We have applied the fast Fourier transform, which allows one to compute efficiently convolution sums, to solve the set of self-consistent T-matrix equations to get the Green function of the two-dimensional attractive-U Hubbard model below , extending previous calculations by the same authors. Using a constant order parameter , we calculated as a function of the electron density and interaction strength U. These global results deviate from the BCS behaviour remarkably.
We have applied the fast Fourier transform, which allows one to compute efficiently convolution sums, to solve the set of self-consistent T-matrix equations to get the Green function of the two-dimensional attractive-U Hubbard model below , extending previous calculations by the same authors. Using a constant order parameter , we calculated as a function of the electron density and interaction strength U. These global results deviate from the BCS behaviour remarkably.
Using the Dyson equation of motion for both the diagonal one-particle Green function, G(k,ω) and off--diagonal Green function, F(k,ω), at the level of the Hubbard--I decoupling scheme, we have found that they have four poles symmetric in pairs, justifying a more elaborated calculation done by the Zürich group by means of the T-Matrix approach (Pedersen et al, Z. Physik B {\bf 103}, 21 (1997)) and the moment approach of Nolting (Z. Physik {\bf 255}, 25 (1972)). We find that the energy spectra and the weights of G(k,ω) and F(k,ω) have to be calculated self-consistently. G(k,ω) satisfies the first two moments while F(k,ω) the first sum rule. Our {\it order parameter} α(T) is given by 1/Ns∑kε(k)Δ(k). Due to the fact that we have a purely local attractive interaction Δ(k) can be of {\it any} s--type wave. However, for a {\it pure s--wave}, for which α(T)=0, we go back to the mean--field BCS results, with a renormalized chemical potential. In this case, the off--diagonal Green function, F(k,ω), satisfies the first two off--diagonal sum rules. We explicitly state the range of validity of our approximation.
We use the moment approach of Nolting (exact sum rules) (Z. Physik 255, 25 (1972)) for the attractive Hubbard model in the superconducting phase. Our diagonal and off - diagonal spectral functions are constructed and evaluated with the sum rules. They reduce to the BCS limit for weak interaction. However, the presence of correlations modify the BCS picture dramatically. For example, due to the presence of correlations we have postulated a three - pole ansatz for the diagonal Green function, G(k,ω), while the off - diagonal one, B(k,ω), is supposed to have two poles. In the paper we present results for the three spectral weights of the diagonal Green function, αj(k), j = 1,2,3. Our results compare reasonably well with more elaborated auto - consistent highly non - linear equations (double fluctuation calculations in the T− Matrix approach of one of the authors). Then, the physical picture which emerges is that the lower Hubbard band is split due to the superconducting gap and the upper Hubbard band remains mostly unmodified.
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.
We extend previous studies of the (BCS) canonical approach for the attractive Hubbard model. A derivation of the BCS formulation is presented for both the Hubbard model and a simpler reduced Hamiltonian. Using direct diagonalization, exact one- and two-dimensional solutions for both Hamiltonians are compared to BCS variational calculations. Approximate and exact ground state energies and energy gaps are computed for different electron number systems as well as correlation observables not previously predicted. Reproducing published one-dimensional findings, the BCS approach is an excellent approximation for the Hubbard ground state energy but not the energy gap, a finding that remains true in two dimensions. Propagators and correlators are found more sensitive to wave functions and appreciable differences are computed with the Hubbard model exhibiting a weaker degree of superconductivity than the BCS. Even for weak coupling pronounced differences are found between the BCS and Hubbard on-site correlators. Significantly, in the extreme strong coupling limit the Hubbard model exhibits no dimensional crossover, or density induced transitions, to long-range order in two dimensions. However, for the reduced Hamiltonian the BCS approach is an excellent approximation for all observables in both one and two dimensions.
We constructed the one-particle spectral functions (diagonal and off-diagonal) which reproduce BCS for weak coupling and which take into account the effect of correlations on superconductivity in the attractive Hubbard model. The diagonal spectral function is composed of three peaks and the off-diagonal one is composed of two peaks. This ansatz satisfies the sum rules for the first six moments. Our solutions are valid for intermediate coupling, i.e., for U/t ≈ −4.0. Our set of analytical equations for the unknown variables is self-consistent and has been solved numerically in lowest order of the order parameter. As a result, we obtain that the presence of the third band, or upper Hubbard band, strongly renormalizes the two lower bands, making that the energy gap be k-dependent while the order parameter is pure s-wave. This shows that the order parameter and the gap are two different quantities.
On the basis of spin-pairing-fluctuation-exchange approximation, we study the superconductivity in the quasi-two-dimensional Hubbard model. The integral equations for the Green's function are self-consistently solved by numerical calculation. Solutions for the order parameter, London penetration depth, density of states, and transition temperature are obtained. Some of the results are compared with the experiments for the cuprate high-temperature superconductors. Numerical techniques are presented in detail. With these techniques, the amount of numerical computations can be greatly reduced.