20: Disentanglement restricted inside spherical regions of \(k\) space¶

LaVO\(_3\)¶

Outline: Obtain disentangled MLWFs for strained LaVO\(_3\).
Directory: tutorials/tutorial20/ Files can be downloaded from here
Input Files
- LaVO3.scf The pwscf input file for ground state calculation
- LaVO3.nscf The pwscf input file to obtain Bloch states on a uniform grid
- LaV03.pw2wan Input file for pw2wannier90
- LaVO3.win The wannier90 input file
Run pwscf to obtain the ground state of LaVO\(_3\).

``bash title="Terminal"pw.x < LaVO3.scf > scf.out`
Run pwscf to obtain the Bloch states on a uniform k-point grid.
Terminal
```
pw.x < LaVO3.nscf > nscf.out
```
Run wannier90 to generate a list of the required overlaps (written into the LaVO3.nnkp file).
Terminal
```
wannier90.x -pp LaVO3
```
Run pw2wannier90 to compute the overlap between Bloch states and the projections for the starting guess (written in the LaVO3.mmn and LaVO3.amn files).
Terminal
```
pw2wannier90.x < LaVO3.pw2wan > pw2wan.out
```
Run wannier90 to compute the MLWFs.
Terminal
```
wannier90.x LaVO3
```

Inspect the output file LaVO3.wout. In the initial summary, you will see that the disentanglement was performed only within one sphere of radius 0.2 arount the point \(A=(0.5, 0.5, 0.5)\) in reciprocal space:

Output file

|  Number of spheres in k-space              :                 1             |
|   center n.   1 :     0.500   0.500   0.500,    radius   =   0.200         |

Compare the band structure that Wannier90 produced with the one obtained using Quantum ESPRESSO. You should get something similar to this. Notice how the \(t_{2g}\)-bands are entangled with other bands at \(A\) and the Wannier-interpolated band structure deviates from the Bloch bands only in a small region around that \(k\)-point. It is important to keep in mind that all symmetry equivalent \(k\)-points within the first Brillouin zone must be written explicitly in the list of sphere centers. For instance, the \(A\) point in the simple tetragonal lattice of this tutorial is non-degenerate, while the \(X\) point has degeneracy two, hence one must specify both \((1/2,0,0)\) and \((0,1/2,0)\) (see the SrMnO\(_3\) example here below).

Further ideas¶

Try to obtain the Wannier functions using the standard disentanglement procedure (without spheres, dis_spheres_num = 0). You will notice that the Wannier-interpolated band structure now shows deviations also in regions of \(k\)-space far away from \(A\), where disentanglement is actually not necessary. If you disable the disentanglement completely, instead, the Wannierisation procedure does not converge.
In order to illustrate all possible cases, it is instructive to apply this method to SrMnO\(_3\), where the \(t_{2g}\) bands are entangled with the above-lying \(e_g\) bands, and also with the deeper O-\(2p\) states. In the SrMnO\(_3\) subfolder, you can find input files for building three different sets of Wannier functions: only \(t_{2g}\) states, only \(e_g\) states, or all V-\(3d\)-derived states (\(t_{2g} + e_g\)). In each case one needs to specify different disentanglement spheres, according to which region(s) in \(k\)-space show entanglement of the targeted bands. Also the index dis_sphere_first_wan needs to be adapted to the new disentanglement window, which here contains also states below the lowest-lying Wannier function (at variance with the
LaVO\(_3\) case).

Image title — Band structure of epitaxially-strained (tetragonal) LaVO₃. Black: Bloch bands; red circles: Wannier-interpolated band structure. The disentanglement was performed only for k-points within a sphere of radius 0.2 Å⁻¹ centered in A.