Perturbative corrections to linear spin-wave theory

[Hao-Phys]

Features we would like to add to go beyond the linear spin-wave theory (will be implemented in the order listed below):

• 1. Compute the correction to any SU(N) order parameters [ordered moment for SU(2)] from the next order in $1/\lambda_1$ [or $1/S$ for SU(2)]. (Now available as energy_per_site_lswt_correction and magnetization_lswt_correction.)
• In progress. 2. Compute the two-particle contribution (aka the "longitudinal" channel) to the dynamical spin structure factor $S(\boldsymbol{q}, \omega)$.
• 3. Compute the corrections from the quartic bosonic Hamiltonian that would renormalize the magnon bands given by the linear spin-wave theory.
• 4. Compute the corrections from the cubic bosonic Hamiltonian that would renormalize the magnon bands (real part of the self-energy operator) and give a finite lifetime (intrinsic broadening in neutron data from the imaginary part of the self-energy operator).
• 5. Incorporate steps 2-4 to calculate the corrected $S(\boldsymbol{q}, \omega)$ (include all the corrections up to the order in $\mathcal{O}(1/\lambda_1)$).
• 6. Support non-perturbative calculations, such as the exact diagonalizations up to n-particle space. We will benefit from step 4.