Models of the Pseudogap State in High - Temperature Superconductors

We present a short review of our basic understanding of the physics of copper - oxide superconductors and formulate the list of "solved" and "unsolved" problems. The main problem remains theoretical description of the properties of the normal state, requiring clarification of the nature of the so called pseudogap state. We review simplified models of the pseudogap state, based on the scenario of strong electron scattering by (pseudogap) fluctuations of "dielectric" (AFM, CDW) short - range order and the concept of "hot" spots (patches) on the Fermi surface. Pseudogap fluctuations are described as appropriate static Gaussian random field scattering electrons. We derive the system of recurrence equations for the one - particle Green's function and vertex parts, taking into account all Feynman diagrams for electron scattering by pseudogap fluctuations. Results of calculations of spectral density, density of states and optical conductivity are presented, demonstrating both pseudogap and localization effects. We analyze the anomalies of superconducting state (both s- and d - wave pairing) forming on the "background" of these pseudogap fluctuations. Microscopic derivation of Ginzburg - Landau expansion allows calculations of critical temperature T_c and other basic characteristics of a superconductor, depending on the parameters of the pseudogap. We also analyze the role of "normal" (nonmagnetic) impurity scattering. It is shown that our simplified model allows semiquantitative modelling of the typical phase diagram of superconducting cuprates.