Spin waves in the Heisenberg ferromagnet

The above discussion also leads us naturally to the low-lying excitations starting from the ground state These excitations will consist of a linear superposition of configurations with the z component of only one spin reduced by one unit. In order to produce states that reflect the crystal periodicity, just as we did for the construction of states based on atomic orbitals in chapter 4, we must multiply the spin configuration which has the reduced spin at site R by the phase factor exp(ik R). The resulting states are 

To apply the Heisenberg ferromagnetic hamiltonian to the spin-wave state 

We wish next to analyze the behavior of the system in a spin-wave state. To this end, we define the transverse spin-spin correlation operator which measures the correlation between the non-z components of two spins at sites R, R from its definition, this operator is 

The expectation value of this operator in state will involve the application 

This shows that spins which are apart by R — R have a difference in transverse orientation of cos[k (R - R )], which provides a picture of what the spin-wave state means spins are mostly oriented in the z direction but have a small transverse component which changes orientation by an angle k (R - R ) for spins separated by a distance R - R. This angle obviously depends on the wave-vector 

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