Magnetism and superconductivity in strongly correlated systems

Current ideas on the interaction between magnetic and superconducting states in strongly magnetized systems are discussed in terms of the Hubbard model and its limiting case in the form of the $t-J$ model. Two approaches to the problem are compared, namely, those of weak and strong Coulomb repulsion, i.e., $U\ll W$ and $U\gg W$, respectively, where $U$ is the repulsion and $W$ the electronic band width. The dynamic magnetic susceptibility of the system is analyzed in both cases, and different types of magnetic instability are identified. Spin fluctuations that grow near the instability boundaries of the paramagnetic phase give rise to Cooper instability. The role of longitudinal and transverse spin fluctuations in the evolution of the superconducting state in a magnetically ordered phase is also investigated. Particular attention is devoted to the two-dimensional model near half-filling. Analytic studies based on the generalized random phase approximation are presented. In addition, a detailed review is given of numerical calculations that involve a single hole in the antiferromagnetic phase and are based on the Monte Carlo method and the exact diagonalization of small clusters. The problem of two interacting holes is also examined. Such studies may provide the conceptual basis for magnetic (correlational) mechanisms of high-T$_c$ superconductivity in copper oxide compounds.