Linear spin wave theory wishlist

[kbarros]

Features we'd like to have in an eventual LSWT implementation:

• [x] General SU(N) result, or projection to dipoles only (i.e. coherent states restricted to $CP^1$).
• [x] Ability to specify fully general interactions between SU(N) coherent states at different sites. One possibility is an $(N^2-1) \times (N^2-1)$ matrix that describes coupling between expectation values at different sites. A perhaps better representation is a sparse decomposition of the form $\sum_k A^k \otimes B^k$, where each of $A^k$ and $B^k$ are local Hermitian operators for the two sites, respectively. Then the expected energy would be $\sum_k \mathbf{z_1}^\dagger A^k \mathbf{z_1} \mathbf{z_2}^\dagger B^k \mathbf{z_2}$, given coherent states $\mathbf{z_1}$ and $\mathbf{z_2}$.
• [x] In the case of "fully general" SU(N) interactions, there should be a way to specify the effective magnetic moment operator that couples to the external field. (Current plan is to create an "entangled units" wrapper over System that stores magnetic moment information.)
• [ ] Efficient support for very large magnetic unit cells (e.g. systems with disorder). In progress with KPM.
• [ ] Efficiency. For dipole mode, SpinGenie is a good target. Looking forward to GPU acceleration, we could consider a batched mode (parallel calculation for many $k$ vectors at a time, used batched linear algebra operations).
• [ ] Ability to calculate gradients through LSWT. For example, suppose we have a "loss" $L$ as a function of intensity data. It will typically be straightforward to calculation the gradient of $L$ with respect to the intensity. Then, using the chain rule, we would like to contract this with the gradient of the intensity with respect to the Hamiltonian parameters. For this, we need to develop a new perturbation theory associated with the para-unitary Bogoliubov transformation.

Non-features:

• Custom support for incommensurate single-Q orderings. (This should be developed out of tree.)

[kbarros]

Update: @XiaojianBai and @AlinNiraula are making great progress on single-Q LSWT!