Question about merged DMI results in a Janus C3v system

[Rabia Çağlayan]

Dear TB2J developers,

I am working on exchange and DMI calculations for a Janus 2D magnetic system with C3v crystal symmetry. My workflow is VASP  > Wannier90 > WannSymm > TB2J, by including SOC.

I performed three separate TB2J calculations with the reference magnetization along Sx, Sy, and Sz, and then used TB2J_merge.py.

From WannSymm, the magnetic symmetries are different for the three reference directions. For Sx I get Gmag = E, sigma_2, for Sy I get Gmag = E, sigma TR, and for Sz I get Gmag = E, C3, C3^2, sigma_1TR, sigma_2TR, sigma3_TR. 
This seems reasonable, since the in-plane magnetization directions reduce the magnetic symmetry.

I have also attached a simple top-view schematic of the structure and the C3v symmetry operators. In this geometry, the mirror planes are not along the Cr-Cr nearest-neighbor bond directions; they are rotated with respect to the Cr-Cr bonds.

My question is about how to interpret the merged result. Below I show only the first-nearest-neighbor Cr1-Cr1 shell. The six bonds have the same distance, about 3.503 Å.

Sx calculation 
R J_iso (meV) DMI (meV) 
(-1,-1,0) 3.4533 ( 0.0001, -0.2112, 0.0000)
 ( 1, 1,0) 3.4533 (-0.0001, 0.2112, -0.0000) 
(-1, 0,0) 3.5750 ( 0.0000, -0.4242, -0.0000) 
( 1, 0,0) 3.5750 (-0.0000, 0.4242, 0.0000) 
( 0,-1,0) 3.4533 ( 0.0001, 0.2112, -0.0000) 
( 0, 1,0) 3.4533 (-0.0001, -0.2112, 0.0000)

Sy calculation 
R J_iso (meV) DMI (meV) 
(-1,-1,0) 3.4525 (-0.0002, -0.0000, -0.2103) 
( 1, 1,0) 3.4525 ( 0.0002, 0.0000, 0.2103) 
(-1, 0,0) 3.5899 ( 0.0000, 0.0002, -0.4231) 
( 1, 0,0) 3.5899 (-0.0000, -0.0002, 0.4231) 
( 0,-1,0) 3.4525 (-0.0002, 0.0000, 0.2103) 
( 0, 1,0) 3.4525 ( 0.0002, -0.0000, -0.2103)

Sz calculation 
R J_iso (meV) DMI (meV) 
(-1,-1,0) 3.7221 ( 0.4067, -0.2348, -0.0640) 
( 1, 1,0) 3.7221 (-0.4067, 0.2348, 0.0640) 
(-1, 0,0) 3.7221 ( 0.0000, -0.4697, 0.0640)
 ( 1, 0,0) 3.7221 (-0.0000, 0.4697, -0.0640) 
( 0,-1,0) 3.7221 ( 0.4067, 0.2348, 0.0640) 
( 0, 1,0) 3.7221 (-0.4067, -0.2348, -0.0640)

For the Sz calculation, the result looks symmetry-consistent to me. All six first-neighbor Cr-Cr bonds have the same Jiso. The in-plane DMI component is perpendicular to the corresponding Cr-Cr bond direction, and the DMI vectors transform consistently under the C3 operations. There is also a finite Dz component. From the geometry, I think this may be allowed because the mirror operation does not contain the directed Cr-Cr bond itself, but maps it to the opposite bond. In that case, the bond-parallel DMI component should be forbidden, while Dz may still be allowed. However, after using TB2J_merge.py with the Sx, Sy, and Sz calculations, the merged result again splits the same first-neighbor shell into two Jiso groups:

Merged Sx/Sy/Sz result 
R J_iso (meV) DMI (meV) 
(-1,-1,0) 3.4928 ( 0.2033, -0.2230, -0.0701)
( 1, 1,0) 3.4928 (-0.2033, 0.2230, 0.0701) 
(-1, 0,0) 3.5386 ( 0.0000, -0.4469, -0.1410) 
( 1, 0,0) 3.5386 (-0.0000, 0.4469, 0.1410) 
( 0,-1,0) 3.4928 ( 0.2033, 0.2230, 0.0701) 
( 0, 1,0) 3.4928 (-0.2033, -0.2230, -0.0701)

This is the point that confuses me. The Sz calculation seems to preserve the expected symmetry of the first-neighbor shell, while the merged result does not. In particular, the merged result gives two different J_iso values for geometrically equivalent Cr1-Cr1 nearest-neighbor bonds, and some DMI vectors contain a bond-parallel component.

Is this expected behavior for TB2J_merge.py? Does the merge procedure simply combine tensor components obtained from the different reference magnetization directions, or should the merged tensor also satisfy the full symmetry constraints of the system?

In this case, since Sx and Sy have lower magnetic symmetry than Sz, can their inclusion in the merge naturally lead to a result that no longer preserves the C3v-equivalent first-neighbor shell?

The Sz calculation looks more symmetry-consistent than the merged result. Would it be reasonable to use the Sz-derived DMI vectors for a symmetry-consistent spin model, or is there a recommended post-processing/symmetrization step for the merged anisotropic exchange and DMI tensors? or am I missing something? 

Thank you in advance. 

Best,
Rabia

[Xu He]

Hello Rabia，

Indeed  the merge script simply mixes the three calculations and does not consider the symmetry. Naturally, if the three have different symmetries, the final result does not follow the symmetry for one of them. 

This is a limitation of the approximation.  

I also think it is incorrect to use the result of the Sz calculation.  A better way is to symmetrize the result. This is under development. Part of the work (isotropic J only) is in TB2J-symmetrize.py script using only the crystal symmetry. We are working on using the magnetic space group to symmetrize the DMI and anisotropic exchange.

Best, 

HeXu 

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