What is the single-electron concept?

Question

In the paper I am reading, single electron concept, single-electron gap, single-electron properties, Single-electron tight-binding description kept coming up and I still don't understand what is this single-electron approximation or single electron concept. It came up in this context:

"There are several mechanisms, which could induce a single-electron gap in a bilayer sample."

and

"Since elementary crystalline unit of graphene includes two carbon atoms A and B, the single-electronic spectrum of the Hamiltonian equation (2) consists of two bands, both of which are doubly degenerate with respect to the spin projection σ."

the paper is:' Electronic properties of graphene-based bilayer systems' https://arxiv.org/abs/1511.06706

Answer 1

There are two interrelated bases for the single-electron concept.

An exact treatment of electronic properties in the condensed matter should deal with a system of $N$ interacting electrons. The exact description of a quantum state for such a system requires an antisymmetric wavefunction of $N$ electrons (fermions). However, it is possible to introduce in many ways approximate theories based on the simplifying assumption that the $N$ electrons wavefunction can be approximated by a Slater determinant or a linear combination of Slater determinants of $N$ one-electron wavefunctions for the $N$ electrons. A prototype for such approximations is the well-known Hartee-Fock approximation. Notice that the Kohn-Sham approach to Density Functional Theory also hinges on a one-electron approximation for obtaining the electron density.

On a more fundamental side, in a quantum many-body system described by the exact formalism, at least for systems like Landau's Fermi liquids (normal states of electrons in metallic systems), it is still possible to introduce the concept of one-particle excitations. That is, it is possible to describe the excited states of the many-electron system as if they were systems of non-interacting decaying particles with a long lifetime (the quasi-particles). A single-electron gap refers to the gap in such quasi-particle energy spectrum.