consistent to all orders of the perturbation is presented and applied to the Hubbard problem as an example. Use is made of the recent Kim and Wilson commutator projection operator to derive the Dyson equation for the one-particle Green's function and an exact set of coupled differential equations for the damping or self-energy term which are solved self-consistently. The Hubbard III result follows in our first approximation and the Esterling-Lange (EL)-type result in the second. Since we keep the hopping finite throughout the calculation, no degeneracy problem is encountered as in EL.
The Hartree-Fock-RPA approach is applied to the 1D anti-ferromagnetic Heisenberg model in the Jordan-Wigner representation. Somewhat contrary to expectation, this leads to reasonable results for spectral functions and sum rules in the symmetry unbroken phase. In a preliminary application of Self-Consistent RPA to finite size chains strongly improved results are obtained. pacs: 75.40.Gb, 75.50.Ee, 75.10.Pq, 75.10.Jm The 1D Anti-Ferromagnetic Heisenberg Model (AFHM) [1] belongs to one of the most classic many body research fields. It was the first many-body model to be solved exactly by Bethe with his famous Bethe ansatz solution [2] and it has been a playground for testing and developing many body approaches ever since. In spite of tremendous progress in the understanding of many facets of the model, it continues to be a very active field of interest and research [3, 4]. In an attempt to apply extended RPA-like theories, as e.g. the recently developed Self-Consistent RPA (SCRPA) approach [5, 6], we found out, somewhat to our surprise, that even the standard RPA theory has so far not fully been developed in its application to the 1D-AFHM. This probably stems from the fact that the HF-RPA scheme is usually considered as unreliable in low dimensions [7]. However, it will be found that RPA shows interesting features even in this 1D-version of the model. On the other hand SCRPA shows very promising results in a number of cases [5] and in a first application for a small numbers of sites we here get very close agreement with results from exact diagonalization. We mostly consider the isotropic Heisenberg model with anti-ferromagnetic coupling, however some consideration will also be given to the anisotropic case, for instance to the
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.
We are interested in the possibility that a generalization of Self Consistent Random Phase Approximation (SCRPA) to the Extended Hubbard Model can lead us to accurate estimations of the ground state energy, for closed chains in one dimension with periodic boundary conditions: N=2 (two states problem). We have considered an extended Hubbard model including on-site and off-site interactions, we have compared the SCRPA with a Direct Analytical(DA) results for the ground state energy, and have shown that SCRPA solves the two states problem exactly for any value of U (on-site interaction energy), V2 (off-site interaction energy with opposite spins) and (off-site interaction energy with same spins).
We present a detailed study of the time-dependent Gutzwiller approximation for the Hubbard model. The formalism, labeled GA+RPA, allows us to compute random-phase-approximation-like (RPA) fluctuations on top of the Gutzwiller approximation (GA). No restrictions are imposed on the charge and spin configurations which makes the method suitable for the calculation of linear excitations around symmetry-broken solutions. Well-behaved sum rules are obeyed as in the Hartree-Fock (HF) plus RPA approach. Analytical results for a two-site model and numerical results for charge-charge and current-current dynamical correlation functions in one and two dimensions are compared with exact and HF+RPA results, supporting the much better performance of GA+RPA with respect to conventional HF+RPA theory.
La RPA Self-Consistante (SCRPA) est appliquée aux fonctions de corrélation particule-trou dans le Modèle de Hubbard. Pour une fonction de Green générale à nnn corps cette méthode se dérive à partir de l'Equation de Dyson où seules sont retenues les contributions instantanées de l'opérateur de masse. La fonction de Green est alors donnée par un système d'équations intégrales non-linéaires que l'on cherchera à résoudre de façon self-consistante. Elle satisfait, parmi d'autres théorèmes, la règle de somme pondérée par l'énergie. Pour les fonctions de Green à une et à deux particules, la SCRPA obéit à un principe variationnel. Dans le Modèle de Hubbard les fonctions de corrélation de charge et de spin sont calculées en SCRPA. En négligeant les densités connectées à deux corps, nous obtenons une théorie self-consistante plus simple, la RPA renormalisée. Les deux méthodes sont étudiées et comparées à la RPA standard. Nous établissons et résolvons numériquement les équations de la RPA renormalisée pour les fonctions de corrélations de densité de charge dans le Modèle de Hubbard à une dimension. Les susceptibilités de charge et de spin longitudinal, la distribution des impulsions et plusieurs propriétés du fondamental sont évaluées et comparées aux résultats exacts. Dans la limite du couplage fort de la bande à moitié remplie, la RPA renormalisée possède une solution analytique qui est, à un facteur près, en accord avec le développement pour fortes interactions de l'ansatz de Bethe. Comme prévu, des particularités liées à la dimension spatiale 111, par exemple un comportement de liquide de Luttinger, n'ont pas pu être retrouvées. Or, la description fournie par notre méthode pourrait être assez réaliste en dimensions plus élevées. Une partie de ces travaux a été publié dans "Dyson Equation Approach to Many-Body Greens Functions and Self-Consistent RPA, First Application to the Hubbard Model" Steffen Schäfer, Peter Schuck, Phys. Rev. B 59, 1712-1733 (1999).