Merge algorithm for non-collinear systems

[antelmor]

Consider the anisotropic exchange tensor $J_{ij}$ describing the interaction between two magnetic sites with local magnetic moments $\mathbf{m}_i$ and $\mathbf{m}_j$. Let $\hat{\mathbf{u}}_{ij}$ be a unit vector that is normal to both $\mathbf{m}_i$ and $\mathbf{m}_j$. Then, we can only obtain the projection $\hat{\mathbf{u}}^T J_{ij} \hat{\mathbf{u}}$ with a single TB2J calculation. The latter can be written as

$\hat{\mathbf{u}}^T J_{ij} \hat{\mathbf{u}} = \hat{J}_{ij}^{xx} u_x^2 + \hat{J}_{ij}^{yy} u_y^2 + \hat{J}_{ij}^{zz} u_z^2 + 2\hat{J}_{ij}^{xy} u_x u_y + 2\hat{J}_{ij}^{yz} u_y u_z + 2\hat{J}_{ij}^{zx} u_z u_x$,

where we considered $J_{ij}$ to be symmetric.

If we perform six calculations such that $\hat{\mathbf{u}}_{ij}$ lies along six different directions, then we obtain six linear equations which can be solved for the six independent unknown components of $J_{ij}$.

When $\mathbf{m}_i$ and $\mathbf{m}_j$ are parallel to each other, then we can use the components of $J_{ij}$ along the plane orthogonal to both $\mathbf{m}_i$ and $\mathbf{m}_j$ . Thus, with six different calculations, we could obtain more than six equations for the six components of $J_{ij}$ . In this case, we can obtain the tensor components by a least squares method.

[mailhexu]

Hello Andres,
@antelmor Thank you!

• Could you keep the legacy TB2J_merge.py and put the new script into TB2J_merge2.py, so that when $m_i$ and $m_j$ are parallel, we don't need to redo all the calculations. Or do you think even in these cases, the six-state method should replace the legacy one?
• Could you add a simple tutorial in doc/src/rotate_and_merge.rst?
• How do we get the infomation of the magnetic moments, by the user input or by reading the siesta/openmx/abacus output? I don't think the Wannier functions work well with the strong non-collinear case in the current implementation so perhaps we don't need to worry about how to read from other codes for the moment. Best wishes, HeXu

[antelmor]

Hello @mailhexu.

• This new script is a generalization of the legacy TB2J_merge.py. It also works for the collinear cases in which we only need three configurations. The script doesn't require a specific number of rotated configurations; it always gives the best result considering all the configurations given to the script. If we give too few configurations, the results will have a large error. In the case of a collinear system, we need to provide at least three non-parallel configurations; whereas for a non-collinear system, we need at least six. In the most recent commit, I added a warning message that tells the user when more configurations are required. Furthermore, this script works the same if the configurations are spin-rotated or structure-rotated. This differs from the legacy script, where the user had to tell which rotation type was used.
• I will add a tutorial considering both a collinear and a non-collinear system.
• The script gets the magnetic moments from the TB2J.pickle of each finished calculation. So far I have only tested the script with Siesta-TB2J calculations. Could this change if we use Wannier-TB2J calculations?

[mailhexu]

Hello Andres, Thanks! Don't worry about the Wannier-TB2J calculation for the non-collinear cases. For now I will add that the Wannier-TB2J interface is dis-encouraged in the documentation if the spin is not almost collinear. I will check again and merge when the documentation is ready. Best regard, HeXu

[antelmor]

Hello @mailhexu.

I have modified the script to rotate the structures; it can now generate the structures required for a noncollinear system. Moreover, I have created an additional script for the TB2J-SIESTA calculations, where you can globally rotate the magnetic moments contained in the density matrix file of an SIESTA calculation. This is helpful when a user wants to rotate the spins instead of the structure. Finally, I have updated the rotate_and_merge.rst file with the required documentation for all the scripts.

Best, Andres.

[mailhexu]

Thanks!