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

\[\begin{equation}\mathcal{H}=\sum_{ij}J_{ij}\hat{S}_i \hat{S}_j\end{equation}\label{eq:1}\]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-forcey* way is to use perturbation theory. Before going into spin waves, I think it is important to get that out of the way and this post would be concerning that.

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

\[\begin{equation}\chi^{+-}\left(\mathbf{r}, \mathbf{r}^{\prime}, t-t^{\prime}\right)=-\mathrm{i}\left\langle T\left(\hat{S}^{+}(\mathbf{r}, t) \hat{S}^{-}\left(\mathbf{r}^{\prime}, t^{\prime}\right)\right)\right\rangle\end{equation}\label{eq:2}\]Here, $T$ denotes the time ordering while $\hat{S}^{\pm}(\mathbf{r}, t)=\hat{S}^{x}(\mathbf{r}, t) \pm \mathrm{i} \hat{S}^{y}(\mathbf{r}, t)$ with $S_i$ being the spin density operators. Further let us impose collinearity in the system with the spin direction along $z$ which gives us \(<S_x>=<S_y>=0\). Focusing on the $z$ component, we have that to be

\[\begin{equation}2\left\langle\hat{S}^{z}(\mathbf{r}, t)\right\rangle=M(\mathbf{r})\end{equation}\label{eq:3}\]where we have $M(\mathbf{r})=n^{\uparrow}(\mathbf{r})-n^{\downarrow}(\mathbf{r})$ is the magnetic moment which for our case would be a vector given by $M_i(\mathbf{r}) \forall i \in$ atoms. (to be continued soonâ€¦)

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