Antiferromagnetic spin wave

An antiferromagnetic spin wave is a small collective variation of the ordered spin pattern in an antiferromagnet. In a bipartite Néel state, neighbouring spins point oppositely; the slowly varying Néel order parameter describes their staggered orientation. A long-wavelength disturbance changes that orientation smoothly rather than independently flipping each site.

For an isotropic ordered antiferromagnet in the usual long-wavelength regime, the low-energy dispersion relation is linear, $\omega \simeq c|\mathbf{k}|$, with spin-wave velocity $c$. This differs from the quadratic low-wavenumber ferromagnetic magnon in a conventional isotropic ferromagnet. Anisotropy or an applied magnetic field can alter the branches or introduce gaps. The starting ordered state also needs justification: strong quantum fluctuations, especially in low dimension, can invalidate a naive linear spin-wave approximation. A computed divergence is then a warning about the approximation, not a negative physical magnetization.

Two sublattices give two bosonic modes for each of N/2 wavevectors in a magnetic Brillouin zone. For a symmetric quadratic pairing block with number coefficient A and anomalous coefficient B, a bosonic Bogoliubov transformation gives two equal positive energies $E_k=\sqrt{A^2-B_k^2}$ and the displayed vacuum shift. There are N modes overall. The quantum depletion of Néel order per site is $(2/N){\sum }_{\mathrm{MBZ}}{v}_k^2$, where $v_k^2=(A/E_k-1)/2$. Equivalently one can use one branch over a full zone with the corresponding factors of one half. Mixing these conventions doubles the vacuum energy or depletion.

The nearest-neighbour isotropic chain has linear-spin-wave energy ${\omega }_k=2JS|\sin k|$ in units $\hslash =a=1$. Gapless points occur at zero and the zone boundary. Integer-spin quantum gaps and strongly fluctuating finite-spin spectra require physics beyond this approximation.

For $H=J{\sum }_{⟨i,j⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$, with $J>0$, dimensionless spin length $S$ and lattice spacing $a$ on a $d$-dimensional hypercubic lattice, the linear spin-wave approximation givesThe angular frequency is ${\omega }_{\mathbf{k}}=E_{\mathbf{k}}/\hslash$. Near a gapless point, $E_{\mathbf{k}}\simeq 2JSa\sqrt{d}|\mathbf{k}|$, so the physical spin-wave velocity is $2JSa\sqrt{d}/\hslash$. Writing the energy formula directly as an angular frequency assumes $\hslash =1$.

The zero-temperature spin-boson occupation is $[1/\sqrt{1-{\gamma }_{\mathbf{k}}^2}-1]/2$, producing quantum depletion of Néel order. Exact zero modes need an infrared cutoff or separate global-rotation treatment. The energy, angular frequency and spin normalization must remain consistent when comparing different conventions.

The spin-wave velocity is the slope $c={\lim }_{k\to 0}\omega (k)/k$ of a linearly dispersing spin wave. For the leading square-lattice Heisenberg antiferromagnet continuum theory, $c=2\sqrt{2}JSa$.

Zeeman splitting is the separation of energy levels or mode frequencies by a background magnetic field. For an antiferromagnetic spin wave, opposite circular spin polarizations receive opposite frequency shifts.