# Heisenberg ferromagnet and spin waves

1. Mar 25, 2012

### JohnGringo

Hey
Given an anisotropic hamiltonian
$\mathcal{H} = -\sum_{j,\rho} \left( J_\rho^z s_j^z s_{j+\rho}^z + \frac{J_\rho^{xy}}{2}\left( s_j^+ s_{j+\rho}^- + s_j^- s_{j+\rho}^+ \right)\right) - g\mu_B H\sum_j s_j^z$
Here $\rho$ is a vector connecting the neighbouring sites.
How do I show that the state
$|k> = \frac{1}{\sqrt{2S}}s_{k}^{-}|0>$
where
$s_{k}^-=\frac{1}{\sqrt{N}}\sum_j\exp(ik\cdot r_j)s_j^-$
is an eigenstate of the hamiltonian?
So the plan is the use the fourier transform some how, but I am kind of lost with this. What do I substitute where and why?

Thanks!