31: Platinum — Selected columns of density matrix algorithm for spinor wavefunctions¶

Note: This tutorial requires a recent version of the pw2wannier90.x post-processing code of Quantum ESPRESSO (v6.3 or above).

Outline: For bulk crystalline platinum with spin-orbit coupling, generate the \(A_{mn}\) matrices via the selected columns of density matrix (SCDM) algorithm and the corresponding spinor-MLWFs. To better understand the input files and the results of these calculations, it is crucial that the Reader has familiarized with the concepts and methods explained in Ref. ¹. More info on the keywords related to the SCDM method may be found in the user_guide.

This tutorial focuses on the use of the SCDM method for spin-noncollinear systems. For the overview of the use of SCDM method to spinless systems, please refer to this tutorial.
Directory: tutorials/tutorial31/ Files can be downloaded from here

The input files for this tutorials are similar to the ones in Tutorial 29, except that a coarser k-point grid is used and that the keywords related to postw90.x are removed.
Input Files:
- Pt.scf The pwscf input file for the ground state calculation
- Pt.nscf The pwscf input file to obtain Bloch states on a uniform grid
- Pt.pw2wan The input file for pw2wannier90 with keywords related to the SCDM method
- Pt.win The wannier90 input file

We will compute 18 localized WFs. Since the band structure of platinum is metallic, the low-lying bands are entangled with other high-energy bands, and the columns of the density matrix are not exponentially localized by construction. Thus, we use a modified density matrix ¹, with the function \(f(\varepsilon_{n,\mathbf{k}})\) defined as a complementary error function. Refer to Tutorial 27 for the definition of the modified density matrix and the functional form of \(f(\varepsilon_{n,\mathbf{k}})\).

Run pwscf to obtain the ground state of platinum
Terminal
```
pw.x < Pt.scf > scf.out
```
Run pwscf to obtain the Bloch states on a uniform \(7\times 7\times 7\) \(k\)-point grid
Terminal
```
pw.x < Pt.nscf > nscf.out
```
Inspect the Pt.win input file and make sure that the auto_projections flag is set to .true.. Also, make sure that no projection block is present.
Run wannier90 to generate a list of the required overlaps (written into the Pt.nnkp file)
Terminal
```
wannier90.x -pp Pt
```
Inspect the Pt.nnkp file and make sure you find the auto_projections block and that no projections have been written in the projections block.
Inspect the Pt.pw2wan input file. You will find four SCDM-related keywords: scdm_proj, scdm_entanglement, scdm_mu and scdm_sigma. In particular, the keyword scdm_proj will instruct pw2wannier90.x to use the SCDM method when generating the \(A_{mn}\) matrix. The remaining three keywords defines the formula and parameters to define the function \(f(\varepsilon_{n\mathbf{k}})\) (see Ref. ¹ and Tutorial 27).
Run pw2wannier90 to compute the overlap between Bloch states and the projections via the SCDM method (written in the Pt.mmn and Pt.amn respectively).
Terminal
```
pw2wannier90.x < Pt.pw2wan > pw2wan.out
```
Inspect the pw2wan.out output file. Compared to the spinless case, you will find the following two additional lines.
Output file
```
         Number of pivot points with spin up  :     9
         Number of pivot points with spin down:     9
```
These lines give information on the pivots obtained by the QR decomposition with column pivoting (QRCP) in the SCDM algorithm. Each pivot determines a point in the real-space grid and a spin state. The basis of the spin state is determined by the basis used in the electronic structure code. In pwscf, the basis states are spin up and down states along the Cartesian \(z\)-axis.
Run wannier90 to compute the MLWFs
Terminal
```
wannier90.x Pt
```

A. Damle and L. Lin. Disentanglement via entanglement: A unified method for Wannier localization. ArXiv e-prints, March 2017. URL: http://adsabs.harvard.edu/abs/2017arXiv170306958D, arXiv:1703.06958. ↩↩↩