where indices i, j denote localized orbitals and σ denotes spin part α or β, operators a † iσ and a iσ are the creation and annihilation operators of an electron on a localized orbital i with spin σ, and n iσ is a particle number operator: n iσ = a † iσ a iσ . Although Hubbard's model parameterizes only one-particle interactions between the configurations, it includes the Coulomb repulsion U between the conducting (active) electrons, making it sufficiently flexible to describe non-trivial physical phenomena, such as Mott-Hubbard phase transition .