Spin Waves from Spin Susceptibility

I have been studying this amazing idea put forth by Takao Kotani and Mark van Schilfgaarde where they come up with a way to calculate spin waves from transverse spin susceptibilities. Before that it is important to first introduce and get familiar with the idea of calculating Heisenberg interaction for a spin Hamiltonian described by

The usual way to calculate this ab-initio is to do super-cell multi-level energy calculations, i.e. one would calculate the total energy of FM and different AFM configurations from which the energy needed to flip a single spin can be extracted. But the second and more less brute-force way is to use perturbation theory.

We start by writing down the time ordered transverse spin susceptibility as

Here, denotes the time ordering while with being the spin density operators. Further let us impose collinearity in the system with the spin direction along which gives us . Focusing on the component, we have

where we have is the magnetic moment which for our case would be a vector given by .

To be continued...