Spin wave

A spin wave is a smooth long-wavelength fluctuation of the orientation of an ordered continuous-symmetry field. In an $O(2)$ phase it is a fluctuation of the angle $\theta$ governed at leading order by the Gaussian gradient energy.

A magnon is a quantum of a spin wave.

A mobile spin-triplet excitation above a singlet background in a quantum magnet. At low density, an appropriate triplet branch can be described as bosonic quasiparticles with an effective mass and repulsive interaction. Applied magnetic field can lower one branch toward condensation. The term does not imply that the total excitation number is conserved on thermodynamic timescales.

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.

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.