Mechanisms of superconductivity and inhomogeneous states in metallic hydrogen and electron systems with attraction

In the Review we discuss anomalous aspects of superconductivity (SC) and normal state, as well as formation of inhomogeneous (droplet-like or cluster-like) states in electron systems with attraction. We consider both the models with the retardation (Eliashberg mechanism of SC for strong electron-phonon interaction in metallic hydrogen) and without retardation (but with local onsite attraction). We concentrate on the mechanism of the BCS-BEC crossover for the Hubbard model with local attraction and diagonal disorder for the two-dimensional films of the dirty metal. We analyze also the model of the inhomogeneous spaceseparated Fermi-Bose mixture for the bismuth oxides BaKBiO, which contains the paired clusters of bosonic states as well as unpaired fermionic clusters. Superconductivity is realized in this system due to local pairs tunneling from one bosonic cluster to the neighboring one via the fermionic barrier. For metallic hydrogen and metallic hydrides, we calculate the critical temperature and discuss important possibility for practical applications how to increase the temperature by decreasing pressure in the framework of the generalized Eliashberg approach. We advocate also interesting analogies with the quantum (vortex) crystal for long-living low-dimensional metastable phases of metallic hydrogen including filamentous phase with proton chains embedded in 3D electron Fermi liquid and planar phase with proton plains. We formulate the concept of two Bose-condensates in SC electron and superfluid (SF) ion subsystems and provide the estimate for the lifetime of the long-living metastable phases at normal pressure. The estimate is connected with the formation and growth of the critical seeds of the new (molecular) phase in the process of quantum under-barrier tunneling.