A solved problem for spin waves

Question:

(a) Making use of the spin commutation relation,$[s_{i}^{\alpha },s_{j}^{\beta }]=i\delta _{ij}\varepsilon ^{\alpha \beta \gamma }s_{i}^{\gamma }$ , apply the identity

$i{\dot {s}}_{i}=[{\hat {s_{i}}},{\hat {H}}]$

To express the equation of motion of a spin in a nearest neighbor spin S one dimensional Heisenberg ferromagnet.

(b)Interpreting the spins as classical vectors, and taking the continuum limit, show that the equation of motion of the hydrodynamics modes takes the form

${\dot {s}}=Ja^{2}S\times {\partial }^{2}S,$

Where a denotes the lattice spacing.

(c) Confirm that the equation of motion is solved by the Ansatz,

$S=(cCos(\omega t-kx),cSin(kx-\omega t)),{\sqrt {S^{2}-c^{2}}})$

and determine the dispersion, sketch a snapshot configuration of the spins in the chain.

Solution:

(a) Making the use of equation of motion $i{\dot {s}}_{i}=[{\hat {s_{i}}},{\hat {H}}]$ and the commutation relation ,$[s_{i}^{\alpha },s_{j}^{\beta }]=i\delta _{ij}\varepsilon ^{\alpha \beta \gamma }s_{i}^{\gamma }$, we obtain

$i{\dot {s}}_{i}=J_{i}{\hat {s}}_{i}\times ({\hat {s}}_{i+1}+{\hat {s}}_{i-1})$

(b) Interpreting the spins as classical vectors, and applying the Taylor expansion

${\hat {S}}_{i+1}=S_{i}+a{\partial }S_{i}+({\frac {a^{2}}{2}}){\partial }^{2}S_{i}$

We obtain the classical equation of motion

${\dot {s}}=Ja^{2}S\times {\partial }^{2}S,$

Substituting, we find that $S=(cCos(\omega t-kx),cSin(kx-\omega t)),{\sqrt {S^{2}-c^{2}}})$, satisfies the equation of motion with $\omega =J(ka^{2}){\sqrt {S^{2}-c^{2}}}$.

(c) The corresponding spin wave solution has the precessional from shown in blow figurer.