Bosonic Nambu normal-ordering shift ($a_k{a}_k^{†}=1+a_k^{†}{a}_k$)

In a bosonic Nambu spinor expression, the second diagonal entry commonly contains $a_{-k}{a}_{-k}^{†}$. The canonical commutation relations add one constant per mode when this is normal ordered. For an antiferromagnetic chain coefficient $JS$, rewriting the quadratic Hamiltonian as a full Nambu sum therefore changes the exterior constant from $-NJ{S}^2$ to $-NJS(S+1)$. Omitting the shift corrupts the ground-state energy even when the dispersion is unchanged.