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Journal of the Physical Society of Japan

Perturbation Theory and Singularities in Two-Dimensional Hubbard Model at Low Electron Density

Hidetoshi Fukuyama, Osamu Narikiyo, Yasumasa Hasegawa

Published Feb 15, 1991• 33 citations
Publisher

Abstract

The two-dimensional Hubbard model has been studied in the region of low electron density based on the t -matrix approximation. This can take proper account of the singular behaviors of the forward scattering whose importance was indicated by Anderson. By performing first the angular integrations in the irreducible particle-particle correlation function, we can prove analytically that all of the poles of the t -matrix lie on the real axis. Hence the t -matrix and then the self-energy of the retarded Green function, Σ( k , z ), are analytic in the upper half-plane of the energy variable, z . This, together with Im Σ( k F , ε+i δ)∝ε 2 ln |ε|, leads to the conclusion that the electrons described by the two-dimensional Hubbard model are Fermi liquid, at least in the region of low electron density.

DOI
10.1143/jpsj.60.372

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The two-dimensional Hubbard model has been studied in the region of low electron density based on the t -matrix approximation. This can take proper account of the singular behaviors of the forward scattering whose importance was indicated by Anderson. By performing first the angular integrations in the irreducible particle-particle correlation function, we can prove analytically that all of the poles of the t -matrix lie on the real axis. Hence the t -matrix and then the self-energy of the retarded Green function, Σ( k , z ), are analytic in the upper half-plane of the energy variable, z . This, together with Im Σ( k F , ε+i δ)∝ε 2 ln |ε|, leads to the conclusion that the electrons described by the two-dimensional Hubbard model are Fermi liquid, at least in the region of low electron density.
The two dimensional Hubbard model with repulsive interaction is studied in the t -matrix approximation, which is the valid approximation in low electron density. The particle-particle correlation function and the t -matrix are obtained analytically. By use of these analytical results it is shown that the t -matrix and hence the self-energy have no singularities in the upper half plane of the complex energy variable. The phase angle δ(2 k F + q , ω), with k F being the Fermi wave vector, in the limit of q =0 and ω=0 is found to depend strongly on the ratio ω/ q and is not always zero. The case of the attractive interaction is also briefly studied.
The low-density regime of the two-dimensional Hubbard model has been investigated by the t-matrix approximation, i.e., by taking into account the particle-particle ladder processes represented in terms of particle-particle correlation function, K(q, x+iδ). The q-dependence of K(q, x+iδ) in the region of small x has been examined in detail and it is found that the imaginary part of self-energy function of electrons depends on excitation energy, ε, as ε2ln|ε| which are due to contributions involving K(q, x+iδ) with q≃2kF, kF being the fermi momentum. The existence of the upper Hubbard band for the case of large U has also been demonstrated.
We consider two- and three-dimensional repulsive Hubbard models at low density, using the gas approximation. The large-U three-dimensional model is shown to behave as a Fermi liquid and the corresponding two-particle scattering length is found. In the two-dimensional case we find that the gas approximation is inconsistent, as indicated by an isolated pole of the two-particle Green function below the occupied states in the upper half-plane. This behaviour is specific to the lattice system as opposed to the continuum two-dimensional Fermi gas with short-range repulsion.
We consider the 2D Hubbard model in the strong-coupling case (U ≫ W) and at low electron density (nd2 ≪ 1). We find an antibound state as a pole in the two-particle T-matrix. The contribution of this pole in the self-energy reproduces a two-pole structure in the dressed one-particle Green-function similar to the Hubbard-I approximation. We also discuss briefly the Engelbrecht-Randeria mode which corresponds to the pairing of two holes below the bottom of the band for U ≫ W and low electron density. Both poles produce nontrivial corrections to Landau Fermi-liquid picture already at low electron density but do not destroy it in 2D.
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 real part of the self-energy of interacting two-dimensional electrons has been calculated in the t-matrix approximation. It is shown that the forward scattering results in an anomalous term leading to the vanishing renormalization factor of the one-particle Green function, which is a non-perturbative effect of the interaction U. The present result is a microscopic demonstration of the claim by Anderson based on the conventional many-body theory. The effect of the damping of the interacting electrons, which has been ignored in reaching above conclusion, has been briefly discussed.