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\pdf_title "Sähköpajan päiväkirja"
\pdf_author "Marek Nečada"
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\begin_body

\begin_layout Title
Symmetry-adapted basis functions for honeycomb lattice at 
\begin_inset Formula $K$
\end_inset

-point
\end_layout

\begin_layout Section
Generation theorem
\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $\mathbf{G}$
\end_inset

 be a group and 
\begin_inset Formula $\Gamma^{i}\left\{ R\to\mathbf{D}^{i}\left(R\right)\right\} $
\end_inset

 some 
\begin_inset Formula $d_{i}$
\end_inset

-dimensional rep of 
\begin_inset Formula $\mathbf{G}$
\end_inset

.
 Let the group ring (corresponding to the given rep indexed by 
\begin_inset Formula $i$
\end_inset

) elements be defined as [Bradley&Cracknell (2.2.2)]
\begin_inset Formula 
\[
W_{ts}^{i}=\frac{d_{i}}{\left|\mathbf{G}\right|}\sum_{R\in\mathbf{G}}\mathbf{D}^{i}\left(R\right)_{ts}^{*}R.
\]

\end_inset


\end_layout

\begin_layout Standard
From [Bradley&Cracknell, theorem 2.2.1]:
\end_layout

\begin_layout Standard
If 
\begin_inset Formula $\phi$
\end_inset

 is an arbitrary function of 
\begin_inset Formula $V$
\end_inset

 (a linear space in which the realisation of the group operation act) such
 that 
\begin_inset Formula $W_{ss}^{i}\phi\ne0$
\end_inset

 (
\begin_inset Formula $s$
\end_inset

 is fixed and is a number in the range 1 to 
\begin_inset Formula $d_{i}$
\end_inset

; 
\begin_inset Formula $i$
\end_inset

 is idx of the rep) then the funs 
\begin_inset Formula $W_{ts}^{i}\phi=\phi_{t}^{i}$
\end_inset

, 
\begin_inset Formula $t=1$
\end_inset

 to 
\begin_inset Formula $d_{i}$
\end_inset

, form a basis for the rep 
\begin_inset Formula $\Gamma^{i}$
\end_inset

.
\end_layout

\begin_layout Section
Particle-centered transformations
\end_layout

\begin_layout Standard
Now let's see what are the point group actions on SVWF in the origin [Schulz]:
\end_layout

\begin_layout Standard
\begin_inset Tabular
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<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
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\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $Z$
\end_inset

-axis rotation by 
\begin_inset Formula $2\pi/N$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $C_{N}M_{l}^{m}=e^{\pm?i2\pi m/N}M_{l}^{m}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $C_{N}N_{l}^{m}=e^{\pm?i2\pi m/N}N_{l}^{m}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
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\begin_inset Text

\begin_layout Plain Layout
Horizontal (xy) reflection
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\sigma_{h}M_{l}^{m}=\left(-1\right)^{m+l+1}M_{l}^{m}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\sigma_{h}N_{l}^{m}=\left(-1\right)^{m+l}N_{l}^{m}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
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\begin_inset Text

\begin_layout Plain Layout
Vertical (yz) reflection
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\sigma_{yz}M_{l}^{m}=-M_{l}^{-m}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\sigma_{yz}N_{l}^{m}=N_{l}^{-m}$
\end_inset


\end_layout

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</cell>
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\begin_inset Text

\begin_layout Plain Layout
Vertical (xz) reflection
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\sigma_{xz}M_{l}^{m}=\left(-1\right)^{m+1}M_{l}^{-m}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\sigma_{xz}N_{l}^{m}=\left(-1\right)^{m}N_{l}^{-m}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Section
Transformations in a lattice
\end_layout

\begin_layout Standard

\end_layout

\end_body
\end_document