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Physical review. B, Condensed matter

Ground-state energy of thed=1,2,3 dimensional Hubbard model in the weak-coupling limit

Walter Metzner, D. Vollhardt

Published Mar 1, 1989• 60 citations
Publisher

Abstract

The correlation energy of the Hubbard model on simple-cubic lattices is calculated in the weak-coupling regime for dimensions d=1,2,3. It is shown that, in spite of the nonanalytic dependences known to arise and in contrast to earlier expectations, the exact second-order contribution to the ground-state energy in d=1 can be obtained by ordinary perturbation theory. This result is argued to hold for any finite order and, in particular, for any dimension.

DOI
10.1103/physrevb.39.4462

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The correlation energy of the Hubbard model on simple-cubic lattices is calculated in the weak-coupling regime for dimensions d=1,2,3. It is shown that, in spite of the nonanalytic dependences known to arise and in contrast to earlier expectations, the exact second-order contribution to the ground-state energy in d=1 can be obtained by ordinary perturbation theory. This result is argued to hold for any finite order and, in particular, for any dimension.
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We develop an analytical expression for the self-energy of the infinite-dimensional Hubbard model that is correct in a number of different limits. The approach represents a generalization of the iterative perturbation theory to arbitrary fillings. In the weak-coupling regime perturbation theory to second order in the interaction U is recovered. The theory is exact in the atomic limit. The high-energy behavior of the self-energy up to order 1/E 2 and thereby the first four moments of the spectral density are reproduced correctly. Referring to a standard strong-coupling moment method, we analyze the limit U ↦ → ∞. Different modifications of the approach are discussed and tested by comparing with the results of an exact diagonalization study. I.
We develop an analytical expression for the self-energy of the infinite-dimensional Hubbard model that is correct in a number of different limits. The approach represents a generalization of the iterative perturbation theory to arbitrary fillings. In the weak-coupling regime perturbation theory to second order in the interaction U is recovered. The theory is exact in the atomic limit. The high-energy behavior of the self-energy up to order 1/E2{\mathrm{E}}^{2}E2 and thereby the first four moments of the spectral density are reproduced correctly. Referring to a standard strong-coupling moment method, we analyze the limit U->\ensuremath{\infty}. Different modifications of the approach are discussed and tested by comparing with the results of an exact diagonalization study.
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.
The half-filled extended Hubbard model (including nearest-neighbor interaction) is considered in perturbation theory at weak coupling on a d-dimensional hypercubic lattice (d\ensuremath{\gtrsim}3). It is shown that the exact critical temperature and the exact order parameter in the weak-coupling limit differ from the corresponding Hartree results by a renormalization factor q of order unity. This renormalization factor q is calculated systematically up to O(1/d) in a 1/d expansion. The results depend sensitively on dimension and on the relative strength of the nearest-neighbor interaction. The renormalization factor in d=3, estimated from the 1/d expansion, is q\ensuremath{\simeq}0.282 for the pure Hubbard model, q\ensuremath{\simeq}0.179 near the spin-density-wave to charge-density-wave transition and q\ensuremath{\simeq}0.830 in the limit of pure nearest-neighbor interaction.