Now we take the following Hamiltonian $$ \hat{H} = \sum_n n \delta \hat{N}_n - G \hat{P}^\dagger \hat{P} $$ where $$ \hat{N}_n = \hat{a}^\dagger_{n, m=+1/2} \hat{a}_{n, m=+1/2} + \hat{a}^\dagger_{n, m=-1/2} \hat{a}_{n, m=-1/2} $$ and $$ \hat{P}^\dagger = \sum_{n} \hat{a}^\dagger_{n, m=+1/2} \hat{a}^\dagger_{n, m=-1/2} $$ We can write down the $ 6 \times 6$ Hamiltonian in the basis from the prior slide: $$ H = \left ( \begin{array}{cccccc} 2\delta -2G & -G & -G & -G & -G & 0 \\ -G & 4\delta -2G & -G & -G & -0 & -G \\ -G & -G & 6\delta -2G & 0 & -G & -G \\ -G & -G & 0 & 6\delta-2G & -G & -G \\ -G & 0 & -G & -G & 8\delta-2G & -G \\ 0 & -G & -G & -G & -G & 10\delta -2G \end{array} \right ) $$ (You should check by hand that this is correct.)

For \( \delta = 0 \) we have the closed form solution of the g.s. energy given by \( -6G \).