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Given an anisotropic hamiltonian

[itex]

\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

[/itex]

Here [itex] \rho [/itex] is a vector connecting the neighbouring sites.

How do I show that the state

[itex]

|k> = \frac{1}{\sqrt{2S}}s_{k}^{-}|0>

[/itex]

where

[itex]

s_{k}^-=\frac{1}{\sqrt{N}}\sum_j\exp(ik\cdot r_j)s_j^-

[/itex]

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!

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# Heisenberg ferromagnet and spin waves

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