STRONGLY CORRELATED ELECTRON-SYSTEMS - SPIN-REFLECTION POSITIVITY ANDSOME RIGOROUS RESULTS

Theory of spin-reflection positivity developed in recent years is revi ewed. This theory makes use of symmetries in an electron system and th eory of matrix to investigate the ground state properties. Existences of anti-and ferromagnetic long-range orders in itinerant electron syst ems, and of off-diagonal long-range order are two successful applicati ons of the theory. In this article, the author attempt to summarize ex act results proved by utilizing this theory and related topics. First a general theory and basic theorems are introduced. Second, based on t he band structures of conduction electrons, existences of a singlet st ate with strongly antiferromagnetic correlation, a state with both ant i- and ferromagnetic long-range orders, and a fully saturated ferromag netic state are proved. The theory is applied to several of the main t heoretical models for strongly correlated electron systems, such as th e Heisenberg model, the Hubbard model, the Anderson model, the single- and multichannel Kondo model, and the generalized Hubbard model, and a series of rigorous results are found in these models. Third, it is pr oved that off-diagonal long-range order and charge-density wave exist in the ground states of the attractive Hubbard model and the generaliz ed Hubbard model. A relation between pseudospin symmetry and the unifo rm density theorem is introduced. Fourth, the theory is applied succes sfully to explain experimental observations of oscillatory interlayer magnetic coupling in ultrathin magnetic films. Finally several unsolve d problems are discussed. All results introduced in this article are m athematically exact.