I need a book that explains the tight binding model in second quantization formalism. for example by telling the meaning of the eigenfunctions of kernel of Hamiltonian in $k$ space. also explains $2$ atom per site or $2$ orbitals per site (is there a difference?). Also there are some lecture notes explains this but, I don't trust them since they are lecture notes.
Mahan's "Many-Particle Physics" explains the tight-binding model in second quantization formalism at pages 20-26 of the third edition. You can find it here: