The pairing model consists of \( 2N \) fermions that occupy \( N \) of \( P \) energy levels. The fermions can only change energy level by pair. It's Hamiltonian is $$ \begin{align} H=\sum_{p\sigma} \delta_pa_{p\sigma}^{\dagger}a_{p\sigma}+\sum_{pq}g_{pq}a_{p+}^{\dagger}a_{p-}^{\dagger}a_{q-}a_{q+} , \tag{19} \end{align} $$ where \( p \) and \( q \) sum over the set \( \{1,2,...,P\} \) and \( \sigma \) sums over the set \( \{+,-\} \). Also, \( a \) and \( a^{\dagger} \) are the fermionic creation and annihilation operators.