Linear spin-wave approximation

The linear spin-wave approximation expands a Holstein–Primakoff transformation about a classical ordered state and retains the quadratic oscillator Hamiltonian. It gives the leading order-$S$ excitation energies above the order-$S^2$ classical energy. In a Heisenberg antiferromagnet, first make a bipartite spin rotation. The approximation also predicts a zero-point reduction of the order parameter; infrared-divergent quantum depletion of Néel order signals that the assumed ordered state is not a valid thermodynamic starting point.

For the XY model with equal ferromagnetic exchange $J$ in its plane and field term $-hS{\sum }_j{S}_j^x$, rotate the local quantization axis along the Classical spin. The Holstein–Primakoff transformation gives the number coefficient ${\alpha }_k=S(|h|+2J-J\cos k)$ and anomalous coefficient ${\Delta }_k=JS\cos k$ with one transverse-axis convention. Bosonic Bogoliubov diagonalization gives ${ϵ}_k=\sqrt{{\alpha }_k^2-{\Delta }_k^2}$ and field gap $S\sqrt{|h|(|h|+2J)}$. A nonzero field stabilizes every oscillator mode.

The Bogoliubov vacuum has original-boson occupation $v_k^2=({\alpha }_k/{ϵ}_k-1)/2$, reducing the ordered spin by its average over momenta. For the zero-field planar ferromagnetic chain, $v_k^2\sim 1/(2\sqrt{2}|k|)$. Even after excluding the exact uniform zero mode, the sum grows as $(2\pi \sqrt{2}{)}^{-1}\log L$. It invalidates a thermodynamic fixed-orientation expansion with finite dilute depletion; it does not represent a negative physical magnetization.

For a polarized spin chain with dispersion relation $ϵ(k)=2S[J(1+u)-J\cos k-D\sin k]$, the first zero mode appears at the maximum over the allowed ring momenta. In the thermodynamic limit this threshold is $\sqrt{1+(D/J{)}^2}-1$, since the maximum of $J\cos k+D\sin k$ is $\sqrt{J^2+D^2}$. At finite size the minimizing wave vector need not be available, so a positive continuum threshold need not give a positive finite-ring threshold. For $L=4$ and $0<D<J$, the latter is zero.

The polarized ground state of an isotropic Heisenberg model with ferromagnetic exchange is an exact state. Its normal-ordered harmonic Hamiltonian has no pair-creation terms, no vacuum magnon occupation and no additional zero-point shift. An antiferromagnetic Néel state instead generates anomalous terms, a squeezed Bogoliubov transformation vacuum, a negative energy correction and reduced staggered magnetization. Neither statement licenses omitting normal-ordering constants from an arbitrary spin-wave model with anisotropy or a different reference state.

For a rotated nearest-neighbour chain, a full-zone sum of $JS(a_k^{†},a_{-k})M_k(a_k,a_{-k}^{†}{)}^T$ contains a normal-ordering constant $JSN$. Rewriting the classical constant as $-JNS(S+1)$ compensates it. One must not additionally halve the sum without changing mode counting.

With unit lattice spacing and $\hslash =1$, a nearest-neighbour Heisenberg model has quadratic low-momentum ferromagnetic magnons and linear antiferromagnetic modes in the linear spin-wave approximation. The ferromagnetic polarized vacuum is an exact ground state, whereas the assumed antiferromagnetic ordered vacuum has quantum depletion of Néel order. In one dimension its logarithmic infrared divergence invalidates finite ordered magnetization and prevents the linear approximation from deciding the exact gap.

For $0<\Delta <1$, choose a planar Néel reference and rotate local axes to align its longitudinal spins. One phase convention gives the displayed full-zone spectrum, linear near $k=0$ with slope $JS\sqrt{2(1+\Delta )}$, and gapped at $k=\pi$ by $2JS\sqrt{2(1-\Delta )}$. Shifting a sublattice phase can interchange the momentum locations. Folding the magnetic zone separates the gapless in-plane mode from the gapped out-of-plane polarization. An easy-axis reference in this regime is unstable and does not supply this physical expansion.

After a bipartite spin rotation, a leading Holstein–Primakoff transformation expansion gives $H_2=-J\Delta N{S}^2+JS{\sum }_k[2\Delta n_k+\cos k(a_k{a}_{-k}+a_k^{†}{a}_{-k}^{†})]$. The bosonic Bogoliubov diagonalization gives the displayed spectrum and vacuum energy $-J\Delta N{S}^2+JS{\sum }_k[\sqrt{{\Delta }^2-{\cos }^{2}k}-\Delta ]$. The anisotropy gap is $2JS\sqrt{{\Delta }^2-1}$. At $\Delta =1$ the finite-angle transformation is singular at its zero modes, and quantum depletion of Néel order is infrared divergent in one dimension.