Abstract:
In order to explain how impurities affect the unconventional superconductivity, we study non-magnetic impurity effect on the transition temperature using on-site U Hubbard model within a fluctuation exchange (FLEX) approximation. We find that in appearance, the reduction of Tc roughly coincides with the well-known Abrikosov-Gor'kov formula. This coincidence results from the cancellation between two effects; one is the reduction of attractive force due to randomness, and another is the reduction of the damping rate of quasi-particle arising from electron interaction. As another problem, we also study impurity effect on underdoped cuprate as the system showing pseudogap phenomena. To the aim, we adopt the pairing scenario for the pseudogap and discuss how pseudogap phenomena affect the reduction of Tc by impurities. We find that 'pseudogap breaking' by impurities plays the essential role in underdoped cuprate and suppresses the Tc reduction due to the superconducting (SC) fluctuation.

Abstract:
In the orbitally degenerate ($J=5/2$) Periodic Anderson Model, the magnetic susceptibility is composed of both the Pauli term and the Van Vleck term, as is well known. The former is strongly enhanced by the strong correlation between $f$-electrons. But, for the latter, the influence of the strong correlation has been obscure for years. In this paper we give the solution of the longstanding problem. With the aid of the $d=\infty$ approximation, we study this problem on the basis of the Fermi liquid theory with degenerate orbitals, taking account of all the vertex corrections in a consistent way. As a result, we obtain the simple expression for the magnetic susceptibility, and show unambiguously that the Van Vleck term is also highly enhanced} in the strong correlation regime. This fact explains naturally the enhanced magnetic susceptibility observed in many insulating systems (i.e., Kondo insulator). Moreover, we show that the Wilson ratio takes a value around 1 in the metallic system, in good agreement with experiments.

Abstract:
We analyze the two-dimensional {\it d-p} model, considering both antiferromagnetic spin fluctuation and $d_{x^2-y^2}$-wave superconducting fluctuation. We adopt the fluctuation-exchange approximation in order to derive both normal and anomalous vertices composed only of the renormalized {\it d}-electron Green function and the on-site repulsive interaction among the {\it d}-electrons. Using these vertices, we derive a $t$-matrix as a superconducting fluctuation propagator. Then, we obtain self-consistent solutions in which the system is close to antiferromagnetic instability. In our solutions, the superconducting fluctuation couples strongly with the quasiparticle state and this causes the anomalous behavior in the temperature dependences of spin susceptibility and specific heat.

Abstract:
We discuss a microscopic mechanism of the spin-tiplet superconductivity in the quasi-two-dimensional ruthenium oxide Sr2RuO4 on the basis of two-dimensional three-band Hubbard model. We solve the linearized Eliashberg equation by taking into account the full momentum-frequency dependence of the order parameter for the spin-triplet and the spin-singlet states, and estimate the transition temperature as a function of the Coulomb integrals. The effective pairing interaction is expanded perturbatively with respect to the Coulomb interaction at the Ru sites up to the third order. As a result, we show that the spin-triplet p-wave state is more stable than the spin-singlet d-wave state for moderately strong Coulomb interaction. Our results suggest that one of the three bands, $\gamma$, plays a dominant role in the superconducting transition, and the pairing on the other two bands($\alpha$ and $\beta$) is induced passively through the inter-orbit couplings. The most significant momentum dependence for the p-wave pairing originates from the vertex correction terms, while the incommensurate antiferromagnetic spin fluctuations, which are observed in inelastic neutron scattering experiments, are expected to disturb the p-wave pairing by enhancing the d-wave pairing. Therefore we can regard the spin-triplet superconductivity in Sr2RuO4 as one of the natural results of the electron correlations, and cannot consider as a result of some strong magnetic fluctuations. We will also mention the normal Fermi liquid properties of Sr2RuO4.

Abstract:
We investigate the superconducting mechanism and the transition temperature of heav y fermion superconductor UPd_2Al_3 on the basis of a single band two-dimensional Hubbard model on triangular lattice, whi ch represents the most heavy band of UPd_2Al_3. Both normal and anomalous self-energies are calculated up to third order with respect to the Coulomb repulsion U between itinerant electrons. The superconducting transition temperature is obtained by solving the Eliashberg's equation. Reasonable transition temperatures are obtained for moderately large U. It is fou nd that the momentum and frequency dependence of spin fluctuations given by RPA-like terms gives rise to the d-wave pairing state, while the vertex correction terms are important for obtaining reasonable transition tempe ratures. These results seem to show that the superconductivity in UPd_2Al_3 can be explained by the perturbation theory with respect to U.

Abstract:
The pseudogap phenomena in High-$T_{{\rm c}}$ cuprates are investigated on the basis of the Hubbard model which includes only the on-site repulsive interaction $U$. We consider the pairing scenario for the pseudogap. The pseudogap arises from the resonance scattering due to the strong superconducting fluctuations. First, the electronic state and the anti-ferromagnetic spin fluctuations are calculated by using the FLEX approximation. The T-matrix (the propagator of the superconducting fluctuations) is calculated by extending the $\acute{{\rm E}}$liashberg equation. The self-energy due to the superconducting fluctuations is calculated by the T-matrix approximation. The pseudogap is shown in the single particle properties and the magnetic properties by the microscopic calculation. A comprehensive explanation of the doping dependence of the pseudogap is obtained. Furthermore, we apply the theory to the electron-doped cuprates and obtain the consistent results with the recent experiments. Finally, the self-consistent calculation for the spin fluctuations, superconducting fluctuations and the single particle properties are carried out within the FLEX and the self-consistent T-matrix approximations. The calculated superconducting critical temperature $T_{{\rm c}}$ is remarkably reduced from the results of the mean field (FLEX) calculation. It is shown that the critical temperature decreases with decreasing doping in the under-doped region with large $U$.

Abstract:
We study the superconducting instabilities of singlet and triplet pairing in a two-dimensional Hubbard model on the basis of the third-order perturbation theory (TOPT). We investigate the effect of the vertex correction that is given by TOPT, comparing with the study with only the second-order effective interaction. In our results, a stable p-wave pairing state spreads from low to intermediate electron density. A dx2-y2-wave pairing is dominant for the high density near a half-filling. It is shown that the vertex correction plays an essential role in making the p-wave pairing state dominant.

Abstract:
Magnetic susceptibility in a heavy fermion systemis composed of the Pauli term (\chi_P) and the Van-Vleck term (\chi_V). The latter comes from the interband excitation, where f-orbital degeneracy is essential. In this work, we study \chi_P and \chi_V in the orbitally degenerate (J=5/2) periodic Anderson model for both the metallic and insulating cases. The effect of the correlation between f-electrons is investigated using the self-consistent second-order perturbation theory. The main results are as follows. (i) Sixfold degenerate model: both \chi_P and \chi_V are enhanced by a factor of 1/z (z is the renormalization constant). (ii) Nondegenerate model: only \chi_P is enhanced by 1/z. Thus, orbital degeneracy is indispensable for enhancement of \chi_V. Moreover, orbital degeneracy reduces the Wilson ratio and stabilizes a nonmagnetic Fermi liquid state.

Abstract:
We discuss the possibility of spin-triplet superconductivity in a two-dimensional Hubbard model on a triangular lattice within the third-order perturbation theory. When we vary the symmetry in the dispersion of the bare energy band from D_2 to D_6, spin-singlet superconductivity in the D_2-symmetric system is suppressed and we obtain spin-triplet superconductivity in near the D_6-symmetric system. In this case, it is found that the vertex terms, which are not included in the interaction mediated by the spin fluctuation, are essential for realizing the spin-triplet pairing. We point out the possibility that obtained results correspond to the difference between the superconductivity of UNi_2Al_3 and that of UPd_2Al_3.

Abstract:
We theoretically investigate the effect of magnetic field on the pseudogap phenomena in High-Tc cuprates. The obtained results well explain the experimental results including their doping dependences. In our previous paper (J. Phys. Soc. Jpn. 68 (1999) 2999.), we have shown that the pseudogap phenomena observed in High-Tc cuprates are naturally understood as a precursor of the strong coupling superconductivity. On the other hand, there is an interpretation for the recent high field NMR measurements to be an evidence denying the pairing scenarios for the pseudogap. In this paper, we investigate the magnetic field dependence of NMR $1/T_{1}T$ on the basis of our formalism and show the interpretation to be inappropriate. The results indicate that the value of the characteristic magnetic field $B_{{\rm ch}}$ is remarkably large in case of the strong coupling superconductivity, especially near the pseudogap onset temperature $T^{*}$. Therefore, the magnetic field dependences can not be observed and $T^{*}$ does not vary when the strong pseudogap anomaly is observed. On the other hand, $B_{{\rm ch}}$ is small in the comparatively weak coupling case and $T^{*}$ varies when the weak pseudogap phenomena are observed. These results properly explain the high magnetic field NMR experiments continuously from under-doped to over-doped cuprates. Moreover, we discuss the transport phenomena in the pseudogap phase. The behaviors of the in-plane resistivity, the Hall coefficient and the c-axis resistivity in the pseudogap phase are naturally understood by considering the d-wave pseudogap.