We consider a space with \( 2\Omega \) single-particle states, with each state labeled by \( k = 1, 2, 3, \Omega \) and \( m = \pm 1/2 \). The convention is that the state with \( k>0 \) has \( m = + 1/2 \) while \( -k \) has \( m = -1/2 \).

The Hamiltonian we consider is $$ \hat{H} = -G \hat{P}_+ \hat{P}_-, $$ where $$ \hat{P}_+ = \sum_{k > 0} \hat{a}^\dagger_k \hat{a}^\dagger_{-{k}}. $$ and \( \hat{P}_- = ( \hat{P}_+)^\dagger \).

This problem can be solved using what is called the quasi-spin formalism to obtain the exact results. Thereafter we will try again using the explicit Slater determinant formalism.