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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.

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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 $E_0^{N-1}+E_0^{N+1}-2E_0^{N}=E_{gap}$.

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