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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. In 2D Hubbard model in the framework of the Bogoliubov-De Gennes (BdG)
approximation for strong interaction and strong diagonal disorder at low electron densities the
inhomogeneous states are realized in the system with the droplets of the order parameter in the
matrix of unpaired states as well as the percolating insulator-superconductor phase transition
when we increase electron density. We analyze also the model of the inhomogeneous space-
separated Fermi-Bose mixture for the bismuth oxides BaKBiO, which contains the paired
clusters of bosonic states as well as unpaired fermionic clusters. This model explains the
unconventional phase diagram of the system containing the anomalous phases of bosonic
insulator, bosonic semiconductor and bosonic metal. 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.