Averaging multiple parameters

When the spins of sites $i$ and $j$ are along the directions $\hat{\mathbf{m}}_i$ and $\hat{\mathbf{m}}_j$, respectively, the components of $\mathbf{J}^{ani}_{ij}$ and $\mathbf{D}_{ij}$ along those directions will be unphysical. In other words, if $\hat{\mathbf{u}}$ is a unit vector orthogonal to both $\hat{\mathbf{m}}_i$ and $\hat{\mathbf{m}}_j$, we can only obtain the projections $\hat{\mathbf{u}}^T \mathbf{J}^{ani}_{ij} \hat{\mathbf{u}}$ and $\hat{\mathbf{u}}^T \mathbf{D}_{ij} \hat{\mathbf{u}}$. Notice that for collinear systems, there will be two orthonormal vectors $\hat{\mathbf{u}}$ and $\hat{\mathbf{v}}$ that are also orthogonal to $\hat{\mathbf{m}}_i$ and $\hat{\mathbf{m}}_j$.

The projection for $\mathbf{J}^{ani}_{ij}$ can be written as

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

where we considered $\mathbf{J}^{ani}_{ij}$ to be symmetric. This equation gives us a way of reconstructing $\mathbf{J}^{ani}_{ij}$ by performing TB2J calculations on rotated spin configurations. If we perform six calculations such that $\hat{\mathbf{u}}$ lies along six different directions, we obtain six linear equations that can be solved for the six independent components of $\mathbf{J}^{ani}_{ij}$. We can also reconstruct the $\mathbf{D}_{ij}$ tensor in a similar way. Moreover, if the system is collinear then only three different calculations are needed.

To account for this, TB2J provides scripts to rotate the structure and merge the results; they are named TB2J_rotate.py and TB2J_merge.py. The TB2J_rotate.py reads the structue file and generates three(six) files containing the rotated structures whenever the system is collinear (non-collinear). The –noncollinear parameters is used to specify wheter the system is noncollinear. The output files are named atoms_i (i = 0, …, 5), where atoms_0 contains the unrotated structure. A large number of file formats is supported thanks to the ASE library and the output structure files format is provided through the –format parameter. An example for using the rotate file with a collinear system is:

TB2J_rotate.py BiFeO3.vasp --ftype vasp

If te system is noncollinear, then we run the following instead:

TB2J_rotate.py BiFeO3.vasp --ftype vasp --noncollinear

The user has to perform DFT single point energy calculations for the generated structures in different directories, keeping the spins along the $z$ direction, and run TB2J on each of them. After producing the TB2J results for the rotated structures, we can merge the DMI results with the following command by providing the paths to the TB2J results of the three cases:

TB2J_merge.py BiFeO3_1 BiFeO3_2 BiFeO3_0

Here the last directory will be taken as the reference structure. Note that the whole structure are rotated w.r.t. the laboratory axis but not to the cell axis. Therefore, the k-points should not be changed in both the DFT calculation and the TB2J calculation.

A new TB2J_results directory is then made which contains the merged final results.

Another method is to do the DFT calculation with spins rotated globally. That is they are rotated with respect to an axis, but their relative orientations remain the same. This can be specified in the initial magnetic moments from a DFT calculation. For calculations done with SIESTA, there is a script that rotates the density matrix file along different directions. We can then use these density matrix files to run single point calculations to obtain the required rotated magnetic configurations. An example is:

TB2J_rotateDM.py --fdf_fname /location/of/the/siesta/*.fdf/file

As in the previous case, we can use the –noncollinear parameter to generate more configurations. The merging process is performed in the same way.