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 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.
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 atomic limit of the Hubbard model is a simple single-site problem which can be solved exactly, and all one- and two-particle Green's functions can be obtained analytically. These solutions can thus serve as a means of critiquing the success of various approximate theories which might be applied to the full Hubbard model. In particular, we have examined the
T-matrix approximation for the attractive Hubbard model in the atomic limit, which should give reasonable results at low electronic densities, if one can avoid the spurious phase transition that results when a fully non-self-consistent
T-matrix approximation is employed---previously we have shown that any level of self-consistency guarantees that this phase transition is correctly suppressed to zero temperature in two dimensions or less. Here, a minimally self-consistent
T-matrix approximation is shown to be successful in reproducing the exact results for the atomic limit, while fully self-consistent
T-matrix results do not agree with the known solutions. Of particular note is that the minimally self-consistent
T-matrix approximation reproduces not only one- and two-particle (static) thermodynamic quantities, but it also exactly reproduces the one-particle spectral function at low but nonzero temperatures. We also make a comparison to the two-particle self-consistent approach of Vilk and Tremblay, and find that the minimally self-consistent
T-matrix theory can give better results over a broader temperature range.