Abstract:
The iron-based LaFeAsO$_{1-x}$F$_x$ recently discovered by Hosono's group is a fresh theoretical challenge as a new class of high-temperature superconductors. Here we describe the electronic structure of the material and the mechanism of superconductivity. We start with constructing a tight-binding model in terms of the maximally localized Wannier orbitals from a first-principles electronic structure calculation, which has turned out to involve all the five Fe 3d bands. This is used to calculate the spin and charge susceptibilities with the random phase approximation. The spin susceptibility has peaks around ${\Vec k} = (\pi, 0), (0, \pi)$ arising from a nesting across disconnected Fermi surface pockets. We have then plugged the susceptibilities into the linearised Eliashberg equation. For the doping concentration $x = 0.1$, we obtain an unconventional s-wave pairing, which is roughly an extended s in that the gap changes sign between the Fermi pockets, but the gap function is actually a 5$\times$5 matrix. Its experimental implications are also discussed.

Abstract:
We propose that one of the best grounds for the materials design from the viewpoint of {\it electron correlation} such as ferromagnetism, superconductivity is the atomically controlled nanostructures and heterointerfaces, as theoretically demonstrated here from three examples with first-principles calculations: (i) Band ferromagnetism in a purely organic polymer of five-membered rings, where the flat-band ferromagnetism due to the electron-electron repulsion is proposed. (ii) Metal-induced gap states (MIGS) of about one atomic monolayer thick at insulator/metal heterointerfaces, recently detected experimentally, for which an exciton-mechanism superconductivity is considered. (iii) Alkali-metal doped zeolite, a class of nanostructured host-guest systems, where ferromagnetism has been experimentally discovered, for which a picture of the "supercrystal" composed of "superatoms" is proposed and Mott-insulator properties are considered. These indicate that design of electron correlation is indeed a promising avenue for nanostructures and heterointerfaces.

Abstract:
While the composite fermion picture is so effective as to describe the excitation spectra including the spin wave for Laughlin's quantum liquid, ``how heavy and how strongly-interacting" remains a formidable question for the composite fermions, to which this article first addresses. The effective mass (purely interaction originated) defined from the excitation spectrum and obtained for various even- as well as odd-fractions exhibits a curious, step-like filling dependence basically determined by the number of flux quanta attached to each fermion, where the non-monotonic behaviour indicates a strong effect of gauge-field fluctuations. The excitation spectrum fits a Fermi liquid, but again a large effect of inter-composite fermion interaction appears as anomalous Landau's parameters. We have then moved on to see how the introduction of three-dimensionality (where the shape of the Fermi surface becomes relevant) affects the interacting electron system, and propose the magnetic-field induced SDW in three-dimensional systems. This should be a good candidate, in entirely realistic magnetic fields, for the integer QHE recently predicted by Koshino et al to occur in 3D on the fractal energy spectrum similar to Hofstadter's. The mechanism for the field-induced phase is an effect of interaction in Landau's quantisation on incompletely-nested (i.e., multiply-connected) Fermi surfaces, so the interplay of many-body physics and the magnetic quantisation on various Fermi surfaces may provide an interesting future avenue for 3D systems.

Abstract:
An overview is given on how superconductivity with anisotropic pairing can be realised from repulsive electron-electron interaction. (i) We start from the physics in one dimension, where the Tomonaga-Luttinger theory predicts that, while there is no superconducting phase for the repulsive case for a single chain, the phase does exists in ladders with the number of legs equal to or greater than two, as shown both by analytically (renormalisation) and numerically (quantum Monte Carlo). (ii) We then show how this pairing has a natural extension to the two-dimensional case, where anisotropic (usually d) pairing superconductivity arises mediated by spin fluctuations (usually antiferromagnetic), as shown both by analytically (renormalisation) and numerically (quantum Monte Carlo). (iii) We finally discuss how the superconductivity from the electron repulsion can be "optimised" (i.e., how $T_C$ can be raised) in 2D and 3D, where we propose that the anisotropic pairing is much favoured in systems having {\it disconnected Fermi surfaces} where $T_C$ can be almost an order of magnitude higher.

Abstract:
A theoretical overview of the classes of superconductors encompassing (a) high-Tc cuprate, (b) iron-based and (c) aromatic superconductors is given. Emphasis is put on the multiband natures of all the three classes, where the differences in the multiorbits and their manifestations in the electronic structures and pairing are clarified. From these, future directions and prospects are discussed.

Abstract:
Photovoltaic Hall effect -- the Hall effect induced by intense, circularly-polarized light in the absence of static magnetic fields -- has been proposed in Phys. Rev. B 79, 081406R (2009) for graphene where a massless Dirac dispersion is realized. The photovoltaic Berry curvature (a nonequilibrium extension of the standard Berry curvature) is the key quantity to understand this effect, which appears in the Kubo formula extended to Hall transport in the presence of strong AC field backgrounds. Here we elaborate the properties of the photovoltaic curvature such as the frequency and field strength dependence in the honeycomb lattice.

Abstract:
A theory is presented for a nonequilibrium phase transition in the two-dimensional Hubbard model coupled to electrodes. Nonequilibrium magnetic and superconducting phase diagram is determined by the Keldysh method, where the electron correlation is treated in the fluctuation exchange approximation. The nonequilibrium distribution function in the presence of electron correlation is evoked to capture a general feature in the phase diagram.

Abstract:
For the iron-based high $T_c$ superconductor LaFeAsO$_{1-x}$F$_x$, we construct a minimal model, where all of the five Fe $d$ bands turn out to be involved. We then investigate the origin of superconductivity with a five-band random-phase approximation by solving the Eliashberg equation. We conclude that the spin fluctuation modes arising from the nesting between the disconnected Fermi pockets realise, basically, an extended s-wave pairing, where the gap changes sign across the nesting vector.

Abstract:
Photo-induced metallic states in a Mott insulator are studied for the half-filled, one-dimensional Hubbard model with the time-dependent density matrix renormalization group. An irradiation of strong AC field is found to create a linear dispersion in the optical spectrum (current-current correlation) in the nonequilibrium steady state reminiscent of the Tomonaga-Luttinger liquid for the doped Mott insulator in equilibrium. The spin spectrum in nonequilibrium retains the des Cloizeaux-Pearson mode with the spin velocity differing from the charge velocity. The mechanism of the photocarrier-doping, along with the renormalization in the charge velocity, is analyzed in terms of an effective Dirac model.

Abstract:
A possibility of the electronic origin of the high-temperature superconductivity in cuprates is probed with the quantum Monte Carlo method by revisiting the three-band Hubbard model comprising Cu$3d_{x^2-y^2}$ and O$2p_\sigma$ orbitals. The $d_{x^2-y^2}$ pairing correlation is found to turn into an increasing function of the repulsion $U_d$ within the $d$ orbitals or the $d$-$p$ level off-set $\Delta \varepsilon$, where the correlation grows with the system size. % and is long-ranged as also seen from a real-space analysis. We have detected this in both the charge-transfer and Mott-Hubbard regimes upon entering the strong-correlation region ($U_d$ or $\Delta \varepsilon >$ bare band width).