Spin Waves and Magnons

Spin-wave theory expands an ordered magnet about a chosen classical state and represents transverse fluctuations as bosons. Its elementary quanta are magnons. Ferromagnets and bipartite antiferromagnets have different quadratic Hamiltonians, dispersions, and vacuum fluctuations; keeping their exchange sign and Brillouin-zone convention explicit prevents common errors.

Required background. Exchange Interactions and Effective Spin Hamiltonians fixes the sign and spin normalization. Helpful background. Finite-Density Goldstone Counting explains why a ferromagnet has one quadratic mode while an antiferromagnet has linearly dispersing modes.

Holstein–Primakoff expansion

For a spin polarized along local $+z$, introduce a canonical boson,

$$S^z=S-a^\dagger a, \qquad S^+=\sqrt{2S}\sqrt{1-\frac{a^\dagger a}{2S}}\,a, \qquad S^-=\sqrt{2S}\,a^\dagger \sqrt{1-\frac{a^\dagger a}{2S}}.$$

Expanding the square roots organizes interactions in powers of $1/S$ provided the boson density is small compared with $2S$. The transformation is exact on the physical subspace $0\le a^\dagger a\le2S$ Holstein and Primakoff 1940, pp. 1098–1113.

Ferromagnetic magnons

Take $H=-J\sum_{\langle ij\rangle}\mathbf S_i\cdot\mathbf S_j$ with $J>0$ on a Bravais lattice of coordination $z$. Define

$$\gamma_{\mathbf k}=\frac1z\sum_{\boldsymbol\delta} e^{i\mathbf k\cdot\boldsymbol\delta},$$

where the sum includes all $z$ nearest-neighbor vectors. To quadratic order,

$$H=E_{\mathrm{cl}}+\sum_{\mathbf k} \varepsilon_{\mathbf k}a_{\mathbf k}^\dagger a_{\mathbf k}, \qquad \varepsilon_{\mathbf k}=JSz(1-\gamma_{\mathbf k}).$$

For an inversion-symmetric lattice $\gamma_{\mathbf k}$ is real. Near $\mathbf k=0$, $1-\gamma_{\mathbf k}\propto k^2$, so the Goldstone mode is quadratic. The fully polarized state is an exact eigenstate and the harmonic vacuum has no zero-point reduction of the magnetization. Magnon interactions matter at finite density and generate the low-temperature corrections analyzed by Dyson 1956, pp. 1217–1230.

Bipartite antiferromagnets

Now take $H=+J\sum_{\langle ij\rangle}\mathbf S_i\cdot\mathbf S_j$. Rotate the local axes on one sublattice so both classical moments point along local $+z$. The quadratic Hamiltonian contains anomalous pair terms,

$$H_2=E_{\mathrm{cl}}+JSz\sum_{\mathbf k} \left[a_{\mathbf k}^\dagger a_{\mathbf k} +\frac{\gamma_{\mathbf k}}2 (a_{\mathbf k}a_{-\mathbf k}+a_{\mathbf k}^\dagger a_{-\mathbf k}^\dagger) \right],$$

where a reduced-zone two-sublattice notation has been combined into one Bogoliubov branch. The transformation $a_{\mathbf k}=u_{\mathbf k}\alpha_{\mathbf k}+v_{\mathbf k}\alpha_{-\mathbf k}^\dagger$ gives

$$\omega_{\mathbf k}=JSz\sqrt{1-\gamma_{\mathbf k}^2}.$$

The mode is linear near the ordering wave vector (or $\mathbf k=0$ in the magnetic Brillouin zone). The anomalous vacuum reduces the ordered moment by

$$\delta m=\frac1{2N}\sum_{\mathbf k} \left(\frac1{\sqrt{1-\gamma_{\mathbf k}^2}}-1\right).$$

For the square lattice this harmonic correction is finite, about $0.197$ per site. In one dimension the integral diverges, correctly warning that an assumed Néel moment is not self-consistent. Anderson’s treatment identified both the zero-point energy and sublattice-magnetization reduction Anderson 1952, pp. 694–701.

Control, interactions, and observables

The formal expansion parameter is $1/S$, but coordination, dimensionality, frustration, and proximity to a phase boundary can amplify corrections. Cubic terms appear for noncollinear orders and allow magnon decay when kinematics permits; quartic terms renormalize velocities and interactions. A positive harmonic spectrum is necessary, not sufficient, for stability. Imaginary frequencies diagnose that the chosen classical state is not even a local harmonic minimum.

Neutron scattering couples to spin components, so the one-magnon intensity includes polarization factors, magnetic form factors, Bogoliubov coherence factors, and domain averaging. A fitted dispersion alone constrains exchange combinations; absolute intensity and sum rules test the ordered moment and missing multiparticle weight.

Exercises

1. Find the long-wavelength square-lattice ferromagnetic dispersion.

Solution

For $z=4$, $\gamma_{\mathbf k}=[\cos(k_xa)+\cos(k_ya)]/2\simeq1-a^2k^2/4$. Hence $\varepsilon_{\mathbf k}=4JS(1-\gamma_{\mathbf k})\simeq JSa^2k^2$.

2. Show that the square-lattice antiferromagnetic velocity is $c=2\sqrt2JSa$.

Solution

Using the same $\gamma_{\mathbf k}\simeq1-a^2k^2/4$, $1-\gamma_{\mathbf k}^2\simeq a^2k^2/2$. Therefore $\omega_{\mathbf k}=4JS\sqrt{1-\gamma_{\mathbf k}^2}\simeq2\sqrt2JSa|\mathbf k|$.

References

• Anderson, P. W. “An Approximate Quantum Theory of the Antiferromagnetic Ground State.” Physical Review 86 (1952): 694–701. DOI.
• Dyson, F. J. “General Theory of Spin-Wave Interactions.” Physical Review 102 (1956): 1217–1230. DOI.
• Holstein, T., and H. Primakoff. “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet.” Physical Review 58 (1940): 1098–1113. DOI.