Spin wave

Theory

The simplest way of understanding spin waves is to consider the Hamiltonian ${\displaystyle \mathcal{H}}$ for the Heisenberg ferromagnet:

${\displaystyle \mathcal{H}=-\frac{1}{2}J\sum _{i,j}{\mathbf{S}}_i\cdot {\mathbf{S}}_j-g{\mu }_{\mathrm{B}}\sum _i\mathbf{H}\cdot {\mathbf{S}}_i}$

where J is the exchange energy, the operators S represent the spins at Bravais lattice points, g is the Landé g-factor, μ_B is the Bohr magneton and H is the internal field which includes the external field plus any "molecular" field. Note that in the classical continuum case and in 1 + 1 dimensions the Heisenberg ferromagnet equation has the form

${\displaystyle {\mathbf{S}}_t=\mathbf{S}\times {\mathbf{S}}_{xx}.}$

In 1 + 1, 2 + 1 and 3 + 1 dimensions this equation admits several integrable and non-integrable extensions like the Landau-Lifshitz equation, the Ishimori equation and so on. For a ferromagnet J > 0 and the ground state of the Hamiltonian ${\displaystyle |0⟩}$ is that in which all spins are aligned parallel with the field H. That ${\displaystyle |0⟩}$ is an eigenstate of ${\displaystyle \mathcal{H}}$ can be verified by rewriting it in terms of the spin-raising and spin-lowering operators given by:

${\displaystyle S^{\pm }=S^x\pm i{S}^y}$

resulting in

${\displaystyle \mathcal{H}=-\frac{1}{2}J\sum _{i,j}{S}_i^z{S}_j^z-g{\mu }_{\mathrm{B}}H\sum _i{S}_i^z-\frac{1}{4}J\sum _{i,j}(S_i^{+}{S}_j^{-}+S_i^{-}{S}_j^{+})}$

where z has been taken as the direction of the magnetic field. The spin-lowering operator S⁻ annihilates the state with minimum projection of spin along the z-axis, while the spin-raising operator S⁺ annihilates the ground state with maximum spin projection along the z-axis. Since

${\displaystyle S_i^z|0⟩=s|0⟩}$

for the maximally aligned state, we find

${\displaystyle \mathcal{H}|0⟩=\left(-J{s}^2-g{\mu }_{\mathrm{B}}Hs\right)N|0⟩}$

where N is the total number of Bravais lattice sites. The proposition that the ground state is an eigenstate of the Hamiltonian is confirmed.

One might guess that the first excited state of the Hamiltonian has one randomly selected spin at position i rotated so that

${\displaystyle S_i^z|1⟩=(s-1)|1⟩,}$

but in fact this arrangement of spins is not an eigenstate. The reason is that such a state is transformed by the spin raising and lowering operators. The operator ${\displaystyle S_i^{+}}$ will increase the z-projection of the spin at position i back to its low-energy orientation, but the operator ${\displaystyle S_j^{-}}$ will lower the z-projection of the spin at position j. The combined effect of the two operators is therefore to propagate the rotated spin to a new position, which is a hint that the correct eigenstate is a spin wave, namely a superposition of states with one reduced spin. The exchange energy penalty associated with changing the orientation of one spin is reduced by spreading the disturbance over a long wavelength. The degree of misorientation of any two near-neighbor spins is thereby minimized. From this explanation one can see why the Ising model magnet with discrete symmetry has no spin waves: the notion of spreading a disturbance in the spin lattice over a long wavelength makes no sense when spins have only two possible orientations. The existence of low-energy excitations is related to the fact that in the absence of an external field, the spin system has an infinite number of degenerate ground states with infinitesimally different spin orientations. The existence of these ground states can be seen from the fact that the state ${\displaystyle |0⟩}$ does not have the full rotational symmetry of the Hamiltonian ${\displaystyle \mathcal{H}}$, a phenomenon which is called spontaneous symmetry breaking.

Magnetization

In this model the magnetization

${\displaystyle M=\frac{N{\mu }_{\mathrm{B}}gs}{V}}$

where V is the volume. The propagation of spin waves is described by the Landau-Lifshitz equation of motion:

${\displaystyle \frac{d\mathbf{M}}{dt}=-\gamma \mathbf{M}\times \mathbf{H}-\frac{\lambda \mathbf{M}\times (\mathbf{M}\times \mathbf{H})}{M^2}}$

where γ is the gyromagnetic ratio and λ is the damping constant. The cross-products in this forbidding-looking equation show that the propagation of spin waves is governed by the torques generated by internal and external fields. (An equivalent form is the Landau-Lifshitz-Gilbert equation, which replaces the final term by a more "simple looking" equivalent one.)

The first term on the right hand side of the equation describes the precession of the magnetization under the influence of the applied field, while the above-mentioned final term describes how the magnetization vector "spirals in" towards the field direction as time progresses. In metals the damping forces described by the constant λ are in many cases dominated by the eddy currents.

One important difference between phonons and magnons lies in their dispersion relations. The dispersion relation for phonons is to first order linear in wavevector k, namely ώ = ck, where ω is frequency, and c is the velocity of sound. Magnons have a parabolic dispersion relation: ώ = Ak² where the parameter A represents a "spin stiffness." The k² form is the third term of a Taylor expansion of a cosine term in the energy expression originating from the S_i ⋅ S_j dot product. The underlying reason for the difference in dispersion relation is that the order parameter (magnetization) for the ground-state in ferromagnets violates time-reversal symmetry. Two adjacent spins in a solid with lattice constant a that participate in a mode with wavevector k have an angle between them equal to ka.