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\backslash
chapter{Band Unfolding}
\backslash
label{chap:unfolding}
\backslash
hypertarget{Unfolding}{}
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else
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\begin_layout Title
FPLO documentation
\begin_inset Newline newline
\end_inset

Band unfolding
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\backslash

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[2ex]
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Logo{}
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\begin_layout Author
Klaus Koepernik
\end_layout

\begin_layout Date
07/01/2011
\end_layout

\begin_layout Standard
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status open

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\backslash
fi
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\begin_layout Section
Citations
\begin_inset Index idx
status open

\begin_layout Plain Layout
unfolding
\end_layout

\end_inset


\end_layout

\begin_layout Standard
When a citation is need it would be 
\begin_inset CommandInset citation
LatexCommand cite
key "FPLOWEB,WeiKu2010,EvHeumen2011"
literal "true"

\end_inset

.
 The second is the idea by Wei Ku and the third the first FPLO implementation/ap
plication.
 
\end_layout

\begin_layout Subsection
Copy right note
\end_layout

\begin_layout Standard
The content of this report cannot be used in another publication without
 contacting the FPLO authors first.
\end_layout

\begin_layout Section
Introduction
\end_layout

\begin_layout Subsection
Original idea
\end_layout

\begin_layout Standard
The idea is due to Wei Ku 
\begin_inset CommandInset citation
LatexCommand cite
key "WeiKu2010"
literal "true"

\end_inset

.
 Let's assume that we have a normal cell (NC) and a super cell (SC) derived
 from it.
 The NC contains 
\begin_inset Formula $N_{s}$
\end_inset

 atoms 
\begin_inset Formula $\boldsymbol{s}_{i}\in\left[1,N_{s}\right]$
\end_inset

 and has 
\begin_inset Formula $N_{r}$
\end_inset

 lattice vectors 
\begin_inset Formula $\boldsymbol{r}$
\end_inset

 in a defined Born von Karman torus.
 The SC contains 
\begin_inset Formula $N_{S}$
\end_inset

 atoms 
\begin_inset Formula $\boldsymbol{S}_{i}\in\left[1,N_{S}\right]$
\end_inset

 and has 
\begin_inset Formula $N_{R}$
\end_inset

 lattice vectors 
\begin_inset Formula $\boldsymbol{R}$
\end_inset

 in the 
\series bold
same
\series default
 BvK torus.
 The SC is a 
\begin_inset Quotes eld
\end_inset

multiple
\begin_inset Quotes erd
\end_inset

 of NC, meaning 
\begin_inset Formula $N_{S}=fN_{s}$
\end_inset

, 
\begin_inset Formula $f\in\mathbb{N}$
\end_inset

 and because of the same BvK torus 
\begin_inset Formula $N_{R}=\frac{1}{f}N_{r}$
\end_inset

, such that both tori contain the same number of sites 
\begin_inset Formula $N_{s}N_{r}=N_{S}N_{R}$
\end_inset

.
 The KS states of both cells fulfill their respective Bloch theorem
\begin_inset Formula 
\[
\Psi_{\boldsymbol{k}N}\left(\boldsymbol{r}-\boldsymbol{p}\right)=\Psi_{\boldsymbol{K}N}\left(\boldsymbol{r}\right)\mathrm{e^{-i\boldsymbol{k}\boldsymbol{p}}}
\]

\end_inset


\begin_inset Formula 
\[
\Psi_{\boldsymbol{K}N}\left(\boldsymbol{r}-\boldsymbol{P}\right)=\Psi_{\boldsymbol{K}N}\left(\boldsymbol{r}\right)\mathrm{e^{-i\boldsymbol{K}\boldsymbol{P}}}
\]

\end_inset


\end_layout

\begin_layout Standard
where we denote reciprocal lattice vectors in the NC Brillouin zone (bz)
 with 
\begin_inset Formula $\boldsymbol{k}$
\end_inset

 and reciprocal lattice vectors in the SC Brillouin zone (BZ) with 
\begin_inset Formula $\boldsymbol{K}$
\end_inset

.
 The Bloch spectral density is defined as
\begin_inset Formula 
\[
\hat{A}\left(\omega\right)=-\frac{1}{\pi}\mbox{Im}\hat{G}\left(\omega\right)
\]

\end_inset

with the single particle Green's function
\begin_inset Formula 
\[
\hat{G}\left(\omega\right)=\frac{1}{\omega^{+}-\hat{H}}
\]

\end_inset

and reads in the SC KS basis
\begin_inset Formula 
\begin{eqnarray*}
A_{\overline{\boldsymbol{K}}\overline{N},\boldsymbol{K}N}\left(\omega\right) & = & \langle\overline{\boldsymbol{K}}\overline{N}\mid\hat{A}\mid\boldsymbol{K}N\rangle\\
 & = & \delta_{\overline{\boldsymbol{K}}\boldsymbol{K}}\delta_{\overline{N}N}\delta\left(\omega-\varepsilon_{\boldsymbol{K}N}\right)
\end{eqnarray*}

\end_inset

It is diagonal in the SC quantum numbers.
 The full Green's function can be written as 
\begin_inset Formula 
\[
\hat{G}^{-1}=\hat{G}_{0}^{-1}-\hat{V}
\]

\end_inset

where 
\begin_inset Formula $G_{0}$
\end_inset

 is the Green's function of the NC and 
\begin_inset Formula $V$
\end_inset

 is the perturbing potential, which makes the difference between the exactly
 duplicated NC and the actual SC.
 Here, we already assume that only the potential is different, meaning that
 all atoms are exactly the same, just moved around a bit.
 We will discuss this later.
 Of course, if 
\begin_inset Formula $V$
\end_inset

 is weak, one can argue that the NC KS states 
\begin_inset Formula $\Psi_{\boldsymbol{k}n}$
\end_inset

 are also a good basis to express the Bloch spectral density.
 When done so, 
\begin_inset Formula $A$
\end_inset

 will no longer be diagonal in the 
\begin_inset Formula $\boldsymbol{k}$
\end_inset

 indices, since the NC Bloch symmetry is broken in SC.
 However, focusing only on the diagonal elements one get's an approximation
 for 
\begin_inset Formula $A$
\end_inset

, which will have approximatly the Bloch symmetry of the NC and hence 
\begin_inset Formula $A_{\boldsymbol{k}n,\boldsymbol{k}n}=\langle\boldsymbol{k}n\mid\hat{A}\mid\boldsymbol{k}n\rangle$
\end_inset

 will be the unfolded Bloch spectral density.
 Of course, 
\begin_inset Formula $\Psi_{\boldsymbol{k}n}$
\end_inset

 can only be a basis of SC, if SC is a 
\begin_inset Quotes eld
\end_inset

multiple
\begin_inset Quotes erd
\end_inset

 of NC (no atom substitution, see discussion in Section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Perturbations"

\end_inset

).
\end_layout

\begin_layout Subsection
The local orbital connection
\end_layout

\begin_layout Standard
Now, one can express the SC 
\begin_inset Formula $\Psi_{\boldsymbol{K}N}$
\end_inset

 in terms of Wannier functions (WFs) and claim that the WFs are the same
 in NC.
 Then one can transfer the WFs from SC to NC and build the KS functions
 in both cells from the same basis.
 This allows to express everything in terms of these WFs.
 In FPLO the local basis forms a complete WF basis, if L
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
"o
\end_layout

\end_inset

wdin orthogonalized.
 (This orthogonalization leads to a new basis 
\begin_inset Formula $\tilde{\Phi}=\Phi S^{-\frac{1}{2}}$
\end_inset

, which has the property of being the one 
\begin_inset Quotes eld
\end_inset

closest
\begin_inset Quotes erd
\end_inset

 to the non-orthogonal basis.
 Hence, an L-ortho FPLO basis is very localized and is a WF basis.
 To avoid unnessecary complications we work in our non-ortho FPLO basis
 and use gross-projection, when needed (
\begin_inset Formula $\frac{1}{2}\left(S*+*S\right)$
\end_inset

).
 Under the assumption that no atom substitution has taken place in SC we
 can do the algebra and calculate the following.
 The KS states are expressed via Bloch sums of WFs/local orbitals 
\begin_inset Formula 
\[
\Phi_{S\mu}^{\boldsymbol{K}}=\frac{1}{\sqrt{N_{R}}}\sum_{R}\Phi_{RS\mu}\mathrm{e}^{i\boldsymbol{K}\left(\boldsymbol{R}+\boldsymbol{S}\right)}
\]

\end_inset

where 
\begin_inset Formula $\Phi_{RS\mu}$
\end_inset

 is the 
\begin_inset Formula $\mu$
\end_inset

 orbital at site 
\begin_inset Formula $\boldsymbol{S}$
\end_inset

 in cell 
\begin_inset Formula $\boldsymbol{R}$
\end_inset

.
 The KS-state is a simple linear combination of these Bloch sums
\begin_inset Formula 
\[
\Psi_{\boldsymbol{K}N}=\sum_{S\mu}\Phi_{S\mu}^{\boldsymbol{K}}C_{S\mu,N}^{\boldsymbol{K}}
\]

\end_inset

Hence, 
\begin_inset Formula 
\begin{eqnarray*}
\hat{A} & = & \sum_{KN}\Psi_{\boldsymbol{K}N}\delta\left(\omega-\varepsilon_{\boldsymbol{K}N}\right)\Psi_{\boldsymbol{K}N}^{+}\\
 & = & \sum_{KN}\sum_{S\mu,\overline{S}\overline{\mu}}\Phi_{S\mu}^{\boldsymbol{K}}n_{S\mu,\overline{S}\overline{\mu}}^{\boldsymbol{K}N}\left(\omega\right)\Phi_{\overline{S}\overline{\mu}}^{\boldsymbol{K}}
\end{eqnarray*}

\end_inset

which introduces the DOS weight matrix in the local basis
\begin_inset Formula 
\[
n_{S\mu,\overline{S}\overline{\mu}}^{\boldsymbol{K}N}\left(\omega\right)=C_{S\mu,N}^{\boldsymbol{K}}\delta\left(\omega-\varepsilon_{\boldsymbol{K}N}\right)C_{\overline{S}\overline{\mu},N}^{\boldsymbol{K}*}
\]

\end_inset

Now, we map the sites of the SC onto the sites of NC.
 The SC consists of 
\begin_inset Formula $f$
\end_inset

 
\begin_inset Quotes eld
\end_inset

copies
\begin_inset Quotes erd
\end_inset

 of NC, translated by NC lattice vectors 
\begin_inset Formula $\boldsymbol{r}_{j}$
\end_inset

, 
\begin_inset Formula $j\in\left[1,f\right]$
\end_inset

.
 For every site 
\begin_inset Formula $\boldsymbol{S}$
\end_inset

 there is a site 
\begin_inset Formula $\boldsymbol{s}$
\end_inset

 in NC, which is in the NC cell translated by 
\begin_inset Formula $\boldsymbol{r}_{j}$
\end_inset

.
 The SC lattice vector and the translation 
\begin_inset Formula $\boldsymbol{r}_{j}$
\end_inset

 give a NC lattice vector 
\begin_inset Formula 
\[
\boldsymbol{r}=\boldsymbol{R}+\boldsymbol{r}_{j}
\]

\end_inset

and the site mapping can be written by labeling the SC sites according to
 their construction out of NC sites and cells (cf.
 Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Cell-mapping"

\end_inset

)
\begin_inset Formula 
\[
\boldsymbol{s}+\boldsymbol{r}_{j}=\boldsymbol{S}_{sj}.
\]

\end_inset


\end_layout

\begin_layout Standard
Consequently, absolute site vectors can be written
\begin_inset Formula 
\[
\boldsymbol{r}+\boldsymbol{s}=\boldsymbol{R}+\boldsymbol{r}_{j}+\boldsymbol{s}=\boldsymbol{R}+\boldsymbol{S}_{sj}
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement H
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pic/cellmapping.pdf
	width 40col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Cell-mapping"

\end_inset

Cell mapping: The SC as the 
\begin_inset Quotes eld
\end_inset

fourfold
\begin_inset Quotes erd
\end_inset

 of the NC.
 Some of the open atoms are displaced against ideal NC positions.
 The small vectors are NC lattice vectors and show how the 4 SC open atoms
 are mapped onto their one NC equivalent.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 If we now neglect eventual atom displacements we can form SC Bloch sums
 out of NC Bloch sums (but at the SC k-vector)
\begin_inset Formula 
\begin{eqnarray}
\Phi_{\left[NC\right]s}^{\boldsymbol{K}} & = & \frac{1}{\sqrt{N_{r}}}\sum_{r}\Phi_{rs\mu}\mathrm{e}^{i\boldsymbol{K}\left(\boldsymbol{r}+\boldsymbol{s}\right)}\nonumber \\
 & = & \frac{\sqrt{N_{R}}}{\sqrt{N_{r}}}\left(\frac{1}{\sqrt{N_{R}}}\sum_{Rj}\Phi_{Rr_{j},s\mu}\mathrm{e}^{i\boldsymbol{K}\left(\boldsymbol{R}+\boldsymbol{r}_{j}+\boldsymbol{s}\right)}\right)\nonumber \\
 & = & \frac{1}{\sqrt{f}}\left(\frac{1}{\sqrt{N_{R}}}\sum_{Rj}\Phi_{RS_{sj}\mu}\mathrm{e}^{i\boldsymbol{K}\left(\boldsymbol{R}+\boldsymbol{S}_{sj}\right)}\right)\nonumber \\
\Phi_{\left[NC\right]s}^{\boldsymbol{K}} & = & \frac{1}{\sqrt{f}}\sum_{j}\Phi_{\left[SC\right]S_{sj}}^{\boldsymbol{K}}\label{eq:contractNCBlochSum}
\end{eqnarray}

\end_inset

The NC Bloch sum is a contraction of the SC Bloch sums.
 The KS states behave as 
\begin_inset Formula 
\begin{eqnarray*}
\Psi_{N}^{\boldsymbol{K}} & = & \sum_{S\mu}\Phi_{S\mu}^{\boldsymbol{K}}C_{S\mu,N}^{\boldsymbol{K}}\\
 & = & \sum_{sj\mu}\Phi_{S_{sj}\mu}^{\boldsymbol{K}}C_{S_{sj}\mu,N}^{\boldsymbol{K}}
\end{eqnarray*}

\end_inset

Now, the coefficients would fulfill the translational symmetry 
\begin_inset Formula 
\[
C_{S_{sj}}=C_{r_{j}s}=C_{s}U
\]

\end_inset

if 
\begin_inset Formula $\boldsymbol{r}_{j}$
\end_inset

 is strictly a lattice vector.
 (
\begin_inset Formula $U$
\end_inset

 denotes a possible unitary transformation, which does not figure in the
 following, where we reverse the argumentation).
 Thus, if the NC translational symmetry were true, we could replace 
\begin_inset Formula 
\[
C_{S_{sj}}=\frac{1}{f_{s}}\sum_{j=1}^{f_{s}}C_{S_{sj}}
\]

\end_inset

Let's do that in the KS state and use Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:contractNCBlochSum"

\end_inset

)
\begin_inset Formula 
\begin{eqnarray*}
\Psi_{N}^{\boldsymbol{K}} & \approx & \sum_{sj\mu}\Phi_{S_{sj}\mu}^{\boldsymbol{K}}\frac{1}{f_{s}}\sum_{i=1}^{f_{s}}C_{S_{si}\mu,N}^{\boldsymbol{K}}\\
 & \approx & \sqrt{f}\sum_{s\mu}\Phi_{\left[NC\right]s\mu}^{\boldsymbol{K}}\frac{1}{f_{s}}\sum_{i=1}^{f_{s}}C_{S_{si}\mu,N}^{\boldsymbol{K}}\\
 & = & \frac{\sqrt{f}}{f_{s}}\sum_{s\mu,i=1}^{f_{s}}\Phi_{\left[NC\right]s\mu}^{\boldsymbol{K}}C_{S_{si}\mu,N}^{\boldsymbol{K}}
\end{eqnarray*}

\end_inset

Hence, we get the coefficients in the NC Bloch sum representation
\begin_inset Formula 
\begin{equation}
C_{\left[NC\right]s\mu,N}^{\boldsymbol{K}}=\frac{\sqrt{f}}{f_{s}}\sum_{j=1}^{f_{s}}C_{S_{sj}\mu,N}^{\boldsymbol{K}}\label{eq:CNCfromNSC}
\end{equation}

\end_inset

and 
\begin_inset Formula 
\[
\Psi_{N}^{\boldsymbol{K}}\approx\sum_{s\mu}\Phi_{\left[NC\right]s\mu}^{\boldsymbol{K}}C_{\left[NC\right]s\mu,N}^{\boldsymbol{K}}
\]

\end_inset

Now, we can write the Bloch spectral density in terms of these approximate
 Bloch sums
\begin_inset Formula 
\begin{eqnarray}
\hat{A} & \approx & \sum_{KN}\sum_{s\mu,\overline{s}\overline{\mu}}\Phi_{\left[NC\right]s\mu}^{\boldsymbol{K}}\left[\frac{f}{f_{s}f_{\overline{s}}}\sum_{j=1}^{f_{s}}\sum_{\overline{j}=1}^{f_{\overline{s}}}n_{S_{sj}\mu,\overline{S}_{\overline{s}\overline{j}}\overline{\mu}}^{\boldsymbol{K}N}\left(\omega\right)\right]\Phi_{\left[NC\right]\overline{s}\overline{\mu}}^{\boldsymbol{K}}\label{eq:ApproxA}
\end{eqnarray}

\end_inset

The un-approximated Bloch spectral function fulfills
\begin_inset Formula 
\[
\int_{-\infty}^{\infty}A_{\boldsymbol{K}N}\left(\omega\right)=\int_{-\infty}^{\infty}\mathrm{Tr}\hat{A}_{\boldsymbol{K}N}\left(\omega\right)=1
\]

\end_inset

with 
\begin_inset Formula $\hat{A}_{\boldsymbol{K}N}=\mid\boldsymbol{K}N\rangle\delta\left(\omega-\varepsilon_{\boldsymbol{K}N}\right)\langle\boldsymbol{K}N\mid$
\end_inset

.
 The Bloch sums yield the overlap matrix via 
\begin_inset Formula 
\[
S_{\overline{s}\overline{\mu},s\mu}^{\boldsymbol{K}}=\left\langle \Phi_{\overline{s}\overline{\mu}}^{\boldsymbol{K}}\mid\Phi_{s\mu}^{\boldsymbol{K}}\right\rangle 
\]

\end_inset

So, we basically get 
\begin_inset Formula $\int A=\int\mathrm{Tr}n^{K}\left(\omega\right)S^{K}$
\end_inset

, which is nothing but the normalization condition 
\begin_inset Formula $C^{+}SC=1$
\end_inset

 for non-orthogonal eigenvalue problems.
 Using Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:CNCfromNSC"

\end_inset

) it is clear that Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:ApproxA"

\end_inset

) is properly normalized if the NC Bloch symmetry is exact.
 (Note, that then 
\begin_inset Formula $f_{s}=f$
\end_inset

.) If we introduce a symmetrized gross projected method, where we use contragradi
ent Bloch sums 
\begin_inset Formula $\tilde{\Phi}^{K}=\Phi^{K}\left(S^{K}\right)^{-1}$
\end_inset

, 
\begin_inset Formula $\left\langle \Phi\mid\tilde{\Phi}\right\rangle =1$
\end_inset

 on one side of 
\begin_inset Formula $A$
\end_inset

 we get the expression 
\begin_inset Formula 
\[
\hat{A}=\frac{1}{2}\left(\Phi nS\tilde{\Phi}^{+}+\tilde{\Phi}Sn\Phi^{+}\right)
\]

\end_inset

and the trace give 
\begin_inset Formula 
\[
\mathrm{Tr}\underline{A}=\frac{1}{2}\mathrm{Tr}\left(nS+Sn\right)
\]

\end_inset

Hence, the diagonal part of 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none

\begin_inset Formula $\underline{A}=\frac{1}{2}\left(nS+Sn\right)$
\end_inset

 gives the unfolded weights and is normalized (
\begin_inset Formula $\mathrm{Tr}\underline{A}=1$
\end_inset

), if the NC Bloch symmetry is strictly true.
 For cases, where atoms got substituted, we have to make a modification
 to Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:contractNCBlochSum"

\end_inset

).
 The 
\begin_inset Formula $f$
\end_inset

 should be the actual multiplicity of the NC site 
\begin_inset Formula $s$
\end_inset

 under consideration 
\begin_inset Formula $f=f_{s}$
\end_inset

.
 This is a definition, which is consistent with any possible choice of NC
 and also with the cases of partial unfolding due to substitutions.
 In any case the normalization is proper.
 Consider the case of perfect NC Bloch symmetry.
 Then 
\begin_inset Formula $n_{\left[SC\right]S_{sj}}=\frac{1}{f}n_{\left[NC\right]s}$
\end_inset

.
 Hence (omitting 
\begin_inset Formula $S$
\end_inset

 for simplicity) the diagonal part of Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:ApproxA"

\end_inset

) reads
\begin_inset Formula 
\begin{eqnarray*}
A_{s} & = & \frac{f}{f_{s}f_{s}}\sum_{j=1}^{f_{s}}\sum_{\overline{j}=1}^{f_{s}}n_{\left[SC\right]S_{sj}S_{s\overline{j}}}\\
 & = & \frac{f}{f_{s}f_{s}}\sum_{j=1}^{f_{s}}\sum_{\overline{j}=1}^{f_{s}}\frac{1}{f}n_{\left[NC\right]ss}\\
 & = & n_{\left[NC\right]ss}
\end{eqnarray*}

\end_inset

And normalization becomes 
\begin_inset Formula 
\[
\sum_{s}\mathrm{Tr}_{\mu}A_{s\mu}=1
\]

\end_inset

which is fulfilled since it is for the NC expression.
 If we now have to partially unfold because out of 
\begin_inset Formula $f$
\end_inset

 sites, which would backfold to a single NC site, several (let's say 
\begin_inset Formula $m_{s}$
\end_inset

) are occupied by a different atom then the 
\begin_inset Formula $f_{s}=f-m_{s}$
\end_inset

 other sites, we cannot fully contract over all 
\begin_inset Formula $f$
\end_inset

 sites.
 But we can contract over 
\begin_inset Formula $f_{s}=f-m_{s}$
\end_inset

 sites and leave the 
\begin_inset Formula $m_{s}$
\end_inset

 other atoms uncontracted (means not unfolded).
 This would restrict the contraction sum to 
\begin_inset Formula $f_{s}\neq f$
\end_inset

.
 Let's assume that still approximately 
\begin_inset Formula $n_{\left[SC\right]S_{sj}}=\frac{1}{f}n_{\left[NC\right]s}$
\end_inset

 (just to discuss the normalization) then we get, letting the first parameter
 
\begin_inset Formula $f=F$
\end_inset

 be choosen later 
\begin_inset Formula 
\begin{eqnarray*}
A_{s} & = & \frac{F}{f_{s}f_{s}}\sum_{j=1}^{f_{s}}\sum_{\overline{j}=1}^{f_{s}}n_{\left[SC\right]S_{sj}S_{s\overline{j}}}\\
 & = & \frac{F}{f}n_{\left[NC\right]ss}
\end{eqnarray*}

\end_inset

Besides this we have the 
\begin_inset Formula $m_{s}$
\end_inset

 un-unfolded weights of the sites 
\begin_inset Formula $s_{\lambda}$
\end_inset

, which have substituted atoms 
\begin_inset Formula 
\begin{eqnarray*}
A_{s_{\lambda}} & = & \frac{1}{f}n_{\left[NC\right]s_{\lambda}s_{\lambda}},\qquad\lambda\in\left[1,m_{s}\right]\\
 & \approx & \frac{1}{f}n_{\left[NC\right]ss},\quad\mbox{(if perfect)}
\end{eqnarray*}

\end_inset

The normalization reads 
\begin_inset Formula 
\[
A_{s}+\sum_{\lambda=1}^{m_{s}}A_{s_{\lambda}}=\frac{F+m_{s}}{f}n_{\left[NC\right]ss}
\]

\end_inset

and it becomes clear that 
\begin_inset Formula $F=f_{s}$
\end_inset

 gives normalization when summed over all sites.
 Hence our final definition is
\begin_inset Formula 
\begin{eqnarray}
w_{s\mu}^{\boldsymbol{K}N}\left(\omega\right) & = & \frac{1}{f_{s}}\sum_{j=1}^{f_{s}}\sum_{\overline{j}=1}^{f_{s}}n_{S_{sj}\mu S_{s\overline{j}}\mu}^{\boldsymbol{K}N}\left(\omega\right)\label{eq:UnfoldWeight}
\end{eqnarray}

\end_inset

Or in gross projection and droping the delta function, which was a placeholder
 for the energy dependence
\begin_inset Formula 
\begin{eqnarray*}
w_{s\mu}^{\boldsymbol{K}N} & = & \frac{1}{f_{s}}\sum_{j=1}^{f_{s}}\sum_{\overline{j}=1}^{f_{s}}\frac{1}{2}\left(C_{S_{sj}\mu,N}^{\boldsymbol{K}}\left[C^{+}S\right]_{N,S_{s\overline{j}}\mu}^{\boldsymbol{K}}+\left[SC\right]_{S_{sj}\mu,N}^{\boldsymbol{K}}C_{S_{s\overline{j}}\mu,N}^{\boldsymbol{K}*}\right)\\
 & = & \frac{1}{f_{s}}\sum_{j=1}^{f_{s}}\sum_{\overline{j}=1}^{f_{s}}\frac{1}{2}\left(C_{S_{sj}\mu,N}^{\boldsymbol{K}}\left[C^{+}S\right]_{N,S_{s\overline{j}}\mu}^{\boldsymbol{K}}+\left[C^{+}S\right]_{N,S_{s\overline{j}}\mu}^{\boldsymbol{K}*}C_{S_{sj}\mu,N}^{\boldsymbol{K}*}\right),\quad j\leftrightarrow\overline{j},\mbox{ in second term}\\
 & = & \frac{1}{f_{s}}\sum_{j=1}^{f_{s}}\sum_{\overline{j}=1}^{f_{s}}\mathrm{Re}\left(C_{S_{sj}\mu,N}^{\boldsymbol{K}}\left[C^{+}S\right]_{N,S_{s\overline{j}}\mu}^{\boldsymbol{K}}\right)
\end{eqnarray*}

\end_inset

Net projection will be discussed below.
\end_layout

\begin_layout Subsection
Alternative considerations
\end_layout

\begin_layout Standard
First let's defined the normal cell problem and derive some symmetries.
 The orbitals 
\begin_inset Formula $\Phi_{rs\mu}$
\end_inset

 form Bloch sums 
\begin_inset Formula 
\[
\Phi_{s\mu}^{\boldsymbol{k}}=\frac{1}{\sqrt{N_{r}}}\sum_{r}\Phi_{rs\mu}\mathrm{e}^{i\boldsymbol{k}\left(\boldsymbol{r}+\boldsymbol{s}\right)}
\]

\end_inset

and wave functions
\begin_inset Formula 
\[
\Psi_{n}^{\boldsymbol{k}}=\sum_{s\mu}\Phi_{s\mu}^{\boldsymbol{k}}C_{s\mu,n}^{\boldsymbol{k}}
\]

\end_inset

where 
\begin_inset Formula $C$
\end_inset

 is determined by (overlap ignored, since it does not change the results
 derived here) 
\begin_inset Formula 
\[
H^{\boldsymbol{k}}=\left(\Phi^{\boldsymbol{k}}\mid H\mid\Phi^{\boldsymbol{k}}\right)
\]

\end_inset


\begin_inset Formula 
\[
H^{\boldsymbol{k}}C^{\boldsymbol{k}}=C^{\boldsymbol{k}}\varepsilon^{\boldsymbol{k}}
\]

\end_inset

At a shifted 
\begin_inset Formula $\boldsymbol{k}$
\end_inset

-vector we get
\begin_inset Formula 
\begin{eqnarray*}
\Phi_{s}^{\boldsymbol{k}+\boldsymbol{g}} & = & \Phi_{s\mu}^{\boldsymbol{k}}\mathrm{e}^{i\boldsymbol{g}\left(\boldsymbol{r}+\boldsymbol{s}\right)}\\
 & = & \Phi_{s\mu}^{\boldsymbol{k}}\mathrm{e}^{i\boldsymbol{g}\boldsymbol{s}}
\end{eqnarray*}

\end_inset

and hence
\begin_inset Formula 
\[
H_{\overline{s}s}^{\boldsymbol{k}+\boldsymbol{g}}=\mathrm{e}^{-i\boldsymbol{g}\overline{\boldsymbol{s}}}H_{\overline{s}s}^{\boldsymbol{k}}\mathrm{e}^{i\boldsymbol{g}\boldsymbol{s}}
\]

\end_inset


\begin_inset Formula 
\begin{eqnarray*}
H^{\boldsymbol{k}+\boldsymbol{g}}C^{\boldsymbol{k}+\boldsymbol{g}} & = & C^{\boldsymbol{k}+\boldsymbol{g}}\varepsilon^{\boldsymbol{k}+\boldsymbol{g}}\\
H^{\boldsymbol{k}}\left(\mathrm{e}^{i\boldsymbol{g}\boldsymbol{s}}C^{\boldsymbol{k}+\boldsymbol{g}}\right) & = & \left(\mathrm{e}^{i\boldsymbol{g}\boldsymbol{s}}C^{\boldsymbol{k}+\boldsymbol{g}}\right)\varepsilon^{\boldsymbol{k}+\boldsymbol{g}}
\end{eqnarray*}

\end_inset

which says that the expression in parenthese is also an eigenvector for
 
\begin_inset Formula $\boldsymbol{k}$
\end_inset

 and hence
\begin_inset Formula 
\[
\varepsilon^{\boldsymbol{k}+\boldsymbol{g}}=\varepsilon^{\boldsymbol{k}}
\]

\end_inset


\begin_inset Formula 
\begin{equation}
\boxed{C^{\boldsymbol{k}+\boldsymbol{g}}=\mathrm{e}^{-i\boldsymbol{g}\boldsymbol{s}}C^{\boldsymbol{k}}U^{\boldsymbol{k}}}\label{eq:c_k_plus_g}
\end{equation}

\end_inset

where 
\begin_inset Formula $U$
\end_inset

 mixes only states in degenerate subspaces (degenerate bands along symmetry
 lines/planes or at band crossings).
 Now, one can argue that there is always a way to adjust the phases (gauge
 freedom) of an actually calculated 
\begin_inset Formula $C^{\boldsymbol{k}+\boldsymbol{g}}$
\end_inset

 (solution to the eigenvalyue problem) such that the result of a shift by
 
\begin_inset Formula $\boldsymbol{g}$
\end_inset

 is at most a reordering of band indices, which allows to set 
\begin_inset Formula $U=1$
\end_inset

 in our considerations.
 In essence we are saying that the set over all band indices 
\begin_inset Formula $\bigcup_{n}\Psi_{n}^{\boldsymbol{k}+\boldsymbol{g}}$
\end_inset

 equals 
\begin_inset Formula $\bigcup_{n}\Psi_{n}^{\boldsymbol{k}}$
\end_inset

.
 We have periodicity of the sets of bands with respect to reciprocal translation
s.
 This is strictly only true for the sets, which happen to form continuous
 and periodic functions of 
\begin_inset Formula $\boldsymbol{k}$
\end_inset

 (yes functions of sets!).
 If an individual band is followed smoothly (e.g.
 by the 
\begin_inset Formula $\boldsymbol{k}\cdot\boldsymbol{p}$
\end_inset

-method) the resulting 
\begin_inset Formula $\Psi_{n}^{\boldsymbol{k}+\boldsymbol{g}}$
\end_inset

 can become another band 
\begin_inset Formula $\Psi_{n^{\prime}}^{\boldsymbol{k}}$
\end_inset

.
 
\end_layout

\begin_layout Standard
Now, we describe the same system by introducing a super cell with 
\begin_inset Formula $f$
\end_inset

 copies of the original cell.
 This leads to new lattice vectors 
\begin_inset Formula $\boldsymbol{R}$
\end_inset

 and sites 
\begin_inset Formula $\boldsymbol{S}_{sj}=\boldsymbol{r}_{j}+\boldsymbol{s}$
\end_inset

, which are formed by shifting the original sites 
\begin_inset Formula $\boldsymbol{s}$
\end_inset

 via a set of 
\begin_inset Formula $f$
\end_inset

 original lattice vectors 
\begin_inset Formula $\boldsymbol{r}_{j}$
\end_inset

, 
\begin_inset Formula $j\in\left[1,f\right]$
\end_inset

.
 Of course, the set of normal cell lattice vectors is obtained from 
\begin_inset Formula $\left\{ \boldsymbol{r}\right\} =\bigcup_{j=1}^{f}\left\{ \boldsymbol{R}+r_{j}\right\} $
\end_inset

, which also leads to 
\begin_inset Formula $\sum_{Rj}F\left(\boldsymbol{R}+\boldsymbol{r}_{j}\right)=\sum_{r}F\left(\boldsymbol{r}\right)$
\end_inset

.
 We also have to require that the BvK torus in both descriptions has the
 same volume or that the number of normal cell lattice vectors is 
\begin_inset Formula $f$
\end_inset

-times the number of super cell lattice vectors: 
\begin_inset Formula $N_{r}=fN_{R}$
\end_inset

.
 Now, we can reformulate the problem in this super cell description by forming
 Bloch sums with proper super cell translational symmetry
\begin_inset Formula 
\[
\Phi_{\left[SC\right]S_{sj}\mu}^{\boldsymbol{k}}=\frac{1}{\sqrt{N_{R}}}\sum_{R}\Phi_{RS_{js}\mu}\mathrm{e}^{i\boldsymbol{k}\left(\boldsymbol{R}+\boldsymbol{S}_{js}\right)}
\]

\end_inset

The corresponding coefficients are 
\begin_inset Formula $C_{\left[SC\right]S_{sj}\mu,n}^{\boldsymbol{k}}$
\end_inset

.
 
\end_layout

\begin_layout Standard
This description cannot lead to a different result compared to the normal
 cell description, since we only changed the artifical choice of a supercell.
 However, by increasing the matrix size (number of sites per unit cell)
 we increased the number of eigen solutions or bands per 
\begin_inset Formula $\boldsymbol{k}$
\end_inset

-point.
 In order to get the same number of physically different solutions the size
 of the reciprocal unit cell must be 
\begin_inset Formula $f$
\end_inset

-times smaller than the NC reciprocal unit cell.
 Well, as everyone knows the excess solutions are bands, which got backfolded
 into this smaller reciprocal cell.
 To understand this mathematically it suffices to note that we already showed
 that in every unit cell the sets of bands form periodic and contiuous functions.
 Hence, the SC wave functions are periodic with repect to translations by
 reciprocal lattice vectors 
\begin_inset Formula $\boldsymbol{G}$
\end_inset

: 
\begin_inset Formula $\bigcup_{n}\Psi_{n}^{\boldsymbol{k}+\boldsymbol{G}}=\bigcup_{n}\Psi_{n}^{\boldsymbol{k}}$
\end_inset

.
 Let's construct the 
\begin_inset Formula $f\cdot m$
\end_inset

 SC solutions out of the 
\begin_inset Formula $m$
\end_inset

 NC solutions.
 The first 
\begin_inset Formula $m$
\end_inset

 solutions are just identical to the NC solutions 
\begin_inset Formula $\bigcup_{n=1}^{m}\Psi_{\left[SC\right]n}^{\boldsymbol{k}}=\bigcup_{n=1}^{m}\Psi_{\left[NC\right]n}^{\boldsymbol{k}}$
\end_inset

.
 Now, for every reciprocal vector 
\begin_inset Formula $\boldsymbol{G}$
\end_inset

 the set 
\begin_inset Formula $\bigcup_{n=1}^{m}\Psi_{\left[NC\right]n}^{\boldsymbol{k}+\boldsymbol{G}}$
\end_inset

 is either equivalent to 
\begin_inset Formula $\bigcup_{n=1}^{m}\Psi_{\left[NC\right]n}^{\boldsymbol{k}}$
\end_inset

 (
\begin_inset Formula $\boldsymbol{G}$
\end_inset

 is a NC vector 
\begin_inset Formula $\boldsymbol{g}$
\end_inset

) or forms a new backfolded set in which case there are exactly 
\begin_inset Formula $f-1$
\end_inset

 different additional sets.
 We identify 
\begin_inset Formula $f-1$
\end_inset

 representative vectors 
\begin_inset Formula $\boldsymbol{G}_{l}\notin\left\{ \boldsymbol{g}\right\} $
\end_inset

, 
\begin_inset Formula $l\in\left[1,f-1\right]$
\end_inset

 and 
\begin_inset Formula $\boldsymbol{G}_{0}=0$
\end_inset

 and define the set of 
\begin_inset Formula $f\cdot m$
\end_inset

 SC solutions via 
\begin_inset Formula $\bigcup_{n=1}^{m}\Psi_{\left[SC\right]n+lm}^{\boldsymbol{k}}=\bigcup_{n=1}^{m}\Psi_{\left[NC\right]n}^{\boldsymbol{k}+\boldsymbol{G}_{l}},\quad\forall\;l\in\left[0,f-1\right]$
\end_inset

.
 This is just fancy talk for all the backfolding.
 Beware, that in a real calculations all coefficients 
\begin_inset Formula $C$
\end_inset

 contain random phase factors from the eigenvalue solver, which means that
 the construction above deviates in practice from the actual solutions by
 phases and unitary mixing of degenerate bands.
 
\end_layout

\begin_layout Standard
However, now we can go a step further by identifying equivalences between
 coefficients.
 Using the construction of the SC sites discussed above we can write
\begin_inset Formula 
\begin{eqnarray*}
\sum_{S}\Phi_{\left[SC\right]S_{sj}\mu}^{\boldsymbol{k}} & = & \frac{1}{\sqrt{N_{R}}}\sum_{RS}\Phi_{RS_{js}\mu}\mathrm{e}^{i\boldsymbol{k}\left(\boldsymbol{R}+\boldsymbol{S}_{js}\right)}\\
 & = & \frac{1}{\sqrt{N_{R}}}\sum_{Rjs}\Phi_{Rr_{j}s\mu}\mathrm{e}^{i\boldsymbol{k}\left(\boldsymbol{R}+\boldsymbol{s}+\boldsymbol{r}_{j}\right)}\\
 & = & \frac{\sqrt{f}}{\sqrt{N_{r}}}\sum_{rs}\Phi_{rs\mu}\mathrm{e}^{i\boldsymbol{k}\left(\boldsymbol{r}+\boldsymbol{s}\right)}
\end{eqnarray*}

\end_inset

which leads to the SC expression
\begin_inset Formula 
\begin{eqnarray*}
\Psi_{\left[SC\right]n+lm}^{\boldsymbol{k}} & = & \sum_{S\mu}\Phi_{\left[SC\right]S\mu}^{\boldsymbol{k}}C_{\left[SC\right]S\mu,n+lm}^{\boldsymbol{k}}\\
 & = & \frac{1}{\sqrt{N_{R}}}\sum_{RS\mu}\Phi_{RS\mu}\mathrm{e}^{i\boldsymbol{k}\left(\boldsymbol{R}+\boldsymbol{S}\right)}C_{\left[SC\right]S\mu,n+lm}^{\boldsymbol{k}}\\
 & = & \frac{\sqrt{f}}{\sqrt{N_{r}}}\sum_{Rjs\mu}\Phi_{Rr_{j}s\mu}\mathrm{e}^{i\boldsymbol{k}\left(\boldsymbol{R}+\boldsymbol{r}_{j}+\boldsymbol{s}\right)}C_{\left[SC\right]S_{js}\mu,n+lm}^{\boldsymbol{k}}
\end{eqnarray*}

\end_inset

and the NC expression, whose equivalence via backfolding we established
 in the argument above.
\begin_inset Formula 
\begin{eqnarray*}
\Psi_{\left[SC\right]n+lm}^{\boldsymbol{k}} & = & \Psi_{\left[NC\right]n}^{\boldsymbol{k}+\boldsymbol{G}_{l}}\\
 & = & \sum_{s\mu}\Phi_{\left[NC\right]s\mu}^{\boldsymbol{k}+\boldsymbol{G}_{l}}C_{\left[NC\right]s\mu,n}^{\boldsymbol{k}+\boldsymbol{G}_{l}}\\
 & = & \frac{1}{\sqrt{N_{r}}}\sum_{rs\mu}\Phi_{rs\mu}\mathrm{e}^{i\left(\boldsymbol{k}+\boldsymbol{G}_{l}\right)\left(\boldsymbol{r}+\boldsymbol{s}\right)}C_{\left[NC\right]s\mu,n}^{\boldsymbol{k}+\boldsymbol{G}_{l}}\\
 & = & \frac{1}{\sqrt{N_{r}}}\sum_{Rjs\mu}\Phi_{Rr_{j}s\mu}\mathrm{e}^{i\left(\boldsymbol{k}+\boldsymbol{G}_{l}\right)\left(\boldsymbol{R}+\boldsymbol{r}_{j}+\boldsymbol{s}\right)}C_{\left[NC\right]s\mu,n}^{\boldsymbol{k}+\boldsymbol{G}_{l}}\\
 & = & \frac{1}{\sqrt{N_{r}}}\sum_{Rjs\mu}\Phi_{Rr_{j}s\mu}\mathrm{e}^{i\boldsymbol{k}\left(\boldsymbol{R}+\boldsymbol{r}_{j}+\boldsymbol{s}\right)}\mathrm{e}^{i\boldsymbol{G}_{l}\left(\boldsymbol{r}_{j}+\boldsymbol{s}\right)}C_{\left[NC\right]s\mu,n}^{\boldsymbol{k}+\boldsymbol{G}_{l}}
\end{eqnarray*}

\end_inset

By comparison with the SC expression we identify
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_layout Standard
\noindent
\align center
\begin_inset Box Boxed
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "60col%"
special "none"
height "1in"
height_special "totalheight"
thickness "0.4pt"
separation "3pt"
shadowsize "4pt"
framecolor "black"
backgroundcolor "none"
status open

\begin_layout Plain Layout
\begin_inset Formula 
\begin{eqnarray}
C_{\left[SC\right]S_{js}\mu,n+lm}^{\boldsymbol{k}} & = & \frac{1}{\sqrt{f}}\mathrm{e}^{i\boldsymbol{G}_{l}\boldsymbol{S}_{sj}}C_{\left[NC\right]s\mu,n}^{\boldsymbol{k}+\boldsymbol{G}_{l}}\nonumber \\
 & = & \frac{1}{\sqrt{f}}\mathrm{e}^{i\boldsymbol{G}_{l}\boldsymbol{r}_{j}}\mathrm{e}^{i\boldsymbol{G}_{l}\boldsymbol{s}}C_{\left[NC\right]s\mu,n}^{\boldsymbol{k}+\boldsymbol{G}_{l}}\label{eq:C_SC_maped_to_NC}
\end{eqnarray}

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
which is (up to unitary mixing, which we neglected) the exact mapping between
 the solutions of the two equivalent descriptions.
 Now, we average the SC coefficients over the equivalent sites with respect
 to NC periodicity (
\begin_inset Formula $j$
\end_inset

-sum with 
\begin_inset Formula $f_{s}$
\end_inset

 terms).
\begin_inset Formula 
\[
\frac{1}{f_{s}}\sum_{j}C_{\left[SC\right]S_{js}\mu,n+lm}^{\boldsymbol{k}}=\frac{1}{\sqrt{f}}\frac{1}{f_{s}}\left(\sum_{j}\mathrm{e}^{i\boldsymbol{G}_{l}\boldsymbol{r}_{j}}\right)\mathrm{e}^{i\boldsymbol{G}_{l}\boldsymbol{s}}C_{\left[NC\right]s\mu,n}^{\boldsymbol{k}+\boldsymbol{G}_{l}}
\]

\end_inset

The sum in parentheses runs over all NC lattice vectors 
\begin_inset Formula $\boldsymbol{r}_{j}$
\end_inset

, which are needed to make the whole NC lattice from the SC lattice vectors
 
\begin_inset Formula $\boldsymbol{R}$
\end_inset

.
 We can introduce a basis in the real space lattice via 
\begin_inset Formula 
\begin{eqnarray*}
\boldsymbol{R} & = & \sum_{I=1}^{3}R_{I}\boldsymbol{A}_{I}\\
\boldsymbol{r} & = & \sum r_{I}\boldsymbol{a}_{I}
\end{eqnarray*}

\end_inset

where in order for the SC to be a commensurate multiple of the NC the relation
 
\begin_inset Formula $\boldsymbol{A}_{I}=\boldsymbol{a}_{J}M_{JI}$
\end_inset

, 
\begin_inset Formula $M_{JI}\in\mathbb{N}$
\end_inset

 must hold.
 The reciprocal lattice then also has a basis
\begin_inset Formula 
\begin{eqnarray*}
\boldsymbol{G} & = & \sum G_{I}\boldsymbol{B}_{I}\\
\boldsymbol{g} & = & \sum g_{I}\boldsymbol{b}_{I}
\end{eqnarray*}

\end_inset

with the defining relation of the reciprocal basis 
\begin_inset Formula $\boldsymbol{B}_{I}\boldsymbol{A}_{J}=\boldsymbol{b}_{I}\boldsymbol{a}_{J}=2\pi\delta_{IJ}$
\end_inset

.
 Hence 
\begin_inset Formula $\boldsymbol{B}_{I}\boldsymbol{a}_{K}M_{KJ}=2\pi\delta_{IJ}$
\end_inset

 or 
\begin_inset Formula $\boldsymbol{B}_{I}\boldsymbol{a}_{J}=2\pi\left(M^{-1}\right)_{IJ}$
\end_inset

 
\begin_inset Formula 
\begin{eqnarray*}
\boldsymbol{G}_{l}\boldsymbol{r}_{j} & = & \sum_{I}G_{l,I}\boldsymbol{B}_{I}\cdot\sum_{J}r_{j,J}\boldsymbol{a}_{J}\\
 & = & \sum_{IJ}G_{l,I}r_{j,J}\boldsymbol{B}_{I}\cdot\boldsymbol{a}_{J}\\
 & = & 2\pi\sum_{IJ}G_{l,I}r_{j,J}\left(M^{-1}\right)_{IJ}
\end{eqnarray*}

\end_inset

The inverse of an integer matrix 
\begin_inset Formula $M$
\end_inset

 is a matrix of rational numbers.
 The coefficients 
\begin_inset Formula $G_{l,I}r_{j,J}$
\end_inset

 in the equation above are integer in such a way that if 
\begin_inset Formula $\boldsymbol{G}_{l}$
\end_inset

 were a reciprocal lattice vector 
\begin_inset Formula $\boldsymbol{g}$
\end_inset

 of the NC 
\begin_inset Formula $G_{l,I}\left(M^{-1}\right)_{IJ}\in\mathbb{N}$
\end_inset

 must be an integer because 
\begin_inset Formula $\boldsymbol{g}\boldsymbol{r}=2\pi h$
\end_inset

, 
\begin_inset Formula $h\in\mathbb{N}$
\end_inset

 always holds for dual lattices.
 Hence for the non trivial 
\begin_inset Formula $\boldsymbol{G}_{l}$
\end_inset

, which actually backfold the original bands onto new bands 
\begin_inset Formula $G_{l,I}\left(M^{-1}\right)_{IJ}$
\end_inset

 must be rationals.
 On the other hand 
\begin_inset Formula $r_{j,J}$
\end_inset

 are also integer but chosen such that they do not represent SC lattice
 vectors, since otherwise 
\begin_inset Formula $r_{j,J}\left(M^{-1}\right)_{IJ}$
\end_inset

 must be integer for 
\begin_inset Formula $\boldsymbol{G}\boldsymbol{R}=2\pi H$
\end_inset

, 
\begin_inset Formula $H\in\mathbb{N}$
\end_inset

.
 Altogether, when thinking hard one realizes that 
\begin_inset Formula $\boldsymbol{G}_{l}\boldsymbol{r}_{j}$
\end_inset

 runs over 
\begin_inset Formula $f$
\end_inset

 fractionals such that the resulting 
\begin_inset Formula $f$
\end_inset

 complex numbers 
\begin_inset Formula $\mathrm{e}^{i\boldsymbol{G}_{l}\boldsymbol{r}_{j}}$
\end_inset

, 
\begin_inset Formula $j\in\left[1,f\right]$
\end_inset

 are equally spaced on the unit circle including the number 
\begin_inset Formula $1$
\end_inset

 (for 
\begin_inset Formula $\boldsymbol{r}_{j}=0$
\end_inset

).
 But then a general sum of unity theorem tells us 
\begin_inset Formula $\sum_{j}\mathrm{e}^{i\boldsymbol{G}_{l}\boldsymbol{r}_{j}}=f_{s}\delta_{\boldsymbol{G}_{l}\boldsymbol{g}}$
\end_inset

, where any NC reciprocal vector 
\begin_inset Formula $\boldsymbol{g}$
\end_inset

 will work.
 If we do partial backfolding, where 
\begin_inset Formula $f_{s}<f$
\end_inset

 the sum of unity is no longer correct, but one can argue that we get something
 with 
\begin_inset Formula $f_{s}$
\end_inset

 terms of order one and an approximate delta function.
 
\end_layout

\begin_layout Standard
Hence, the average becomes (using Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:c_k_plus_g"

\end_inset

))
\begin_inset Formula 
\begin{equation}
\boxed{\frac{1}{f_{s}}\sum_{j}C_{\left[SC\right]S_{js}\mu,n+lm}^{\boldsymbol{k}}=\frac{1}{\sqrt{f}}\delta_{\boldsymbol{G}_{l}\boldsymbol{g}}C_{\left[NC\right]s\mu,n}^{\boldsymbol{k}}U^{\boldsymbol{k}}}\label{eq:averaged_C}
\end{equation}

\end_inset

The unitary mixing only happens in degenerate subspaces and drops out of
 the final weight expressions if an average over the degenerate bands is
 taken.
\end_layout

\begin_layout Standard
In words: if the bands are from the non backfolded set, 
\begin_inset Formula $l=0$
\end_inset

, 
\begin_inset Formula $\boldsymbol{G}_{l}=\boldsymbol{g}$
\end_inset

 we get 
\begin_inset Formula $C_{\left[SC\right]}=\frac{1}{\sqrt{f}}C_{\left[NC\right]}$
\end_inset

.
 If it is a backfolded set 
\begin_inset Formula $l\neq0$
\end_inset

 and 
\begin_inset Formula $\boldsymbol{G}_{l}\neq\boldsymbol{g}$
\end_inset

 we get exactly(approximately) zero.
 Hence, the average defined above is shown to differentiate between backfolded
 and original bands.
 Of course one can continue this relation to 
\begin_inset Formula $\boldsymbol{k}$
\end_inset

-points outside of the first SC reciprocal unit cell and hence recover the
 full band structure of the larger NC unit cell.
\end_layout

\begin_layout Standard
What is left is the normalization condition.
 We will discuss net weights in order to stay consistent with the FPLO scheme
 of things.
 The sum over all standard SC weights must equal one.
 Using Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:C_SC_maped_to_NC"

\end_inset

) we can write
\begin_inset Formula 
\begin{eqnarray*}
1=\sum_{S\mu}w_{S\mu}^{k,n+lm} & = & \frac{1}{\Omega}\sum_{S\mu}\left\langle \left|C_{\left[SC\right]S\mu,n+lm}^{\boldsymbol{k}}\right|^{2}\right\rangle _{\mathrm{deg}\;n}\\
 & = & \frac{1}{\Omega f}\sum_{S\mu}\left\langle \left|C_{\left[NC\right]s\mu,n}^{\boldsymbol{k}+\boldsymbol{G}_{l}}\right|^{2}\right\rangle _{\mathrm{deg}\;n}\\
 & = & \frac{1}{\Omega f}\sum_{js\mu}\left\langle \left|C_{\left[NC\right]s\mu,n}^{\boldsymbol{k}+\boldsymbol{G}_{l}}\right|^{2}\right\rangle _{\mathrm{deg}\;n}\\
 & = & \frac{1}{f\Omega}\sum_{s\mu}f_{s}\left\langle \left|C_{\left[NC\right]s\mu,n}^{\boldsymbol{k}+\boldsymbol{G}_{l}}\right|^{2}\right\rangle _{\mathrm{deg}\;n}
\end{eqnarray*}

\end_inset

The proposed unfolded weights using the 
\begin_inset Formula $r_{j}$
\end_inset

 averages read (with an undetermined factor 
\begin_inset Formula $X_{s}$
\end_inset

) 
\begin_inset Formula 
\begin{eqnarray*}
w_{s\mu}^{\boldsymbol{k}n+lm} & = & X_{s}\left\langle \left|\frac{1}{f_{s}}\sum_{j}C_{\left[SC\right]S_{js}\mu,n+lm}^{\boldsymbol{k}}\right|^{2}\right\rangle _{\mathrm{deg}\;n}\\
 & = & X_{s}\frac{1}{f}\left\langle \left|C_{\left[NC\right]s\mu,n}^{\boldsymbol{k}}\right|^{2}\right\rangle _{\mathrm{deg}\;n}\delta_{\boldsymbol{G}_{l}\boldsymbol{g}}
\end{eqnarray*}

\end_inset

The normalization sum reads
\begin_inset Formula 
\[
1=\sum_{s\mu}w_{s\mu}^{\boldsymbol{k}n+lm}=\frac{1}{f}\sum_{s\mu}X_{s}\left\langle \left|C_{\left[NC\right]s\mu,n}^{\boldsymbol{k}}\right|^{2}\right\rangle _{\mathrm{deg}\;n}\delta_{\boldsymbol{G}_{l}\boldsymbol{g}}
\]

\end_inset

Comparing to the sum of SC net weights above assuming that we have an original
 band (
\begin_inset Formula $\delta=1$
\end_inset

) we get 
\begin_inset Formula $X_{s}=\frac{f_{s}}{\Omega}$
\end_inset

 and the final normalized unfolded net weight result is
\begin_inset Formula 
\begin{equation}
\boxed{w_{s\mu}^{\boldsymbol{k}n}=\frac{\frac{1}{f_{s}}\left\langle \left|\sum_{j}C_{\left[SC\right]S_{js}\mu,n}^{\boldsymbol{k}}\right|^{2}\right\rangle _{\mathrm{deg}\;n}}{\sum_{S\mu}\left\langle \left|C_{\left[SC\right]S\mu,n}^{\boldsymbol{k}}\right|^{2}\right\rangle _{\mathrm{deg}\;n}}}\label{eq:unfoldednetweights}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Finally, after having discussed everything for the exact supercell, we postulate
 that the same procedure is also applicable for slightly distorted supercells,
 which gives the unfolding procedure.
\end_layout

\begin_layout Subsection

\series bold
Summary
\series default
: 
\end_layout

\begin_layout Standard
Band unfolding is a fat-band method.
 By projecting the Bloch spectral density operator onto Bloch sums of the
 NC periodicity, the resulting weights will be strongest for the bands,
 which belong to the original NC band structure.
 The other bands of the SC, which are obtained by backfolding NC bands,
 will have smaller or zero weight, depending on the amount of perturbation,
 which differentiates the SC from a perfect duplication of NC cells.
 The philosophy of unfolding contradicts atom substitution.
 However, atom substituion can be handled too at least formally.
\end_layout

\begin_layout Section
Perturbations
\begin_inset CommandInset label
LatexCommand label
name "sec:Perturbations"

\end_inset


\end_layout

\begin_layout Standard
There are two kinds of perturbations, moving atoms and replacing atoms.
 Moving atoms means that the overlap matrix between the NC and the SC orbitals/W
Fs gets approximated in this unfolding technique and that the phase factors
 of the NC Bloch states are only correct on average, which usually is a
 small thing, if atoms are moved only slighty.
 However, one should keep in mind that the projections used assume perfect
 matching (which only exists in perfect multiples of NC).
 Moving physically unimportant atoms (no contribution to the energy window
 under consideration, e.g.
 Fermi level) is off course a potential-only perturbation for the important
 atoms (which contribute to the energy window) and hence unfolding for the
 important atoms makes sense.
\end_layout

\begin_layout Standard
Replacing atoms comes in two modes.
 Replacing unimportant atoms (charge donors, buffer atoms) really just changes
 the potential of the important (other) atoms and the unfolding for the
 important atoms is rather meaningful.
 Example: replacing cations in the pnictides while leaving the FeAs planes
 intact, gives meaningfull unfolding for the Fe bands around the Fermi level.
 Replacing the important atoms gives problems for the following reason.
 Assume the NC and SC as shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Example-for-replacing"

\end_inset

a.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement H
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pic/ncsc_replace.pdf
	width 40col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Example-for-replacing"

\end_inset

Example for replacing atoms
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
The sites 
\begin_inset Formula $s=1$
\end_inset

 gets unfolded by equating sites 
\begin_inset Formula $S=1,3$
\end_inset

.
 This will lead to nicely unfolded bands.
 The unfolding of sites 
\begin_inset Formula $S=2,4$
\end_inset

 cannot be done in the technique used here, since the atoms and hence the
 basis of these atoms are different and there is no way of forming approximate
 Bloch sums of sites 
\begin_inset Formula $S=2$
\end_inset

 and 
\begin_inset Formula $S=4$
\end_inset

.
 Instead one has to request unfolding with site list 
\begin_inset Formula $S=2$
\end_inset

 and 
\begin_inset Formula $S=4$
\end_inset

 separately.
 This just means that the unfolded band weigths of these two sites are identical
 to the not-unfolded weights and hence show the SC periodicity (backfolding
 in the BZ).
 This makes some sense, since there really is no approximate NC symmetry
 for these atoms.
 The total unfolded weight (sum over all orbitals) for unfolded 
\begin_inset Formula $S=2$
\end_inset

 is half as big as the total unfolded weight of 
\begin_inset Formula $S=1,3$
\end_inset

 due to the site count.
 In that sense the unfolding is still visible to a certain extend.
 In Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Example-for-replacing"

\end_inset

b the situation is a bit better.
 Now, unfolded site weights can be defined from 
\begin_inset Formula $S=1,2,3$
\end_inset

 and 
\begin_inset Formula $S=4$
\end_inset

.
 The weights will now play out better, since only one out of four sites,
 which form a full NC Bloch sum is missing.
 So we can plot the 
\begin_inset Formula $S=1,2,3$
\end_inset

 unfolded weights and the unfolded=not-unfolded 
\begin_inset Formula $S=4$
\end_inset

 bands.
\end_layout

\begin_layout Section
Brillouine zones
\end_layout

\begin_layout Standard
The NC bz is larger than the SC BZ.
 Example from Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Example-for-replacing"

\end_inset

a: the bz is shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:bz-and-BZ-replacing"

\end_inset

.
 
\begin_inset Float figure
placement H
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pic/bzBZ_replace.pdf
	width 50col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:bz-and-BZ-replacing"

\end_inset

bz and BZ of Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Example-for-replacing"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 In the NC we have the high symmetry points
\begin_inset Formula 
\begin{eqnarray*}
x & = & \left(\frac{\pi}{a_{\left[NC\right]}},0,0\right)=\frac{2\pi}{a_{\left[NC\right]}}\left(\frac{1}{2},0,0\right)\\
m & = & \left(\frac{\pi}{a_{\left[NC\right]}},\frac{\pi}{a_{\left[NC\right]}},0\right)=\frac{2\pi}{a_{\left[NC\right]}}\left(\frac{1}{2},\frac{1}{2},0\right)\\
y & = & \left(0,\frac{\pi}{a_{\left[NC\right]}},0\right)==\frac{2\pi}{a_{\left[NC\right]}}\left(0,\frac{1}{2},0\right)
\end{eqnarray*}

\end_inset

In the SC the equivalent points are at the same cartesian coordinates,but
 using 
\begin_inset Formula $a_{\left[SC\right]}=2a_{\left[NC\right]}$
\end_inset

 we get
\begin_inset Formula 
\begin{eqnarray*}
X & = & \left(\frac{2\pi}{a_{\left[SC\right]}},0,0\right)=\frac{2\pi}{a_{\left[SC\right]}}\left(1,0,0\right)\\
M & = & \left(\frac{2\pi}{a_{\left[SC\right]}},\frac{2\pi}{a_{\left[SC\right]}},0\right)=\frac{2\pi}{a_{\left[SC\right]}}\left(1,1,0\right)\\
y & = & \left(0,\frac{2\pi}{a_{\left[SC\right]}},0\right)==\frac{2\pi}{a_{\left[SC\right]}}\left(0,1,0\right)
\end{eqnarray*}

\end_inset

Although, only the 
\begin_inset Formula $x$
\end_inset

-direction changes in the new BZ, all fedit coordinates change.
 This is because the cartesian coordinates are the same in NC and SC, but
 the unit 
\begin_inset Formula $\frac{2\pi}{a_{\left[SC\right]}}$
\end_inset

 scales for all directions.
 
\end_layout

\begin_layout Section
Caveats
\end_layout

\begin_layout Itemize
The unfolded weights might look much smaller than the weights in the NC
 along some directions.
 This can happen, when the bands are degenerate in the NC and is due to
 a splitting in the SC combined with the way of counting.
 So, a width 1 band becomes, let's say, 2 width 0.5 bands shifted slightly
 against each other.
 Now, they appear to be half as broad, when the splitting is small and hence
 the band are more or less plotted on top of each other.
\end_layout

\begin_layout Itemize
Take care, when determining the NC bz in the SC setup.
\end_layout

\begin_layout Itemize
Unfolded weights do not show the full information, because the Bloch spectral
 density is not diagonal in the approximate NC Bloch states.
\end_layout

\begin_layout Standard
\begin_inset Note Note
status open

\begin_layout Plain Layout
To have normalization even without specifying all sites, the gross projection
 is used for unfolding.
 Normal FPLO fatbands are net-projected.
 The difference is the larger the higher the energy and manifests itself
 mostly for unoccupied bands.
 
\end_layout

\end_inset


\end_layout

\begin_layout Section
User input/output
\end_layout

\begin_layout Standard
The user defines unfolding by creating the file 
\family typewriter
=.unfold
\family default
.
 If the file is detected the fatbands (
\family typewriter
+bweights_unfolded/ +bweights_kp_unfolded
\family default
) are created according to formula 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:unfoldednetweights"

\end_inset

.
 
\end_layout

\begin_layout Subsection
=.unfold
\end_layout

\begin_layout Standard
The file looks like this
\end_layout

\begin_layout Standard
\begin_inset listings
lstparams "basicstyle={\ttfamily}"
inline false
status open

\begin_layout Plain Layout

# NCsite SCsites
\end_layout

\begin_layout Plain Layout

 1        1 2 3
\end_layout

\begin_layout Plain Layout

# some comment
\end_layout

\begin_layout Plain Layout

 2        4
\end_layout

\end_inset


\end_layout

\begin_layout Standard
There can be any number of comment lines starting with '#', or lines only
 containing whitespace.
 They are ignored.
\end_layout

\begin_layout Standard
Each line, which is not a comment line and not whitespace-only is an unfolding
 defintion.
\end_layout

\begin_layout Standard
An unfolding definition contains 
\end_layout

\begin_layout Description
the
\begin_inset space ~
\end_inset

NC
\begin_inset space ~
\end_inset

site
\begin_inset space ~
\end_inset

number: which has no other use than labeling the contracted atoms in the
 label information in 
\family typewriter
+bweights_unfold
\family default
.
 Maybe it's best to use the actual site number in the NC.
 
\end_layout

\begin_layout Description
the
\begin_inset space ~
\end_inset

SC
\begin_inset space ~
\end_inset

site
\begin_inset space ~
\end_inset

list: which is the list of sites in the SC, which get contracted onto the
 NC site.
\end_layout

\begin_layout Standard
The number of SC sites, which are contracted to this NC site (in the formulas
 above it is 
\begin_inset Formula $f_{s}$
\end_inset

) is determined from the input.
 Usually it equals the number of times the NC fits into the SC 
\begin_inset Formula $f_{s}=f$
\end_inset

.
 In cases where atom substitutions took place 
\begin_inset Formula $f_{s}<f$
\end_inset

 as in the example file above: 4 NC cells form one SC, hence 
\begin_inset Formula $f=4$
\end_inset

.
 Atom 
\begin_inset Formula $4$
\end_inset

 got substituted and has a different atom than sites 
\begin_inset Formula $1\ldots3$
\end_inset

.
 So, we contract/unfold sites 
\begin_inset Formula $1\ldots3$
\end_inset

 and leave site 
\begin_inset Formula $4$
\end_inset

 by itself.
 Note, that one can contract different atoms, as long as they have the same
 basis, i.e.
 in most VCA cases and when substituting similar elements (e.g.
 Fe with Ni).
\end_layout

\begin_layout Subsection
choosing k-points
\end_layout

\begin_layout Standard
Beware of choosing the correct 
\begin_inset Formula $k$
\end_inset

-points.
 The unfolded fatbands are calcuated in the SC BZ.
 So, first you determine the cartesian representation of the special points
 in the NC bz.
 Then you find the relation between bz and BZ and finally transform the
 bz points int BZ points and put them into the fedit menu in units of 
\begin_inset Formula $\frac{2\pi}{a_{SC}}$
\end_inset

.
 That means that a k-point in cartesian coordinates 
\begin_inset Formula $\vec{k}$
\end_inset

 is entered as 
\begin_inset Formula $\tilde{k}$
\end_inset

 in fedit with 
\begin_inset Formula $\vec{k}=\frac{2\pi}{a}\tilde{k}$
\end_inset

, where 
\begin_inset Formula $a$
\end_inset

 is the first lattice constant.
\end_layout

\begin_layout Subsection
Fermi surfaces
\end_layout

\begin_layout Standard
Fermi surfaces can be created with xfsf.
 Xfsf uses the symmetry to create an irreducible mesh and hence to save
 calculational time.
 The NC bz is larger than the SC BZ.
 Therefore, it is impossible to create enough NC k-points by using the SC
 BZ.
 There is a menu input->handmade symmetry, which allows the user to enter
 the NC cell and symmetry operations (only some generators are needed).
 This overwrites the default mesh symmetry.
 After this the process is straight forward, except for the file suffix,
 of the unfolded fatbands, which requires to set a non-default filename
 in input->files.
 Note that the new convention is to name the band file 
\family typewriter
+band_kp
\family default
 for the Fermi surface (in general if 
\family typewriter
=.kp
\family default
 was used), in order to not overwrite 
\family typewriter
+band
\family default
.
 So, the unfolded file is 
\family typewriter
+bweights_kp_unfold
\family default
.
 
\end_layout

\begin_layout Standard
The bandweights can be used for coloring: plot->coloring: extern, state.
 The extern checkbox switches from Fermi velocity coloring to extern coloring
 from the file given in input->files.
 State selects the data column from the file.
 (Transparent coloring as in Ref.
 
\begin_inset CommandInset citation
LatexCommand cite
key "EvHeumen2011"
literal "true"

\end_inset

 is not officially available, since it was a hard coded hack.)
\end_layout

\begin_layout Subsection
Examples
\end_layout

\begin_layout Standard
There are some examples in the FPLO
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
fullversion{}
\end_layout

\end_inset

/DOC/Unfolding_example directory.
 Use 
\end_layout

\begin_layout LyX-Code
fplo
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fullversion{}
\end_layout

\end_inset

-x86_64
\end_layout

\begin_layout Standard
to calculated +bweights....
 Then use 
\end_layout

\begin_layout LyX-Code
faddwei
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fullversion{}
\end_layout

\end_inset

-x86_64
\end_layout

\begin_layout Standard
and if the file =.addwei_unfold exist (supercell sub directories)
\end_layout

\begin_layout LyX-Code
faddwei
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fullversion{}
\end_layout

\end_inset

-x86_64 -s =.addwei_unfold
\end_layout

\begin_layout Standard
to extract +bwsums...
 and finally 
\end_layout

\begin_layout LyX-Code
xfbp bw.cmd
\end_layout

\begin_layout Standard
to show the results.
\end_layout

\begin_layout Standard
\begin_inset Note Note
status collapsed

\begin_layout Section
TODO: think about normalization if incomplete atoms
\end_layout

\begin_layout Plain Layout
The Taylor expansion of a function reads
\begin_inset Formula 
\begin{eqnarray*}
f\left(\boldsymbol{r}-\boldsymbol{P}\right) & = & \mathrm{e}^{-\boldsymbol{\nabla}\boldsymbol{P}}f\left(\boldsymbol{r}\right)\\
 & = & \mathrm{e}^{-i\frac{1}{i}\boldsymbol{\nabla}\boldsymbol{P}}f\left(\boldsymbol{r}\right)\\
 & = & \mathrm{e}^{-i\hat{p}\boldsymbol{P}}f\left(r\right)
\end{eqnarray*}

\end_inset

(
\begin_inset Formula $\hat{\boldsymbol{p}}=\frac{1}{i}\boldsymbol{\nabla}$
\end_inset

) which defines the translation operator
\begin_inset Formula 
\[
T_{\boldsymbol{R}}=\mathrm{e}^{-i\hat{\boldsymbol{p}}\boldsymbol{R}}
\]

\end_inset

which acts as
\begin_inset Formula 
\begin{eqnarray*}
T_{\boldsymbol{R}}\Psi_{\boldsymbol{k}}\left(\boldsymbol{r}\right) & = & \left\langle \boldsymbol{r}\mid T_{\boldsymbol{R}}\mid\Psi_{\boldsymbol{k}}\right\rangle \\
 & = & \left\langle T_{\boldsymbol{R}}^{+}\boldsymbol{r}\mid\Psi_{\boldsymbol{k}}\right\rangle \\
 & = & \left\langle T_{-\boldsymbol{R}}\boldsymbol{r}\mid\Psi_{\boldsymbol{k}}\right\rangle \\
 & = & \left\langle \boldsymbol{r}-\boldsymbol{R}\mid\Psi_{\boldsymbol{k}}\right\rangle \\
 & = & \Psi_{\boldsymbol{k}}\left(\boldsymbol{r}-\boldsymbol{R}\right)
\end{eqnarray*}

\end_inset

The one dimensional irreducible representations are defined via
\begin_inset Formula 
\[
T_{\boldsymbol{R}}\Psi_{\boldsymbol{k}}=\Psi_{\boldsymbol{k}}\mathrm{e}^{-i\boldsymbol{k}\boldsymbol{R}}
\]

\end_inset

which is just the Bloch theorem.
\end_layout

\begin_layout Plain Layout
Projection on irreducible representations via the projection operator 
\begin_inset Formula $P_{\mu\nu}^{j}=\frac{l_{j}}{g}\sum_{g}\left[D^{j}\right]_{\nu\mu}^{-1}\left(g\right)\hat{g}$
\end_inset

, which reads in this case
\begin_inset Formula 
\[
P_{\boldsymbol{k}}=\frac{1}{N}\sum_{R}\mathrm{e}^{i\boldsymbol{k}\boldsymbol{R}}\hat{T}_{\boldsymbol{R}}
\]

\end_inset

Example: take a local orbital 
\begin_inset Formula $\Phi_{s}\left(\boldsymbol{r}-\boldsymbol{s}\right)$
\end_inset


\begin_inset Formula 
\begin{eqnarray*}
P_{\boldsymbol{k}}\Phi_{s}\left(\boldsymbol{r}-\boldsymbol{s}\right) & = & \frac{1}{N}\sum_{R}\mathrm{e}^{i\boldsymbol{k}\boldsymbol{R}}\Phi_{s}\left(\boldsymbol{r}-\boldsymbol{R}-\boldsymbol{s}\right)\\
 & = & \frac{1}{N}\sum_{R}\mathrm{e}^{i\boldsymbol{k}\boldsymbol{R}}\Phi_{Rs}
\end{eqnarray*}

\end_inset

which is a Bloch sum of local orbitals up to a factor 
\begin_inset Formula $\frac{1}{\sqrt{N}}$
\end_inset

 and a phase factor 
\begin_inset Formula $\mathrm{e}^{i\boldsymbol{k}\boldsymbol{s}}$
\end_inset

.
\end_layout

\begin_layout Plain Layout
Now, we can project a SC Bloch sum onto NC Bloch symmetry
\begin_inset Formula 
\begin{eqnarray*}
P_{\boldsymbol{q}}^{\left[NC\right]}\Phi_{\boldsymbol{k}S}^{\left[SC\right]} & = & \frac{1}{N_{r}}\sum_{r}\mathrm{e}^{i\boldsymbol{q}\boldsymbol{r}}\hat{T}_{\boldsymbol{r}}\frac{1}{\sqrt{N_{R}}}\sum_{R}\Phi_{RS}\mathrm{e}^{i\boldsymbol{k}\boldsymbol{R}\boldsymbol{S}}\\
 & = & \frac{1}{N_{r}}\frac{\sqrt{f}}{\sqrt{N_{r}}}\sum_{r}\sum_{R}\mathrm{e}^{i\boldsymbol{q}\boldsymbol{r}}\Phi_{RSr}\mathrm{e}^{i\boldsymbol{k}\boldsymbol{R}\boldsymbol{S}}\\
 & = & \frac{1}{N_{r}}\delta_{\boldsymbol{q},\boldsymbol{k}+\boldsymbol{G}}\sum_{Rj}\mathrm{e}^{i\boldsymbol{q}\boldsymbol{r}_{j}}\Phi_{Rr_{j}S}\mathrm{e}^{i\boldsymbol{k}\boldsymbol{R}\boldsymbol{S}}\\
 & = & \frac{1}{N_{r}}\delta_{\boldsymbol{q},\boldsymbol{k}+\boldsymbol{G}}\sum_{Rj}\mathrm{e}^{i\boldsymbol{G}\boldsymbol{r}_{j}}\Phi_{Rr_{j}S}\mathrm{e}^{i\boldsymbol{k}\boldsymbol{R}\boldsymbol{r}_{j}\boldsymbol{S}}
\end{eqnarray*}

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ifx
\backslash
mystandalone
\backslash
undefined
\end_layout

\begin_layout Plain Layout


\backslash
else
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "ref"
options "plain"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fi
\end_layout

\end_inset


\end_layout

\end_body
\end_document
