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