I investigate corrections to the mean-field approximation for a superconductor at zero temperature using a single-band Hubbard model with an on-site attractive interaction U. These corrections arise from treating the order parameter as an operator rather than as a number. The quasiparticle spectra may be characterized by a renormalized gap and an effective mass that fall rapidly as \ensuremath{\Vert}k-
kF\ensuremath{\Vert} increases. For ${\mathrm{\ensuremath{\Delta}}}_{0}$\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}2}$ of the bandwidth, which estimates suggest is the appropriate range of values for the high-temperature superconductors, one finds that ${\mathrm{\ensuremath{\Delta}}}_{0}$ is reduced by \ensuremath{\sim}20% both for a parabolic band and for the tight-binding band structure at half-filling. This is comparable to the magnitude of renormalization found for pairing bags.
Possible ordered states in the 2D extended Hubbard model with on-site (U>0) and nearest-neighbor (V) interaction are examined near half filling, with emphasis on the effect of finite V. First, the phase diagram at absolute zero is determined in the mean field approximation. For
V<0, a state where d_{x^{2}-y^{2}}-wave superconductivity (dSC), commensurate spin-density-wave (SDW) and
π-triplet pair coexist is seen to be stabilized. Here, the importance of
π-triplet pair on the coexistence of dSC and SDW is indicated. This coexistent state is hampered by the phase separation (PS), which is generally expected to occur in the presence of finite-range attractive interaction, but survives. For V>0, a state where commensurate charge-density-wave (CDW), SDW and ferromagnetism (FM) coexist is seen to be stabilized. Here, the importance of FM on the coexistence of CDW and SDW is indicated. Next, in order to examine the effects of fluctuation on each mean field ordered state, the renormalization group method for the special case that the Fermi level lies just on the saddle points, (
π,0) and (0,
π), is applied. The crucial difference from the mean field result is that superconductivity can arise even for U>0 and
V≥0, where the superconducting gap symmetry is d_{x^{2}-y^{2}}-wave for U>4V and s-wave for U<4V. Finally, the possibilities that the mean field coexistent states survive in the presence of fluctuation are discussed.
We investigate the Hubbard model on a two-dimensional square lattice by the perturbation expansion to the fourth order in the on-site Coulomb repulsion U. Numerically calculating all diagrams up to the fourth order in self-energy, we examine the convergence of perturbation series in the lattice system. We indicate that the coefficient of each order term rapidly decreases as in the impurity Anderson model for T > 0.1t in the half-filled case, but it holds in the doped case even at lower temperatures. Thus, we can expect that the convergence of perturbation expansion in U is very good in a wide parameter region also in the lattice system, except for T < 0.1t in the half-filled case. We next calculate the density of states in the fourth-order perturbation. In the half-filled case, the shape in a moderate correlation regime is quite different from the three peak structure in the second-order perturbation. Remarkable upper and lower Hubbard bands locate at w = +(-)U/2, and a pseudogap appears at the Fermi level w=0. This is considered as the precursor of the Mott-Hubbard antiferromagnetic structure. In the doped case, quasiparticles with very heavy mass are formed at the Fermi level. Thus, we conclude that the fourth-order perturbation theory overall well explain the asymptotic behaviors in a strong correlation regime.
By use of a diagrammatic method we determine the density-density response functions in the random-phase approximation (RPA) and the collective-excitation spectrum for the extended Hubbard model with on-site attraction and arbitrary electron density, in the superconducting ground state. The energy of the collective modes, given by the poles of these response functions, is found to be a linear function of the wave vector (for small k and short-range intersite interaction) with the velocity interpolating smoothly between the weak- (\ensuremath{\propto}2zt, \ensuremath{\Vert}U\ensuremath{\Vert}\ensuremath{\ll}2zt) and strong-coupling (\ensuremath{\propto}
zt2/\ensuremath{\Vert}U\ensuremath{\Vert}, \ensuremath{\Vert}U\ensuremath{\Vert}\ensuremath{\gg}2zt) limits. The latter agrees with the results obtained from an effective pseudospin Hamiltonian valid in the strong-coupling limit. The numerical analysis for a two-dimensional (2D) square lattice shows the occurrence of a rotonlike minima near the zone boundary. In the weak-coupling regime we find that apart from a commensurate charge-density-wave (CDW) instability, an increase in the intersite Coulomb repulsion can give rise to a CDW incommensurate with the lattice period, away from half-filling. The resulting phase diagram for the 2D square lattice, including a singlet superconducting ground state, electronic-droplet formation, and CDW's, is determined. Finally, the effects of a long-range Coulomb interaction are analyzed briefly and it is shown that the energy of collective excitations evolves smoothly from the weak- to the strong-coupling limit for a 2D lattice.
The two-dimensional Hubbard model is studied for small values of the interaction strength (
U of the order of the hopping amplitude
t), using a variational ansatz well suited for this regime. The wave function, a refined Gutzwiller ansatz, has a BCS mean-field state with
d-wave symmetry as its reference state. Superconducting order is found for densities
n<1 (but not for
n=1). This resolves a discrepancy between results obtained with the functional renormalization group, which do predict superconducting order for small values of
U, and numerical simulations, which did not find superconductivity for $U\ensuremath{\lesssim}4t$. Both the gap parameter and the order parameter have a domelike shape as a function of
n with a maximum for $n\ensuremath{\approx}0.8$. Expectation values for the energy, the particle number, and the superconducting order parameter are calculated using a linked-cluster expansion up to second order in
U. In this way large systems (millions of sites) can be readily treated and well converged results are obtained. A big size is indeed required to see that the gap becomes very small at half filling and probably tends to zero in the thermodynamic limit, whereas away from half filling a finite asymptotic limit is reached. For a lattice of a given size the order parameter becomes finite only above a minimal coupling strength
Uc. This threshold value decreases steadily with increasing system size, which indicates that superconductivity exists for arbitrarily small
U for an infinite system. For moderately large systems the size dependence is quite irregular, due to variations in level spacings at the Fermi energy. The fluctuations die out if the gap parameter spans several level spacings.
We investigate with the aid of the 1/N expansion the dispersion relation of the collective mode in the infinite-U Hubbard model in the presence of a long-range Coulomb term. We calculate this numerically in three dimensions and for a parabolic dispersion for the full range of fillings. For moderate fillings the results are consistent with Landau theory and weak-coupling approaches, but for fillings close to one-half the nature of the collective mode is completely changed. In this limit it becomes equivalent to a movement of the electron system as a whole, thereby avoiding the U=\ensuremath{\infty} restriction. This mode never decays into single-particle excitations, and the mass scale on which it appears is that of the free-electron system.