1\section{Disentanglement restricted inside spherical regions of \textit{k}-space LaVO$_3$.} 2\label{sec20:LaVO3} 3 4\begin{itemize} 5 \item Outline: {\it Obtain disentangled MLWFs for strained $\mathrm{LaVO}_3$.} 6\end{itemize} 7 8\begin{figure}[h!] 9\centering 10\subfloat[LaVO$_3$]{\includegraphics[width=0.45\columnwidth,trim={300pt 10pt 300pt 150pt},clip]{figure/example20/LaVO3.png}} 11\centering 12\subfloat[SrMnO$_3$]{\includegraphics[width=0.45\columnwidth,trim={300pt 10pt 300pt 150pt},clip]{figure/example20/SrMnO3.png}} 13\label{fig20} 14\caption{Left: atomic structure of epitaxially-strained (tetragonal) LaVO$_3$. Right: atomic structure of epitaxially-strained (tetragonal) SrMnO$_3$. Both structures have been plotted with the \xcrysden{} program.} 15\end{figure} 16\begin{itemize} 17 \item [1-5] These are the usual steps to generate MLWFs and are not reported here. 18 19 \item {\it Inspect the output file {\tt LaVO3.wout}. In the initial summary, you will see that the disentanglement was 20performed only within one sphere of radius 0.2 around the point {\tt A = (0.5, 0.5, 0.5)} in reciprocal space:} 21 22{\small 23\begin{tcolorbox}[sharp corners,boxrule=0.5pt] 24\begin{verbatim} 25 *------------------------------- DISENTANGLE --------------------------------* 26 | Using band disentanglement : T | 27 28 29 ... 30 31 | Number of spheres in k-space : 1 | 32 | center n. 1 : 0.500 0.500 0.500, radius = 0.200 | 33\end{verbatim} 34\end{tcolorbox} 35} 36 37\item {\it Compare the band structure that \Wannier{} produced with the one obtained using Quantum ESPRESSO.} 38 39To obtain the band structure from the Quantum ESPRESSO calculation we can use the {\tt bands.x} program available at \url{http://www.tcm.phy.cam.ac.uk/~jry20/bands.html}, see mini-tutorial at the end of Ex.~\ref{sec6:copper}. Here, we only report the {\tt .inp} file used to generate the $k$-point mesh for the non-scf calculation 40{\small 41\begin{tcolorbox}[title=bands.x input file LaVO3.inp,sharp corners,boxrule=0.5pt] 42\begin{verbatim} 437.03 0.00 0.00 440.00 7.03 0.00 450.00 0.00 7.6627 46 4730 48 49G 0.00000 0.00000 0.00000 M 0.50000 0.50000 0.00000 50M 0.50000 0.50000 0.00000 X 0.50000 0.00000 0.00000 51X 0.50000 0.00000 0.00000 G 0.00000 0.00000 0.00000 52G 0.00000 0.00000 0.00000 Z 0.00000 0.00000 0.50000 53Z 0.00000 0.00000 0.50000 A 0.50000 0.50000 0.50000 54A 0.50000 0.50000 0.50000 R 0.50000 0.00000 0.50000 55R 0.50000 0.00000 0.50000 X 0.50000 0.00000 0.00000 56\end{verbatim} 57\end{tcolorbox} 58} 59Remember to add the following line to the {\tt .bands} file in order to show the eigenvalues at each k-point. 60{\tt 61\begin{quote} 62verbosity = 'high' 63\end{quote} 64} 65Plot of the interpolated band structure is shown in Fig.~(\ref{fig20.1}). In the top panel, the full band structure is shown. In the bottom panel a magnification around the Fermi energy is shown (similar to Fig. 9 in the Tutorial). 66\end{itemize} 67 68\subsection*{Further ideas} 69\begin{itemize} 70 \item {\it Try to obtain the Wannier functions using the standard disentanglement procedure \dots} 71 72 Plots of the band structure of LaVO$_3$ with full disentanglement and no disentanglement are shown in Fig.~(\ref{fig20.2}). These are plotted against the quantum ESPRESSO band structure (solid black lines) and the \Wannier{}-interpolated one with disentanglement performed only within a sphere centred in A (red dots). We see that the other two methods diverge from the DFT calculation in region of $k$-space where the bands of interest are not entangled with other unwanted bands. For example, in the zone between $\Gamma$ and M and Z and A the interpolated bands with full disentanglement and no disentanglement diverge substantially from the DFT calculation. 73 74 \item {\it In order to illustrate all possible cases, it is instructive to apply this method to SrMnO$_3$ \dots} 75 76 Plots of the interpolated bands for the different cases are shown in Fig.~(\ref{fig20.4}). In this case, the disentanglement for all the Mn-3d-derived states (empty red circles in Fig.~(\ref{fig20.4})) is only necessary around the $\Gamma$ point, as for all the other points and lines the bands of interest are well separated from other bands lower in energy. However, if we only consider the $e_g$ states (solid blue circles in Fig.~(\ref{fig20.4})) then the situation is different as these states are entangled with the $t_{2g}$ states around X. Of course the $t_{2g}$ states (solid green cones in Fig.~(\ref{fig20.4})) are entangled with $e_{g}$ states around X and with lower-lying states at $\Gamma$. 77 78\end{itemize} 79 80\begin{figure}[t!] 81\centering 82\subfloat[Full BS]{\includegraphics[width=0.7\columnwidth]{figure/example20/LaVO3_full_bandstructure.pdf}}\\ 83\subfloat[BS around Fermi energy]{\includegraphics[width=0.7\columnwidth]{figure/example20/LaVO3_bandstructure.pdf}} 84\caption{Top panel: full band structure of epitaxially-strained (tetragonal) LaVO$_3$ along the $\Gamma$-M-X-$\Gamma$-Z-A-R-X from DFT calculation (solid black) and interpolation from \Wannier{} (red dots). Bottom panel: magnification around Fermi energy $16.6049$ (dashed line). The disentanglement was performed only for $k$-points within a sphere of radius 0.2 $\si{\angstrom}^{-1}$ centred in A.} 85\label{fig20.1} 86\end{figure} 87 88\begin{figure}[h!] 89\centering 90\includegraphics[width=0.7\columnwidth]{figure/example20/LaVO3_bandstructure_all.pdf} 91\caption{Comparison of interpolated band structure of epitaxially-strained (tetragonal) LaVO$_3$ with disentanglement on a sphere of radius 0.2 $\si{\angstrom}^{-1}$ centred in A (red dots), full disentanglement (blue dots) and no disentanglement (green dots). Fermi energy is shown with a dashed line.}\label{fig20.2} 92\end{figure} 93 94\begin{figure}[h!] 95\centering 96\includegraphics[width=0.7\columnwidth]{figure/example20/SrMnO3_allbands.pdf} 97\caption{\Wannier{}-interpolated bands of SrMnO$_3$. From only $t_{2g}$ states (solid green cones), from only $e_g$ states (solid blue circles), or all Mn-3d-derived states ($t_{2g} + e_g$) (empty red circles).} 98\label{fig20.4} 99\end{figure} 100