[WIP] Add ADR proposal and LSWT draft

[mducle]

Adds some developer documentation:

• A proposal to record architectural design decisions in the documentation in the repository
• A draft (WIP) of Linear Spin Wave Theory with reference to the (Matlab) SpinW code.

[lucas-wilkins]

Something that's been a real blocker trying to understand all this is how you can do a square root on these objects. I guess if they're matrices you can do a Cholesky decomposition, but later you talk about series expansions. This seems to be pretty key.

[mducle]

Ah sorry the $\sqrt{2S}$ terms refer to the scalar "spin length" or total spin quantum number $S$ rather than the vector $\mathbf{S}=(S^x, S^y, S^z)$. I'll make that clear in the text.

[lucas-wilkins]

You still need to square root the bs though,

[mducle]

@lucas-wilkins Ah, I get your meaning now - it's the $S+ = b\sqrt{2S - b^{\dagger}}$ term you mean.

Right, so in our particular case we only ever calculate the square of these operators in the $\mathbf{S}_i \mathrm{J}_{ij} \mathbf{S}_j$ and $\mathbf{S}_i \mathrm{A}_{i} \mathbf{S}_i$ terms and in the Zeeman term we take the field to be along $z$ and use only the $S^z$ operator.

If we ever need to calculate the matrix elements of odd powers of these operators, then we would need to perform a Taylor expansion of the square root - which is tricky and requires partial differentiation of the operators. However, the authors of this paper worked out a way around this.

[lucas-wilkins]

You still need to square root the bs though,