The band gap in many-body theory I would like some help in understanding the definition of the band gap in many-body theory. I have seen several books state that the fundamental band gap is given by
\begin{align*}
E_{\text{gap}} = \varepsilon_{0+}^{QP} - \varepsilon_{0-}^{QP} = E_{0}^{N+1}+E_{0}^{N-1} - 2E_{0}^N.
\end{align*}
Here $\varepsilon_{0\pm}^{QP}$ denote the quasiparticle energies, which measures the energy cost of the addition or removal of an electron from the system of N electrons. How does this definition make sense in terms of the usual understanding of the band gap as the energy difference between the highest occupied valence band state and lowest unoccupied conduction band state? From that definition I would expect the band gap to be given as the $\varepsilon_{0+}^{QP}$, i.e. the energy cost of adding an extra electron to the many-body system.
 A: 
How does this definition make sense in terms of the usual understanding of the band gap as the energy difference between the highest occupied valence band state and lowest unoccupied conduction band state?

Gaps in single electron spectrum
In a crystal gaps appear due to the underlying periodic lattice, as solutions of a one-particle Schrödinger equation. There are actually many alternating bands and gaps (that is ranges of energy with continua of states, and without any states.) These exist regardless of how many electrons we put into the system, and are determined by the underlying potential. Thus, the only case where the cost of adding an electron equals to the gap size is an insulator/semiconductor at zero temperature - in which the last band with electrons (the valence band) is completely filled, and the next band is completely empty.
Gaps in many-body systems
In many-body theory one deals with the gaps emerging due to the interactions between the particles (not due to the underlying potential) - these are gaps of different nature. The gap here characterizes the cost of exciting the system to a higher energy state, which means removing a quasiparticle from below the gap and placing it above the gap. Note that this definition works as well for the insulator/semiconductor - where taking an electron from the top of the valence band and moving it to the bottom of the conduction band costs gap energy.
A: can you provide the books you mentioned at the beginning? About your question, consider a non-interaction two-band case with band gap. Assume we can fill each band with $N$ particles (thermaldynamics limit). If the state we consider has $N$ particles, to add a particle we need energy $E_0^{N+1}-E_0^{N}-\mu=E_{gap}$, and to remove a particle we need energy $E_0^{N-1}-E_0^{N}+\mu=0$. On the other hand, If the state we consider has $N+1$ particles, to add a particle we need energy $E_0^{N+1}-E_0^{N}-\mu=0$, and to remove a particle we need energy $E_0^{N-1}-E_0^{N}+\mu=E_{gap}$. In each case the gap can be expressed by $E_0^{N-1}+E_0^{N+1}-2E_0^{N}=E_{gap}$.
