Bosonic Bogoliubov zero-point shift ($\Delta E_0=\frac{1}{2}{\sum }_k(E_k-A_k)$)

Canonical diagonalization of $A_k{b}_k^{†}{b}_k$ plus an anomalous pair term gives the vacuum shift $(E_k-A_k)/2$ per momentum. Thus $H=C+\frac{1}{2}{\sum }_k(E_k-A_k)+{\sum }_k{E}_k{\alpha }_k^{†}{\alpha }_k$. The subtraction of $A_k/2$ accompanies the quasiparticle zero-point term; omitting it changes the ground-state energy.