= Solution
Use dimensionless <spin> operators and nearest-neighbour exchange $J_H>0$ on a regular <bipartite graph> with N sites and <coordination number of a lattice> z. Count each bond once and define $\gamma_{\mathbf k}=z^{-1}\sum_\delta e^{i\mathbf k\cdot\delta}$, real for the inversion-symmetric examples considered here.
For the ferromagnetic <Heisenberg model>, $H_F=-J_H\sum_{\langle ij\rangle}\mathbf S_i\cdot\mathbf S_j$. Expanding the <Holstein–Primakoff transformation> about all spins along +z gives
$$
S_i^z=S-a_i^\dagger a_i,\qquad S_i^+\simeq\sqrt{2S}\,a_i,\qquad
H_F=E_{\rm cl}+J_HS\sum_{\langle ij\rangle}(a_i^\dagger-a_j^\dagger)(a_i-a_j),
$$
up to interactions of higher order in boson density, with $E_{\rm cl}=-NzJ_HS^2/2$. The <isotropic spin-wave vacuum energy comparison> starts with this normal ordering. The <linear spin-wave approximation> then gives
$$
\boxed{E_F(\mathbf k)=zJ_HS(1-\gamma_{\mathbf k}),\qquad
H_F=E_{\rm cl}+\sum_{\mathbf k}E_F(\mathbf k)a_{\mathbf k}^\dagger a_{\mathbf k}.}
$$
Each <ferromagnetic magnon> lowers total $S^z$ by one. The fully aligned state is an exact eigenstate and ground state; there is no harmonic vacuum depletion or added zero-point energy in this normal-ordered Hamiltonian. The global spin-rotation multiplet remains degenerate. On a hypercubic lattice of spacing a, $z=2d$ and $\gamma_{\mathbf k}=d^{-1}\sum_j\cos k_ja$, so $E_F\simeq J_HSa^2k^2$. The exact one-magnon sector agrees with this result even at finite S; interactions enter for multiple magnons.
For the <Heisenberg antiferromagnet>, take $H_A=+J_H\sum_{\langle ij\rangle}\mathbf S_i\cdot\mathbf S_j$ and expand around a <Néel state>. On A, $S^z=S-a^\dagger a$ and $S^+\simeq\sqrt{2S}a$; on B, $S^z=-S+b^\dagger b$ and $S^+\simeq\sqrt{2S}b^\dagger$. A bond becomes
$$
\mathbf S_i\cdot\mathbf S_j\simeq-S^2+S(a_i^\dagger a_i+b_j^\dagger b_j+a_ib_j+a_i^\dagger b_j^\dagger).
$$
With N/2 momenta in the magnetic <Brillouin zone>,
$$
H_A=E_{\rm cl}+zJ_HS\sum_{\mathbf k\in{\rm MBZ}}\left[a_{\mathbf k}^\dagger a_{\mathbf k}+b_{\mathbf k}^\dagger b_{\mathbf k}+\gamma_{\mathbf k}(a_{\mathbf k}b_{-\mathbf k}+a_{\mathbf k}^\dagger b_{-\mathbf k}^\dagger)\right].
$$
The bosonic <Bogoliubov transformation> has $u_k^2-v_k^2=1$ and $\tanh2\theta_k=\gamma_k$. For example $a_k=u_k\alpha_k-v_k\beta_{-k}^\dagger$ and $b_{-k}=u_k\beta_{-k}-v_k\alpha_k^\dagger$. It gives two branches in this reduced zone,
$$
\boxed{E_A(\mathbf k)=zJ_HS\sqrt{1-\gamma_{\mathbf k}^2},\quad
E_{A,0}=E_{\rm cl}+\sum_{{\rm MBZ}}[E_A(\mathbf k)-zJ_HS],\quad
v_k^2=\frac12\left[\frac1{\sqrt{1-\gamma_k^2}}-1\right].}
$$
The <reduced-zone counting of bipartite spin waves> gives N oscillators in total, not 2N. The negative vacuum correction and <quantum depletion of Néel order>, $\delta S=(2/N)\sum_{{\rm MBZ}}v_k^2$, distinguish the antiferromagnet from the ferromagnet. At nonzero temperature replace $v_k^2$ by $v_k^2+(u_k^2+v_k^2)n_B(E_k)$. Hypercubic small-momentum modes have $E_A\simeq2J_HSa\sqrt d\,|k|$, hence linear physical frequency $E_A/\hbar$.
Self-consistency requires depletion much smaller than S and respects the <Holstein–Primakoff occupation constraint>. In one dimension the antiferromagnetic vacuum depletion diverges logarithmically, invalidating ordered spin-wave theory; it does not alone prove an exact gap or its absence. At positive temperature, isotropic short-range models in d≤2 cannot sustain such magnetic order by the <Mermin-Wagner theorem>. In three dimensions the quadratic ferromagnetic modes give the low-temperature magnetization reduction proportional to $T^{3/2}$, whereas the antiferromagnetic thermal reduction scales as $T^2$. Anisotropy, fields, frustration and magnon interactions alter these conclusions. <Physics 215C: Quantum Field Theory, Part 3 by John McGreevy>[§4.1, contrasting ferromagnetic and antiferromagnetic low-energy dynamics]{ref}
\Image[/past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2012/paper-1-qcm2012-spin-waves.png]
{title=Spin-wave comparison}
{source=paper-1-qcm2012-spin-waves.py}
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