Heisenberg ferromagnet and spin waves

[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!

 

[azntoon]

To show that the state |k> is an eigenstate of the Hamiltonian, we need to calculate the action of the Hamiltonian on the state and show that it is equal to an eigenvalue. We can do this by first writing out the Hamiltonian as a matrix in the basis of the spin operator states, then using the Fourier transform to express the state |k> in terms of these basis states. We then calculate the action of the Hamiltonian on the state |k> in terms of the matrix elements, and finally show that this is equal to an eigenvalue.Let's start by writing out the Hamiltonian in the basis of the spin operator states. For an N-site chain, the Hamiltonian matrix will be a NxN matrix with entries given by_{ij} = -J_z^z s_i^z s_j^z -J_{xy}^{xy} (s_i^{+}s_j^{-} + s_i^{-}s_j^{+}) -g\mu_B H\delta_{ij}s_i^zwhere J_z^z and J_{xy}^{xy} are the exchange parameters and H is the applied magnetic field.Now we can use the Fourier transform to express the state |k> in terms of the basis states. Recall that the Fourier transform is defined as:s_k^- = \frac{1}{\sqrt{N}} \sum_j \exp(ik\cdot r_j) s_j^-where r_j is the position vector of site j. Plugging this into the definition of |k>, we have:|k> = \frac{1}{\sqrt{2S}} \frac{1}{\sqrt{N}} \sum_j \exp(ik\cdot r_j) s_j^- |0>Now we can calculate the action of the Hamiltonian on the state |k>. Using the definition of the Hamiltonian matrix above, we have|k> = \frac{1}{\sqrt{2S}} \frac{1}{\sqrt{N}} \sum_j \sum_i \exp(ik\cdot r_j