Find the excitation spectrum for a weakly interacting bosonic superfluid


 * 1) Identify the Hamiltonian for the superfluid: \(\mathcal{H}=\sum_k \frac{|k|^2}{2m}\hat{C}_k^{\dagger} \hat{C}_k + \frac{g}{V}\sum_{k,p,q}\hat{C}_{k+q/2}^{\dagger}\hat{C}_{-k+q/2}^{\dagger}\hat{C}_{p+q/2}\hat{C}_{-p+q/2}\)
 * 2) Apply the mean field approximation, \(\langle \hat{C}_0^{\dagger} \rangle = \langle \hat{C}_0 \rangle = \left (Vn_0\right)^{1/2}\) and identify all the cases with terms of \(\hat{C}_0^{\dagger}\) and \(\hat{C}_0\) in the interaction term:
 * 3) \(p = q = k = 0\): \(\frac{g}{V}\sum_{k,p,q}\hat{C}_{0}^{\dagger}\hat{C}_{0}^{\dagger}\hat{C}_{0}\hat{C}_{0} = Vgn_0^2\)
 * 4) \(p=q=0\): \(\hat{C}_{k}^{\dagger}\hat{C}_{-k}^{\dagger}\hat{C}_{0}\hat{C}_{0} = gn_0\hat{C}_{k}^{\dagger}\hat{C}_{-k}^{\dagger}\)
 * 5) \(k=q=0\): \(\hat{C}_{0}^{\dagger}\hat{C}_{0}^{\dagger}\hat{C}_{p}\hat{C}_{-p} = gn_0\hat{C}_{p}\hat{C}_{-p}\)
 * 6) \(\frac{q}{2}=p=k\): \(\hat{C}_{q}^{\dagger}\hat{C}_{0}^{\dagger}\hat{C}_{q}\hat{C}_{0} = gn_0\hat{C}_{q}^{\dagger}\hat{C}_{q}\)
 * 7) \(\frac{-q}{2}=p=k\): \(\hat{C}_{0}^{\dagger}\hat{C}_{-q}^{\dagger}\hat{C}_{0}\hat{C}_{-q} = gn_0\hat{C}_{-q}^{\dagger}\hat{C}_{-q}\)
 * 8) Convert all the dummy variables to \(k\) (noting that \(\sum_q \hat{C}^{\dagger}_{-q} \hat{C}_{-q} = \sum_q \hat{C}^{\dagger}_{2k} \hat{C}_{2k}\): \(\mathcal{H}\approx Vgn_0^2 + \sum_k \left(\frac{|k|^2}{2m}\hat{C}_k^{\dagger}\hat{C}_k + gn_0\hat{C}_{k}^{\dagger}\hat{C}_{-k}^{\dagger} + gn_0\hat{C}_{k}^{\dagger}\hat{C}_{k} + gn_0\hat{C}_{-k}^{\dagger}\hat{C}_{-k}\right)\)
 * 9) Rearrange: \(\mathcal{H}\approx Vgn_0^2 + \frac{1}{2}\sum_k \left( \left( \frac{|k|^2}{2m} + 2gn_0\right) \left(\hat{C}_k^{\dagger} \hat{C}_k + \hat{C}_{-k}^{\dagger} \hat{C}_{-k} \right) + 2gn_0\left(\hat{C}_{k}^{\dagger}\hat{C}_{-k}^{\dagger} + \hat{C}_{k}\hat{C}_{-k}\right)\right) \)
 * 10) Identify \(k\) and \(-k\) with separate bosons: \(\hat{C}_k \sim \hat{a}_k\) \(\hat{C}_{-k} \sim \hat{b}_k\)
 * 11) Substitute these identities into the Hamiltonian, making sure that only values of \(k\) where \(k_x > 0\) are counted: \(\mathcal{H}\approx Vgn_0^2 + \sum_k^{\prime} \left( \left( \frac{|k|^2}{2m} + 2gn_0\right) \left(\hat{a}_k^{\dagger} \hat{a}_k + \hat{b}_k^{\dagger} \hat{b}_k \right) + 2gn_0\left(\hat{a}_k^{\dagger}\hat{b}_k^{\dagger} + \hat{a}_{k}\hat{b}_k\right)\right) \)
 * 12) Diagonalise with a Bogoliubov transformation as described here to find the excitation spectrum: \(\hat{H} = \sum_k^{\prime}\left(\sqrt{F^2 - G^2}\left(\hat{\alpha}^{\dagger}\hat{\alpha} + \hat{\beta}^{\dagger}\hat{\beta}\right) + \sqrt{F^2 - G^2} - F\right) + Vgn_0^2\) \(\sqrt{F^2 - G^2} = \sqrt{\left( \frac{|k|^2}{2m} + 2gn_0\right)^2 - \left(2gn_0\right)^2} = \sqrt{\frac{|k|^2}{2m}\left( \frac{|k|^2}{2m} + 2gn_0\right)}\)