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

Physica Scripta

Analytic Properties of the Self-Energy of the Hubbard Model in One and Two Dimensions

V. Zlatić, Berislav Horvatić

Published Jan 1, 1991• 8 citations
Publisher

Abstract

A new method of calculating the properties of correlated electrons propagting on D -dimensional lattices with N D sites is used to study the analytic properties of the Hubbard model in one and two dimensions. The single-particle self-energy is evaluated in the weak-coupling limit around the Fermi energy, ω F , and momentum, p F , at T = 0. For D = 1, the imaginary part of the self-energy, around p F and ω F , behaves in the limit as Im Σ p F (ω F + ω) ~ |ω| for |ω| ~ 1/ N . For D = 2 at half-filling and along the (1,0) direction in the Brillouin zone, the dominate contribution to the self-energy around ω F is estimated as Im Σ p F (ω F + ω) ~ |ω| ln|ω|. For one and two dimensional lattices, the excitation spectra of the U = 0 and U → 0 models become qualitatively different as N → ∞ and can not be simply connected by the perturbation theory.

DOI
10.1088/0031-8949/1991/t39/023

Related papers

Some remarks on second-order perturbation theory for the Hubbard model

V. Zlati, B. Horvati, Gudrun Schliecker +1 more

The single-particle second-order self-energy is calculated numerically for the one-dimensional Hubbard model in the weak-coupling limit. On finite lattices the time- and momentum-dependent self-energy can be computed exactly. As an efficient and simple computer code we use the fast Fourier transform technique. For one and two dimensions and for half-filling, analytical results in the limit ω → 0 are given.

Philosophical Magazine B•Jun 1, 1992•9 citations

WEAK-COUPLING TREATMENT OF THE HUBBARD-MODEL IN THE HIGH-DIMENSIONAL AND LOW-DIMENSIONAL SYSTEMS

Kyuhwe Kang, Byung Il Min

Employing the equation of motion method for the Green's function, we have obtained the self-energy for the Hubbard model in the weak correlation limit up to the 2nd order in U, the on-site interaction strength. The quasi-particle spectral density of states is determined numerically, using the local approximation which is known to be exact for the infinite dimensional Hubbard model. Further, assuming a linear dispersion for the bare electronic spectra for the 1D Hubbard model, we have evaluated analytically the imaginary part of the self-energy, Im Sigma, near the Fermi energy (E(F)) and the renormalization coefficient at E(F), Z(F). It is found that the 1D Hubbard model system exhibits a non-Fermi liquid behavior within the 2nd order perturbational treatment.

•Jan 1, 1995•0 citations

Quasi-Particle Density of States of Two-Dimensional Hubbard Model

Takashi Hotta

The density of states (DOS) of the two-dimensional Hubbard model is calculated in second-order perturbation with respect to the on-site Coulomb interaction U . The direct analytic continuation formula is presented to calculate the self-energy on the real-frequency axis. The momentum as well as the energy integral in the self-energy is seriously carried out with use of the Fast-Fourier-Transformation algorithm. Near the half-filling, the imaginary part of the self-energy becomes proportional to the energy in the vicinity of the Fermi level in our weak-coupling calculation. In the interacting case, the van Hove singularity (VHS) in DOS is smeared by the self-energy effect, but there still exists a sharp peak structure due to VHS.

Journal of the Physical Society of Japan•Aug 15, 1995•6 citations

Second-order self-energy of the Hubbard Hamiltonian: Absence of quasiparticle excitations near half-filling

J. Galán, José Antonio Vergés, A. Martín‐Rodero

The electron self-energy Sigma(k,\ensuremath{\omega}) corresponding to the one-band Hubbard Hamiltonian on clusters of the square lattice has been obtained as a function of the band filling up to second order in the interaction parameter U. The k dependence of the self-energy is completely taken into account. As a consequence, the imaginary part of the self-energy shows a linear behavior for fillings close to one electron per site whereas for large doping rates the quadratic \ensuremath{\omega} dependence characterizing Fermi liquids is recovered. The origin of this behavior has been investigated analytically: linear terms associated with nesting are shown to exist in any finite-dimensional lattice although its numerical relevance decreases very rapidly for space dimensionality larger than two. Implications of these results on the renormalization factor Z have been analyzed. From a practical point of view, we conclude that a standard second-order perturbative treatment is quantitatively precise for U values of the on-site Coulomb interaction smaller than the bandwidth of the noninteracting spectrum.

Physical review. B, Condensed matter•Nov 1, 1993•15 citations

Self-energy calculations in the Hubbard model

Christopher J. Rhodes, R L Jacobs

The authors have calculated the electronic self-energy of the two-dimensional Hubbard model as a function of frequency and momentum for various band fillings. They present this up to second order in the interaction parameter and also up to infinite order for ladder diagrams in the two-particle scattering channel. The authors have carried out direct k-space integrations using a triangle method. For a half-filled perfectly nested band, the imaginary part, lm Sigma (k f , omega ), has a linear- omega dependence for nested Fermi surfaces which is consistent with the marginal Fermi liquid phenomenology for high-T c superconductors. At, and near, half-filling there is a strong low-energy peak in lm Sigma (2) (k, omega ) when k is off the Fermi surface. The physical origin of this peak is discussed. The peak is enhanced in the ladder approximation. The corresponding structure in Re Sigma (2) (k, omega ) leads to multiple solutions of the Dyson equation for a wide range of values of the parameters. This leads to an appealing explanation of observed structure in some experimental angle-resolved photoemission spectra of high-T c materials. The authors also find anti-bound states split off from the band in the two-particle scattering ladders consistent with the suggestion of Anderson.

Journal of Physics Condensed Matter•Aug 2, 1993•4 citations

Interaction-energy functional for lattice density functional theory: Applications to one-, two-, and three-dimensional Hubbard models

Román López-Sandoval, G. M. Pastor

The Hubbard model is investigated in the framework of lattice density functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable of the many-body problem. A new approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ is proposed which is based on its scaling properties and which recovers exactly the limit of strong electron correlations at half-band filling. In this way, a more accurate description of W is obtained throughout the domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}},$ including the crossover from weak to strong correlations. As examples of applications results are given for the ground-state energy, charge-excitation gap, and charge susceptibility of the Hubbard model in one-, two-, and three-dimensional lattices. The performance of the method is demonstrated by comparison with available exact solutions, with numerical calculations, and with LDFT using a simpler dimer ansatz for W. Goals and limitations of the different approximations are discussed.

arXiv:cond-mat/0311470•Physical Review B•Feb 13, 2004•29 citations
A new method of calculating the properties of correlated electrons propagting on D -dimensional lattices with N D sites is used to study the analytic properties of the Hubbard model in one and two dimensions. The single-particle self-energy is evaluated in the weak-coupling limit around the Fermi energy, ω F , and momentum, p F , at T = 0. For D = 1, the imaginary part of the self-energy, around p F and ω F , behaves in the limit as Im Σ p F (ω F + ω) ~ |ω| for |ω| ~ 1/ N . For D = 2 at half-filling and along the (1,0) direction in the Brillouin zone, the dominate contribution to the self-energy around ω F is estimated as Im Σ p F (ω F + ω) ~ |ω| ln|ω|. For one and two dimensional lattices, the excitation spectra of the U = 0 and U → 0 models become qualitatively different as N → ∞ and can not be simply connected by the perturbation theory.
Employing the equation of motion method for the Green's function, we have obtained the self-energy for the Hubbard model in the weak correlation limit up to the 2nd order in U, the on-site interaction strength. The quasi-particle spectral density of states is determined numerically, using the local approximation which is known to be exact for the infinite dimensional Hubbard model. Further, assuming a linear dispersion for the bare electronic spectra for the 1D Hubbard model, we have evaluated analytically the imaginary part of the self-energy, Im Sigma, near the Fermi energy (E(F)) and the renormalization coefficient at E(F), Z(F). It is found that the 1D Hubbard model system exhibits a non-Fermi liquid behavior within the 2nd order perturbational treatment.
The density of states (DOS) of the two-dimensional Hubbard model is calculated in second-order perturbation with respect to the on-site Coulomb interaction U . The direct analytic continuation formula is presented to calculate the self-energy on the real-frequency axis. The momentum as well as the energy integral in the self-energy is seriously carried out with use of the Fast-Fourier-Transformation algorithm. Near the half-filling, the imaginary part of the self-energy becomes proportional to the energy in the vicinity of the Fermi level in our weak-coupling calculation. In the interacting case, the van Hove singularity (VHS) in DOS is smeared by the self-energy effect, but there still exists a sharp peak structure due to VHS.
The electron self-energy Sigma(k,\ensuremath{\omega}) corresponding to the one-band Hubbard Hamiltonian on clusters of the square lattice has been obtained as a function of the band filling up to second order in the interaction parameter U. The k dependence of the self-energy is completely taken into account. As a consequence, the imaginary part of the self-energy shows a linear behavior for fillings close to one electron per site whereas for large doping rates the quadratic \ensuremath{\omega} dependence characterizing Fermi liquids is recovered. The origin of this behavior has been investigated analytically: linear terms associated with nesting are shown to exist in any finite-dimensional lattice although its numerical relevance decreases very rapidly for space dimensionality larger than two. Implications of these results on the renormalization factor Z have been analyzed. From a practical point of view, we conclude that a standard second-order perturbative treatment is quantitatively precise for U values of the on-site Coulomb interaction smaller than the bandwidth of the noninteracting spectrum.
The authors have calculated the electronic self-energy of the two-dimensional Hubbard model as a function of frequency and momentum for various band fillings. They present this up to second order in the interaction parameter and also up to infinite order for ladder diagrams in the two-particle scattering channel. The authors have carried out direct k-space integrations using a triangle method. For a half-filled perfectly nested band, the imaginary part, lm Sigma (k f , omega ), has a linear- omega dependence for nested Fermi surfaces which is consistent with the marginal Fermi liquid phenomenology for high-T c superconductors. At, and near, half-filling there is a strong low-energy peak in lm Sigma (2) (k, omega ) when k is off the Fermi surface. The physical origin of this peak is discussed. The peak is enhanced in the ladder approximation. The corresponding structure in Re Sigma (2) (k, omega ) leads to multiple solutions of the Dyson equation for a wide range of values of the parameters. This leads to an appealing explanation of observed structure in some experimental angle-resolved photoemission spectra of high-T c materials. The authors also find anti-bound states split off from the band in the two-particle scattering ladders consistent with the suggestion of Anderson.
The Hubbard model is investigated in the framework of lattice density functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable of the many-body problem. A new approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ is proposed which is based on its scaling properties and which recovers exactly the limit of strong electron correlations at half-band filling. In this way, a more accurate description of W is obtained throughout the domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}},$ including the crossover from weak to strong correlations. As examples of applications results are given for the ground-state energy, charge-excitation gap, and charge susceptibility of the Hubbard model in one-, two-, and three-dimensional lattices. The performance of the method is demonstrated by comparison with available exact solutions, with numerical calculations, and with LDFT using a simpler dimer ansatz for W. Goals and limitations of the different approximations are discussed.