Formation of droplets of the order parameter and superconductivity in inhomogeneous Fermi–Bose mixtures (brief review)

The studies of a number of systems treated in terms of an inhomogeneous (spatially separated) Fermi–Bose mixture with superconducting clusters or droplets of the order parameter in a host medium with unpaired normal states are reviewed. A spatially separated Fermi–Bose mixture is relevant to superconducting Ba-KBiO$_3$ bismuth oxides. Droplets of the order parameter can occur in thin films of a dirty metal, described in the framework of the strongly attractive two-dimensional Hubbard model at a low electron density with a clearly pronounced diagonal disorder. The Bose–Einstein condensate droplets are formed in mixtures and dipole gases with an imbalance in the densities of the Fermi and Bose components. The Bose–Einstein condensate clusters also arise at the center or at the periphery of a magnetic trap involving spin-polarized Fermi gases. Exciton and plasmon collapsing droplets can emerge in the presence of the exciton–exciton or plasmon–plasmon interaction. The plasmon contribution to the charge screening in MgB$_2$ leads to the formation of spatially modulated inhomogeneous structures. In metallic hydrogen and metal hydrides, droplets can be formed in shock-wave experiments at the boundary of the first-order phase transition between the metallic and molecular phases. In a spatially separated Fermi–Bose mixture arising in an Aharonov–Bohm interference ring with a superconducting bridge in a topologically nontrivial state, additional Fano resonances may appear and collapse due to the presence of edge Majorana modes in the system.