Abstract:
Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the non-interacting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical soutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second order calculation yeilds s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.

Abstract:
Using large-scale dynamical cluster quantum Monte Carlo simulations, we explore the unconventional superconductivity in the hole-doped Hubbard model on the triangular lattice. Due to the interplay of electronic correlations, geometric frustration, and Fermi surface topology, we find a doubly degenerate singlet pairing state at an interaction strength close to the bare bandwidth. Such an unconventional superconducting state is mediated by antiferromagnetic spin fluctuations along the $\Gamma$-$K$ direction, where the Fermi surface is nested. An exact decomposition of the irreducible particle-particle vertex further confirms the dominant component of the effective pairing interaction comes from the spin channel. Our findings provide support for chiral $d +i d$ superconductivity in water-intercalated sodium cobaltates Na$_{x}$CoO$_{2} \cdot y$H$_{2}$O, as well as insight into the superconducting phases of the organic compounds $\kappa$-(ET)$_{2}$X and Pd(dmit)$_{2}$.

Abstract:
Possibility of superconductivity from electron repulsion in the Shastry-Sutherland lattice, which has a spin gap at half filling, is explored with the repulsive Hubbard model in the fluctuation-exchange approximation. We find that, while superconductivity is not favored around the half-filling, superconductivity is favored around the quarter-filling. Our results suggest that the Fermi surface nesting is more important than the spin dimerization for superconductivity.

Abstract:
We study the electronic states of the anisotropic triangular lattice Hubbard model at half filling, which is a simple effective model for the organic superconducting $\kappa$-BEDT-TTF compounds. We treat the effect of the Coulomb interaction by the fluctuation exchange (FLEX) method, and obtain the phase diagram of this model for various sets of parameters. It is shown that the d-wave superconductivity is realized in the wide region of the phase diagram, next to the antiferromagnetic states. The obtained phase diagram explains the characters of the experimental results very well.

Abstract:
A half-filled-band Hubbard model on an anisotropic triangular lattice (t in two bond directions and t' in the other) is studied using an optimization variational Monte Carlo method, to consider the Mott transition and superconductivity arising in \kappa-BEDT-TTF_2X. Adopting wave functions with doublon-holon binding factors, we reveal that a first-order Mott (conductor-to-nonmagnetic insulator) transition takes place at U=U_c approximately of the band width, for a wide range of t'/t. This transition is not directly connected to magnetism. Robust d-wave superconductivity appears in a restricted parameter range: immediately below U_c and moderate strength of frustration (0.4\lsim t'/t\lsim 0.7), where short-range antiferromagnetic correlation sufficiently develops but does not come to a long-range order. The relevance to experiments is also discussed.

Abstract:
We report exact calculations of magnetic and superconducting pair-pair correlations for the half-filled band Hubbard model on an anisotropic triangular lattice. Our results for the magnetic phases are similar to those obtained with other techniques. The superconducting pair-pair correlations at distances beyond nearest neighbor decrease monotonically with increasing Hubbard interaction U for all anisotropy, indicating the absence of frustration-driven superconductivity within the model.

Abstract:
In order to clarify whether the odd-frequency superconductivity can be realized or not, we study a quasi-one-dimensional triangular lattice in the Hubbard model using the random phase approximation (RPA) and the fluctuation exchange (FLEX) approximation. We find that odd-frequency spin-singlet p-wave pairing can be enhanced on a quasi-one-dimensional isosceles triangular lattice.

Abstract:
We propose theoretically that a magnetic field can realize spin-triplet superconductivity in repulsively interacting electron systems having strong ferromagnetic spin fluctuations. We confirm the general idea for the low-density Hubbard model on a triangular lattice, whose Fermi surface consists of disconnected pieces, by calculating the pairing susceptibility in a moderate magnetic field with the quantum Monte-Carlo method combined with the dynamical cluster approximation.

Abstract:
Stimulated by the recent finding of Na$_{0.35}$CoO$_2$.1.3H$_2$O superconductor, we investigate superconducting instabilities on a 2D triangular lattice in the repulsive Hubbard model. Using the third-order perturbation expansion with respect to the on-site repulsion $U$, we evaluate the linearized Dyson-Gor'kov equation. We find that an $f$-wave spin-triplet pairing is the most stable in a wide range of the next nearest neighbor hopping integral $t'$ and an electron number density $n$. The introduction of $t'$ is crucial to adjust the van Hove singularities to the neighborhood of the Fermi surface crossing around K point. In this case, the bare spin susceptibility shows the broad peak around $\Gamma$ point. These conditions stabilize the $f$-wave pairing. Although the $f$-wave pairing is also given by the fluctuation-exchange approximation, the transition temperature is too low to be observed. This is because the depairing effect by the spin fluctuation is over-estimated. Thus, the third-order vertex corrections are important for the spin-triplet superconductivity, like the case in Sr$_2$RuO$_4$.

Abstract:
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.