  Based on the metastable electron-pair energy band in a two-dimensional (2D)
periodic potential obtained previously by Hai and Castelano [J. Phys.: Condens.
Matter 26, 115502 (2014)], we present in this work a Hamiltonian of many
electrons consisting of single electrons and electron pairs in the 2D system.
The electron-pair states are metastable of energies higher than those of the
single-electron states at low electron density. We assume two different
scenarios for the single-electron band. When it is considered as the lowest
conduction band of a crystal, we compare the obtained Hamiltonian with the
phenomenological model Hamiltonian of a boson-fermion mixture proposed by
Friedberg and Lee [Phys. Rev. B 40, 6745 (1989)]. Single-electron-electron-pair
and electron-pair-electron-pair interaction terms appear in our Hamiltonian and
the interaction potentials can be determined from the electron-electron Coulomb
interactions. When we consider the single-electron band as the highest valence
band of a crystal, we show that holes in this valence band are important for
stabilization of the electron-pair states in the system.
