Superconducting and normal properties of the lowdimensional phases of the metallic hydrogen
In the first part of the paper we consider properties of a two-dimensional low density (n =1) electron system with strong onsite Hubbard attraction U > W (W is the bandwidth) in the presence of a strong random potential V uniformly distributed in the range from -V to +V. In the framework of the Bogoliubov-de Gennes approach we observed an appearance of inhomogeneous state of spatially separated Fermi-Bose mixture of Cooper pairs and unpaired electrons with the formation of bosonic droplets of different size in the unpaired fermionic matrix. Software package to solve the system of Eliashberg equations that describes the transition of a metal to a superconducting state is presented. We use a version of the Eliashberg system of equations written on the imaginary axis. The system is presented in two versions: without taking into account the corrections for the chemical potential of electrons and with taking them into account. The calculations were performed for the I41/amd phase of metallic hydrogen at a pressure of 500 GPa.