= Bosonic Nambu normal-ordering shift
{title2=$a_ka_k^\dagger=1+a_k^\dagger a_k$}
In a bosonic <Nambu spinor> expression, the second diagonal entry commonly contains $a_{-k}a_{-k}^\dagger$. 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 $-NJS^2$ to $-NJS(S+1)$. Omitting the shift corrupts the <ground-state energy> even when the dispersion is unchanged.
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