Speed

In this section we will compare the calculation time for the calculations of anomalous Hall conductivity using Wannier90 and WannierBerri. We will take the example of bcc Fe and vary different parameters. Calculations were performed on identical 32-core virtual nodes of the ScienceCloud cluster at University of Zurich. The nodes are based on AMD EPYC 7702 64-Core Processors with frequency 2GHz and 128 GB RAM per node, and one node was used per task.

The computation consists of two phases. First, some preliminary operations are done. Those include reading the input files and performing Fourier transform from ab initio grid \({\bf q}\) to real-space vectors \({\bf R}\) : eqs. [eq:fourier_q_to_R_H] and [eq:fourier_q_to_R]. This operation takes in WannierBerri(postw90) between 2 (3) seconds for the small ab-initio grid 4x4x4 and 2 (3) minutes for a large grid of 16x16x16. This time is mostly taken by reading the large formatted text file Fe.mmn, and it is done only once and does not scale with the density of the interpolation grid. In WannierBerrithis is done in the constructor of the System\_w90 class, and the object can be saved on disk using a pickle module, so that this operation does not repeat for further calculations.

Next comes the interpolation part itself, for which the evaluation time scales linearly with the number of \({\bf k}\)-points used. Further the time for an interpolation grid 200x200x200 is given, which is a rather good grid to make an accurate calculation for this material.

Fig. 3 Computational time for AHC using WannierBerri(triangles) and postw90.x(circles) for different ab initio grids. For postw90.xthe calculations are done with (yellow) and without (purple) MDRS. Fpr WannierBerrithe calculations are done with (cyan) and without (red) use of symmetries.

We start with comparing time with the MDRS switched off (use_ws_distance=False) and without use of symmetries in WannierBerri. As can be seen in Fig. 3, for a small abinitio \({\bf q}\)-grid 4x4x4 WannierBerriis just slightly faster then postw90.x. However, for dense \({\bf q}\)-grids the computational time of postw90.xgrows linearly with the number of \({\bf q}\) points, while in WannierBerriit stays almost the same. This happens because in postw90.xthe Fourier transform is major time-consuming routine. On the other hand, in WannierBerri, although cost of the mixed Fourier transform is expected to grow logarithmically with the ab-initio grid (see sec-FFT), we do not see it because Fourier transform amounts only to \(\sim 10\)% of the computational time.

Next, we switch on the MDRS method (use_ws_distance=True) in postw90.x, and the computational time grows by a factor of 5. On the other hand the computational time does not change (not shown), just by construction, as described in sec-replica.

Finally let’s switch on the use of symmetries in \({ \tt WannierBerri }\). Thus the computational time decreases by a factor of 8. In the ultra-dense grid limit one would expect the speedup to be approximately equal to the number of elements in the group — 16 in the present example, due to exclusion of symmetry-equivalent \({\bf K}\)-points. But this does not happen, because we use an FFT grid of 25x25x25 \(\boldsymbol{\kappa}\)-points, hence the \({\bf K}\)-grid is only \(8x8x8\), and a considerable part of \({\bf K}\)-points are at high-symmetry positions. Therefore they do not have symmetric partners to be excluded from the calculation.

Fig. 4 Computational time for scanning multiple chemical potentials using WannierBerriand postw90.xfor different ab initio grids. MDRS method and symmetries are disabled here.

Thus we can see that the difference in computational time with postw90.xand WannierBerrireaches 3 orders of magnitude for this example. Note that The examples above were performed only for the pristene Fermi level. Now let’s see what happens upon scanning the chemical potential (Fig. 4). In WannierBerrithe computational time remains practically unchanged when we use upto \(N_\mu\approx1000\) chemical potentials, and only start to grow considerably at \(N_\mu\sim 10^4\). On the other hand in postw90.xthe computational time significantly grows with \(N_\mu\), which is especially remarkable for small \({\bf q}\)-grids.

In this section we did not use the adaptive refinement procedure. However when on starts from a rather dense grid of \({\bf K}\)-points, the new \({\bf K}\)-points coming from the refinement procedure constitute only a small portion of the initial grid, and hence do not contribute much into computational time.