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arXiv:cond-mat/9802107 · Journal of Physics Condensed Matter

A self-consistent, conserving theory of the attractive Hubbard model in two dimensions

Martin Letz, R J Gooding

Published Aug 10, 1998• 17 citations
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Abstract

We have investigated the attractive Hubbard model in the low-density limit for the 2D square lattice using the ladder approximation for the vertex function in a self-consistent, conserving formulation. In the parameter region where the on-site attraction is of the order of the bandwidth, we found no evidence of a pseudo-gap. Furthermore, we have observed that the suppression of the Fermi surface known to destroy superconductivity in one and two dimensions, when these systems are treated using a non-self-consistent theory (Schmitt-Rink, Varma C M and Ruckenstein A E 1989 Phys. Rev. Lett. 63 445), does not occur when pair-pair interactions are included. However, we do find a quasi-particle lifetime that varies linearly with temperature, which is similar to the findings from many experiments. Thus, although this system has a Fermi surface, it shows non-Fermi-liquid-type behaviour over a wide temperature range. We stress that our work uses thermal Green's functions along the real-time axis, and thus allows for a more accurate determination of the dynamical properties of a model than theories that require extrapolations from the imaginary-frequency axis.

DOI
10.1088/0953-8984/10/31/011

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We have investigated the attractive Hubbard model in the low-density limit for the 2D square lattice using the ladder approximation for the vertex function in a self-consistent, conserving formulation. In the parameter region where the on-site attraction is of the order of the bandwidth, we found no evidence of a pseudo-gap. Furthermore, we have observed that the suppression of the Fermi surface known to destroy superconductivity in one and two dimensions, when these systems are treated using a non-self-consistent theory (Schmitt-Rink, Varma C M and Ruckenstein A E 1989 Phys. Rev. Lett. 63 445), does not occur when pair-pair interactions are included. However, we do find a quasi-particle lifetime that varies linearly with temperature, which is similar to the findings from many experiments. Thus, although this system has a Fermi surface, it shows non-Fermi-liquid-type behaviour over a wide temperature range. We stress that our work uses thermal Green's functions along the real-time axis, and thus allows for a more accurate determination of the dynamical properties of a model than theories that require extrapolations from the imaginary-frequency axis.
The attractive Hubbard model on a 2-D square lattice is studied at low electronic densities using the ladder approximation for the pair susceptibility. This model includes (i) the short coherence lengths known to exist experimentally in the cuprate superconductors, and (ii) two-particle bound states that correspond to electron pairs. We study the quasiparticle lifetimes in both non self-consistent and self-consistent theories, the latter including interactions between the pairs. We find that if we include the interactions between pairs the quasiparticle lifetimes vary approximately linearly with the inverse temperature, consistent with experiment.
The authors consider the negative-U Hubbard model on a square lattice in the self-consistent T-matrix approximation. They investigate the influence of the Fermi sea on the bound states which always occur in two spatial dimensions in the presence of an attractive interaction. As long as the binding energy is smaller than the Fermi energy, these bound states are found to be broadened sufficiently that Fermi liquid behaviour is restored. Only in the limit of large binding energy does one expect coexisting stable Bose pairs and dissociated fermions leading to a breakdown of Fermi liquid 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 TTT-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 TTT-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 TTT-matrix approximation is shown to be successful in reproducing the exact results for the atomic limit, while fully self-consistent TTT-matrix results do not agree with the known solutions. Of particular note is that the minimally self-consistent TTT-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 TTT-matrix theory can give better results over a broader temperature range.
A frequency- and momentum-renormalization-group acceleration together with an analytical approach is used to obtain the retarded Green's function in the self-consistent and conserving fluctuation-exchange (FLEX) approximation for the two-dimensional Hubbard model in the normal state and in the superconducting state. The analytical expressions for this approach are given. For band-fillings near half filling the self-energy in the normal state exhibits Fermi-liquid behaviour for, low temperatures and frequencies near the chemical potential, if the momentum is chosen near the Fermi-surface. However due to the presence of large many body effects the observed Fermi-liquid region near the chemical potential and near the Fermi-surface is very small. Results for the single particle density of states, the occupation number and the spectral function are presented. The superconducting state with symmetry is obtained for U = 2 to U = 6 and a (U, n)-phase diagram for the transition temperature Tc is presented. A maximum Tc/t of 0.0275 is obtained for U = 6 near half filling.
In this thesis, I present a non-perturbative approach to the single-band attractive Hubard model which is an extension of previous work by Vilk and Tremblay on the repulsive model. Exact results are derived in the general context of functional derivative approaches to many-body theories. The first step of the approximation is based on a local field type ansatz. All physical quantities can be expressed as a function of double-occupancy (in addition to temperature and filling). Double-occupancy is determined without adjustable parameter by imposing the Pauli principle and a crucial sum-rule, making the first step of the approximation Two-Particle Self-Consistent. The final expression for the self-energy is obtained by calculating the low-frequency part of the exact expression with the two-particle correlation, Green function and renormalized vertex obtained in the first step of the approximation. The Mermin-Wagner theorem in two dimensions is automatically satisfied. Application of this non-perturbative many-body approach to the intermediate coupling regime shows quantitative agreement with quantum Monte Carlo calculations. Both approaches predict the existence of a pseudogap in the single-particle spectral weight. I present some physical properties, such as correlation lengths, superfluid density, and characteristic pair fluctuation energy, to highlight the origin of the pseudogap in the weak to intermediate coupling regime. These results suggest that two-dimensional systems that are described by a symmetry group larger than SO(2) could have a larger region of pseudogap behavior. High-temperature superconductors may belong to that category of systems.
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.