We briefly overview the importance of Hubbard and Anderson-lattice models as applied to explanation of high-temperature and heavy-fermion superconductivity. Application of the models during the last two decades provided an explanation of the paired s
tates in correlated fermion systems and thus extended essentially their earlier usage to the description of itinerant magnetism, fluctuating valence, and the metal-insulator transition. In second part, we also present some of the new results concerning the unconventional superconductivity and obtained very recently in our group. A comparison with experiment is also discussed, but the main emphasis is put on rationalization of the superconducting properties of those materials within the real-space pairing mechanism based on either kinetic exchange and/or Kondo-type interaction combined with the electron correlation effects.
In this brief overview we discuss the principal features of real space pairing as expressed via corresponding low-energy (t-J or periodic Anderson-Kondo) effective Hamiltonian, as well as consider concrete properties of those unconventional supercond
uctors. We also rise the basic question of statistical consistency within the so-called renormalized mean-field theory. In particular, we provide the phase diagrams encompassing the stable magnetic and superconducting states. We interpret real space pairing as correlated motion of fermion pair coupled by short-range exchange interaction of magnitude J comparable to the particle renormalized band energy $sim tx$, where $x$ is the carrier number per site. We also discuss briefly the difference between the real-space and the paramagnon - mediated sources of superconductivity. The paper concentrates both on recent novel results obtained in our research group, as well as puts the theoretical concepts in a conceptual as well as historical perspective. No slave-bosons are required to formulate the present approach.
The coexistence of antiferromagnetism with superconductivity is studied theoretically within the t-J model with the Zeeman term included. The strong electron correlations are accounted for by means of the extended Gutzwiller projection method within
a statistically-consistent approach proposed recently. The phase diagram on the band filling - magnetic field plane is shown, and subsequently the system properties are analyzed for the fixed band filling n=0.97. In this regime, the results reflect principal qualitative features observed recently in selected heavy fermion systems. Namely, (i) with the increasing magnetic field the system evolves from coexisting antiferromagnetic-superconducting phase, through antiferromagnetic phase, towards polarized paramagnetic state, and (ii) the onset of superconducting order suppresses partly the staggered moment. The superconducting gap has both the spin-singlet and the staggered-triplet components, a direct consequence of a coexistence of the superconducting state with antiferromagnetism.
The method used earlier for analysis of correlated nanoscopic systems is extended to infinite (periodic) s-band like systems described by the Hubbard model and its extensions. The optimized single-particle wave functions contained in the parameters o
f the Hubbard model (the hopping textit{t} and the magnitude of the intraatomic interaction textit{U}) are determined explicitly in the correlated state for the electronic systems of various symmetries and dimensions: Hubbard chain, square and triangular planar lattices, and the three cubic lattices (SC, BCC, FCC). In effect, the evolution of the electronic properties as a function of interatomic distance $R$ is obtained. The model parameters in most cases do not scale linearly with the lattice spacing and hence, their solution as a function of microscopic parameters reflects only qualitatively the system evolution. Also, the atomic energy changes with $R$ and therefore should be included in the model analysis. The solutions in one dimension (textit{D} = 1) can be analyzed both rigorously (by making use of the Lieb--Wu solution) and compared with the approximate Gutzwiller treatment. In higher dimensions (textit{D} = 2, 3) only the latter approach is possible to implement within the scheme. The renormalized single particle wave functions are almost independent of the choice of the scheme selected to diagonalize the Hamiltonian in the Fock space in D=1 case. The method can be extended to other approximation schemes as stressed at the end.
