Antiferromagnetic spin wave

An antiferromagnetic spin wave is a long-wavelength fluctuation of staggered magnetic order. In an isotropic ordered antiferromagnet its low-energy dispersion is linear.

For a nearest-neighbour Heisenberg antiferromagnet on a hypercubic lattice, the linear spin-wave approximation gives ${\gamma }_{\mathbf{k}}=d^{-1}{\sum }_j\cos (k_{j}a)$ and the displayed frequency. Near a gapless point its spin-wave velocity is $2JSa\sqrt{d}$. Its ground-state boson occupation is $[1/\sqrt{1-{\gamma }_{\mathbf{k}}^2}-1]/2$, producing quantum depletion of Néel order. Exact zero modes need an infrared regulator or separate global-rotation treatment.

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. Opposite circular spin polarizations receive opposite frequency shifts.