Number-conserving Bogoliubov quadratic Hamiltonian ($A_k={ϵ}_k+n{V}_k,B_k=n{V}_k$)

At fixed total particle number, $N_0=N-{\sum }_{k\ne 0}{b}_k^{†}{b}_k$. Expanding the condensate interaction energy subtracts $n{V}_0$ from the normal excited-particle coefficient, cancelling its direct Hartree term. The resulting quadratic coefficients are $A_k={ϵ}_k+n{V}_k$ and $B_k=n{V}_k$, with an opposite-momentum anomalous pair term. Replacing $N_0$ by $N$ before making this correction loses the cancellation.