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.