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arXiv:1105.4428 · Low Temperature Physics

Manifestation of the upper Hubbard band in the 2D Hubbard model at low electron density

M. Yu. Kagan, Valerii Vladimirovich Val'kov, Peter Woelfle

Published Oct 1, 2011• 6 citations
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Abstract

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.

DOI
10.1063/1.3670026

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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.
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.
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.
An improved version of Hubbard's treatment of correlation in a nondegenerate narrow band is obtained by the use of a new Green's-function decoupling scheme. The resulting one-electron Green's function has two poles on the real axis corresponding to a splitting of the electron bands due to the strong correlations between electrons on the same site. This is the same result that Hubbard obtained, but in our case the poles are shifted and their positions agree with the results of Harris and Lange, who used a moment technique. The theory is applied to a simple cubic lattice with nearest-neighbor interaction, and the lattice is found to be ferromagnetic in the strongly correlated limit for a sufficiently large number of electrons per atom. An examination of the low-density limit shows that the two-pole approximation does not reduce to Kanamori's TTT-matrix result, a failing which it shares with Hubbard's theory. A method is outlined for improving the theory so as to give the correct low-density result. In addition, the possibility of obtaining a minimum principle for the theory is explored.
Quasiparticle bands of the two-dimensional Hubbard model are calculated using the Roth two-pole approximation to the one-particle Green's function. Excellent agreement is obtained with recent Monte Carlo calculations, including an anomalous volume of the Fermi surface near half-filling, which can possibly be explained in terms of a breakdown of Fermi liquid theory. The calculated bands are very flat around the (\ensuremath{\pi},0) points of the Brillouin zone in agreement with photoemission measurements of cuprate superconductors. With doping there is a shift in spectral weight from the upper band to the lower band. The Roth method is extended to deal with superconductivity within a four-pole approximation allowing electron-hole mixing. It is shown that triplet p-wave pairing never occurs. A self-consistent solution with singlet ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is found and optimal doping occurs when the van Hove singularity, corresponding to the flat band part, lies at the Fermi level. Nearest-neighbor antiferromagnetic correlations play an important role in flattening the bands near the Fermi level and in favoring superconductivity. However, the mechanism for superconductivity is a local one, in contrast to spin-fluctuation exchange models. For reasonable values of the hopping parameter the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ is in the range 10--100 K. The optimum doping ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$ lies between 0.14 and 0.25, depending on the ratio U/t. The gap equation has a BCS-like form and 2${\mathrm{\ensuremath{\Delta}}}_{\mathrm{max}}$/kTc{\mathit{kT}}_{\mathit{c}}kTc​\ensuremath{\simeq}4.