Questions Regarding TB2J for Non-Collinear Systems with SOC

[Ana Pedro Fontes]

Dear TB2J Developers,

I hope you are doing well.

I am working on defective systems with SOC that are slightly non-collinear, and I aim to obtain the isotropic exchange interactions in these systems using TB2J. From the documentation, for non-collinear systems with SOC, obtaining physically meaningful tensor components (J_xx, J_yy, J_zz, J_xy, J_yz, J_xz) for both the anisotropic exchange (J_ani) and Dzyaloshinskii-Moriya interaction (DMI, D) requires averaging over calculations performed either on rotated spin structures or on rotated atomic structures while keeping the spins fixed. This ensures the proper reconstruction of the full exchange and DMI tensors.

For the isotropic exchange J, my understanding is that it corresponds to the trace:

Jiso​=1/3*​(J_xx​+J_yy​+J_zz​)

and, in principle, rotation should yield J_xx = J_yy = J_zz by construction, with the off-diagonal terms being zero. Given this, I would like to clarify:

• Is it necessary to rotate the structure/spins to obtain the isotropic J, or would a single calculation be sufficient?
• Would the assumption that the electronic structure remains nearly unchanged upon spin rotation still hold for strong SOC systems?

Additionally, I am considering the systematic convergence of J when using supercells. Apart from rescaling the k-mesh (e.g., using 6×6×1 for a 2×2×1 supercell if the primitive cell used 12×12×1), is there any other way to ensure systematic convergence? I understand that reducing the cutoff radius (r_c) would eliminate interactions beyond r_c, but if r_c is smaller than the supercell size, longer-range interactions would not be captured.

I appreciate your insights and look forward to your advice.

Best regards,
Ana Fontes

[matthieu verstraete]

Hi Ana,

I'll let He Xu complement this, but basically:

* for the isotropic J you get everything in 1 go, including remote neighbors

* for the DMI, there is an additional dependency on the quantization axis of the SOC, which means that with a single calculation you can only get 1 of the DMI vector components. What is done in this case is _not_ to rotate the spins wrt the structure, but rather to rotate both spin and structure wrt the lab frame (z axis chosen in your DFT input file). The bands etc will be identical (unless your code is not well written...) but the spin vectors arriving in TB2J will be along x or y

* the issue with strong SOC and non collinear ground states in general is that the LKAG formula presumes a reference state which is collinear (FM typically) along z (the axis can be relaxed formally, but TB2J uses z) and it's non trivial to go to an arbitrary reference state. Usually things are ok, even if the true ground state is non collinear, but it's an approximation.

* for the convergence I don't have much else to propose, the Green's function formalism allows you to renormalize out and calculate J to arbitrary neighbors, but in practice the numerical noise is quite strong and you are near 0 by the 10th or 15th neighbor. Some systems legitimately do need more precision, and then it's tough to converge everything (starting with the DFT SCF and the Wannier functions in particular - atomic orbital bases are nice because they retain more symmetry)

Best

Matthieu

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Professor Matthieu J Verstraete

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[Ana Pedro Fontes]

Dear Matthieu,

Thanks for the detailed explanation! It clarifies my questions. I’ll keep this in mind while analysing my results.

Regarding convergence, I understand the challenges with numerical noise at longer distances. I’ve also tested both Wannier and atomic orbital bases, but I’m noticing some inconsistencies between the two methods, particularly in the sign of J₁. That said, for the non-defective system, the atomic orbital results appear more physically reasonable and align better with previous DFT studies. In my case, J values are small (< 1 meV). I’m still working on improving the SCF and Wannier steps to see if I can improve consistency.

Thanks again!

Best regards,
Ana

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From: matthieu verstraete <Matthieu....@uliege.be>
Sent: 13 March 2025 07:50
To: Ana Pedro Fontes <ana.ped...@uclouvain.be>
Cc: tb...@googlegroups.com <tb...@googlegroups.com>; Jean-Christophe Charlier <jean-christo...@uclouvain.be>; Jiaqi Zhou <jiaqi...@uclouvain.be>
Subject: Re: Questions Regarding TB2J for Non-Collinear Systems with SOC

 

[matthieu verstraete]

Ok, good to hear. For the wannier case, there can be issues if they are not properly symmetric, though flipping the sign of J1 is a bit extreme... You can try to not localize (nstep 0 in w90), which can give better (more symmetric) wannier functions.

M

[Xu He]

Dear Ana, 

A few comments:  

- There should be no dependency on the size of the supercell in principle except numerical noise. Using a unitcell with a dense k-point mesh is recommended. To ensure convergence, sometimes a very dense kpoint mesh is required. This will by default give very long distance J values, which are close to 0. This will leads to long computation time. If you have an estimation of the distance where J approaches 0, setting a r_c can reduce the time significantly. 

- From what I see, the numerical noise at long distance is often not a problem. On the contrary, because of the approximation used in  strong coupling limit (U>>t) works better at large distances where the effective t is small, the convergence is often much better than the short-distance J. This perhaps explains that there is discrepancy in 1NN J, but is more consistent at longer distances. 

- For the sign of the J, sometimes it is related to the ligand if the Wanier functions are well localized. If you see a large magnetic moment in the ligands, you might consider using the ligand correction (see https://tb2j.readthedocs.io/en/latest/src/downfold.html). But as Matthieu suggested, the first thing to check is the localization and the symmetry of the Wannier functions. 

Best wishes, 

HeXu

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