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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 manyelectrons consisting of single electrons and electron pairs in the 2D system.The electron-pair states are metastable of energies higher than those of thesingle-electron states at low electron density. We assume two differentscenarios for the single-electron band. When it is considered as the lowestconduction band of a crystal, we compare the obtained Hamiltonian with thephenomenological model Hamiltonian of a boson-fermion mixture proposed byFriedberg and Lee [Phys. Rev. B 40, 6745 (1989)]. Single-electron-electron-pairand electron-pair-electron-pair interaction terms appear in our Hamiltonian andthe interaction potentials can be determined from the electron-electron Coulombinteractions. When we consider the single-electron band as the highest valenceband of a crystal, we show that holes in this valence band are important forstabilization of the electron-pair states in the system.

Hamiltonian of a many-electron system with single-electron and electron-pair states in a two-dimensional periodic potential