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arXiv:cond-mat/0010313 · Physical review. B, Condensed matter

Perturbation expansion for the two-dimensional Hubbard model

V. Zlatić, Berislav Horvatić, Branimir Dolički, S Grabowski, P. Entel, K. D. Schotte

Published Dec 29, 2000• 13 citations
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Abstract

We develop an efficient method to calculate the third-order corrections to the self-energy of the hole-doped two-dimensional Hubbard model in space-time representation. Using the Dyson equation we evaluate the renormalized spectral function in various parts of the Brillouin zone and find significant modifications with respect to the second-order theory even for rather small values of the coupling constant U. The spectral function becomes unphysical for $U\ensuremath{\simeq}W,$ where W is the half-width of the conduction band. Close to the Fermi surface and for $U<W,$ the single-particle spectral weight is reduced in a finite energy interval around the Fermi energy. The increase of U opens a gap between the occupied and unoccupied parts of the spectral function.

DOI
10.1103/physrevb.63.035104

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We develop an efficient method to calculate the third-order corrections to the self-energy of the hole-doped two-dimensional Hubbard model in space-time representation. Using the Dyson equation we evaluate the renormalized spectral function in various parts of the Brillouin zone and find significant modifications with respect to the second-order theory even for rather small values of the coupling constant U. The spectral function becomes unphysical for U\ensuremath≃W,U\ensuremath{\simeq}W,U\ensuremath≃W, where W is the half-width of the conduction band. Close to the Fermi surface and for U<W,U<W,U<W, the single-particle spectral weight is reduced in a finite energy interval around the Fermi energy. The increase of U opens a gap between the occupied and unoccupied parts of the spectral function.
The self-energy of the hole-doped two-dimensional (2D) Hubbard model is calculated to second order in the interaction U and the ensuing renormalization of the spectral properties and of the Fermi surface is discussed. In uncorrelated systems the square-shaped Fermi surface separates the Fermi surface closed around the \ensuremath{\Gamma} point (with more holes than particles) from the Fermi surface closed around the Z point (with more particles than holes). In a correlated system a topological change from a Fermi surface centered around the \ensuremath{\Gamma} point to a Fermi surface closed around a Z point is induced, either by increasing the interaction or by diminishing the concentration of holes. The shape of the renormalized spectral function Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) is momentum dependent. Using Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) we evaluate dispersion of single-particle excitations. At low energies the band of quasiparticles with reduced bandwidth is clearly seen. At high energies and far away from the Fermi surface the spectra acquire an additional peak that describes excitations in the Hubbard band. The dispersion in Hubbard bands is weaker along the \ensuremath{\Gamma}-Z, than \ensuremath{\Gamma}-X or X-Z direction. The density of states of a correlated system, as given by the perturbation theory, remains logarithmically singular but the singular weight is reduced with respect to the uncorrelated one. In addition, the correlations transfer the spectral weight out of the low-energy region, where only a narrow Kondo-like structure seems to remain. The spectral properties of the 2D Hubbard model obtained by the truncated perturbation expansion resemble in many ways the recent experimental data on metallic cuprates.
The temperature-dependent self-energy of the hole-doped two-dimensional Hubbard model is calculated perturbatively up to second order and used to find the spectral function for all points in the Brillouin zone. Single-particle dispersion, density of states, and Fermi surface topology are thus obtained and the appearance of the strong-coupling features studied. The renormalized quantities are temperature dependent due to an interplay between the electron correlation and topology of free-electron bands close to the Fermi energy.
We generate the perturbative expansion of the single particle Green's function and related self-energy for a half-filled single-band Hubbard model on a square lattice. We invoke algorithmic Matsubara integration to evaluate single-particle quantities for real and Matsubara frequencies and verify results through comparison to existing data on the Matsubara axis. With low-order expansions at weak coupling we observe a number of outcomes expected at higher orders: the opening of a gap, pseudogap behavior, and Fermi-surface reconstruction. Based on low-order perturbations, we consider the phase diagram that arises from truncated expansions of the self-energy and Green's function and their relation via the Dyson equation. From Matsubara axis data, we observe insulating behavior in direct expansions of the Green's function, whereas the same order of truncation of the self-energy produces metallic behavior. This observation is supported by additional calculations for real frequencies. We attribute this difference to the order in which diagrams are implicitly summed in the Dyson series. By separating the reducible and irreducible contributions at each order we show that the reducible diagrams implicitly summed in the Dyson equation lead to incorrect physics in the half-filled Hubbard model. Our observations for this particular case lead us to question the utility of the Dyson equation for any problem that shows a disparity between reducible and irreducible contributions to the expansion of the Green's function.
Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the noninteracting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical solutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with a broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second-order calculation yields an s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.
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.
Calculations of the one-electron spectral function of the $t\ensuremath{-}J$ model on a two-dimensional square lattice are described. A large-N\mathcal{N}N generalization of the $t\ensuremath{-}J$ model is used to develop 1/N1/\mathcal{N}1/N expansions, and the $t\ensuremath{-}J$ model constraint is enforced by slave bosons. The saddle point about which the 1/N1/\mathcal{N}1/N expansion is carried out is chosen to be uniform in both space and time, and self-energies are calculated to next-to-leading order in 1/N1/\mathcal{N}1/N as in the work of Z. Wang et al. [Phys. Rev. Lett. 67, 2733 (1991)]. The results of computations of the one-electron electron spectral functions for wave vectors throughout the Brillouin zone are presented for several values of the exchange coupling J/tJ/tJ/t and doping $\ensuremath{\delta}$. A sharp dispersive quasiparticle peak is found near the Fermi energy. Additional spectral weight occurs at lower frequencies, and this arises from features in the normal self-energy. These features are qualitatively similar to the self-energy obtained by Kampf and Schrieffer [Phys. Rev. B 41, 6399 (1990)] from a parametrized model of an electron coupled to antiferromagnetic spin fluctuations that leads to shadow band structure in the one-electron spectral function.