Symmetries text in progress.

Former-commit-id: ecf71fe4af190cb6a3c1ed2c876183d01fbe6448

lepaper/arrayscat.lyx

@@ -723,7 +723,11 @@ Truncation notation.
 \end_layout
 
 \begin_layout Itemize
-Example results.
+Example results!
+\end_layout
+
+\begin_layout Itemize
+Figures.
 \end_layout
 
 \begin_layout Itemize
@@ -738,7 +742,7 @@ Fix and unify notation (mainly indices) in infinite lattices section.
 Carefully check the transformation directions in sec.
  
 \begin_inset CommandInset ref
-LatexCommand eqref
+LatexCommand ref
 reference "sec:Symmetries"
 plural "false"
 caps "false"
@@ -749,6 +753,11 @@ noprefix "false"
 
 \end_layout
 
+\begin_layout Itemize
+The text about symmetries is pretty dense.
+ Make it more explanatory and human-readable.
+\end_layout
+
 \begin_layout Standard
 \begin_inset CommandInset include
 LatexCommand include

lepaper/symmetries.lyx

@@ -180,7 +180,7 @@ noprefix "false"
 \end_layout
 
 \begin_layout Subsection
-Finite systems
+Excitation coefficients under point group operations
 \end_layout
 
 \begin_layout Standard
@@ -397,7 +397,7 @@ noprefix "false"
  of the electric field around origin in a rotated/reflected system,
 \begin_inset Formula 
 \[
-\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}lm\left(k\vect r\right)+D_{m,m'}^{\tau l}\left(g\right)\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\left(k\vect r\right)\right),
+\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\vect r\right)+\outcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\vect r\right)\right),
 \]
 
 \end_inset
@@ -491,8 +491,297 @@ literal "false"
 \end_layout
 
 \begin_layout Standard
-With these point group transformation properties in hand, we can proceed
+If the field expansion is done around a point 
- to rotating (or mirror-reflecting) the whole many-particle system.
+\begin_inset Formula $\vect r_{p}$
+\end_inset
+
+ different from the global origin, as in 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:E field expansion multiparticle"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, we have
+\lang english
+
+\begin_inset Formula 
+\begin{align}
+\vect E\left(\omega,R_{g}\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)+\outcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right).\label{eq:rotated E field expansion around outside origin}
+\end{align}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Scatterer orbits under 
+\begin_inset Formula $D_{2}$
+\end_inset
+
+ symmetry.
+ Particles 
+\begin_inset Formula $A,B,C,D$
+\end_inset
+
+ lie outside of origin or any mirror planes, and together constitute an
+ orbit of the size equal to the order of the group, 
+\begin_inset Formula $\left|D_{2}\right|=4$
+\end_inset
+
+.
+ Particles 
+\begin_inset Formula $E,F$
+\end_inset
+
+ lie on the 
+\begin_inset Formula $xz$
+\end_inset
+
+ plane, hence the corresponding reflection maps each of them to itself,
+ but the 
+\begin_inset Formula $yz$
+\end_inset
+
+ reflection (or the 
+\begin_inset Formula $\pi$
+\end_inset
+
+ rotation around the 
+\begin_inset Formula $z$
+\end_inset
+
+ axis) maps them to each other, forming a particle orbit of size 2
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+=???
+\end_layout
+
+\end_inset
+
+.
+ The particle 
+\begin_inset Formula $O$
+\end_inset
+
+ in the very origin is always mapped to itself, constituting its own orbit.
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:D2-symmetric structure particle orbits"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO restructure this
+\end_layout
+
+\end_inset
+
+With these transformation properties in hand, we can proceed to the effects
+ of point symmetries on the whole many-particle system.
+ Let us have a many-particle system symmetric with respect to a point group
+ 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ A symmetry operation 
+\begin_inset Formula $g\in G$
+\end_inset
+
+ determines a permutation of the particles: 
+\begin_inset Formula $p\mapsto\pi_{g}(p)$
+\end_inset
+
+, 
+\begin_inset Formula $p\in\mathcal{P}$
+\end_inset
+
+.
+ For a given particle 
+\begin_inset Formula $p$
+\end_inset
+
+, we will call the set of particles onto which any of the symmetries maps
+ the particle 
+\begin_inset Formula $p$
+\end_inset
+
+, i.e.
+ the set 
+\begin_inset Formula $\left\{ \pi_{g}\left(p\right);g\in G\right\} $
+\end_inset
+
+, as the 
+\emph on
+orbit
+\emph default
+ of particle 
+\begin_inset Formula $p$
+\end_inset
+
+.
+ The whole set 
+\begin_inset Formula $\mathcal{P}$
+\end_inset
+
+ can therefore be divided into the different particle orbits; an example
+ is in Fig.
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:D2-symmetric structure particle orbits"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+ The importance of the particle orbits stems from the following: in the
+ multiple-scattering problem, outside of the scatterers 
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+< FIXME
+\end_layout
+
+\end_inset
+
+ one has 
+\lang english
+
+\begin_inset Formula 
+\begin{align}
+\vect E\left(\omega,R_{g}\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{\pi_{g}(p)}\right)\right)+\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right)\label{eq:rotated E field expansion around outside origin-1}\\
+ & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)+\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right).
+\end{align}
+
+\end_inset
+
+This means that the field expansion coefficients 
+\begin_inset Formula $\rcoeffp p,\outcoeffp p$
+\end_inset
+
+ transform as 
+\begin_inset Formula 
+\begin{align}
+\rcoeffptlm p{\tau}lm & \mapsto\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right),\nonumber \\
+\outcoeffptlm p{\tau}lm & \mapsto\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right).\label{eq:excitation coefficient under symmetry operation}
+\end{align}
+
+\end_inset
+
+Obviously, the expansion coefficients belonging to particles in different
+ orbits do not mix together.
+ As before, we introduce a short-hand block-matrix notation for 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:excitation coefficient under symmetry operation"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ (TODO avoid notation clash here in a more consistent and readable way!)
+\end_layout
+
+\begin_layout Standard
+
+\lang english
+\begin_inset Formula 
+\begin{align}
+\rcoeff & \mapsto J\left(g\right)a,\nonumber \\
+\outcoeff & \mapsto J\left(g\right)\outcoeff.\label{eq:excitation coefficient under symmetry operation block form}
+\end{align}
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+
+\lang english
+The matrices 
+\begin_inset Formula $D\left(g\right)$
+\end_inset
+
+, 
+\begin_inset Formula $g\in G$
+\end_inset
+
+ will play a crucial role blablabla
+\end_layout
+
+\end_inset
+
+If the particle indices are ordered in a way that the particles belonging
+ to the same orbit are grouped together, 
+\begin_inset Formula $J\left(g\right)$
+\end_inset
+
+ will be a block-diagonal matrix, each block representing one particle orbit.
+\end_layout
+
+\begin_layout Subsection
+Irrep decomposition
+\end_layout
+
+\begin_layout Standard
+Knowledge of symmetry group actions 
+\begin_inset Formula $D\left(g\right)$
+\end_inset
+
+ on the field expansion coefficients give us the possibility to construct
+ a symmetry adapted basis in which we can block-diagonalise the multiple-scatter
+ing problem matrix 
+\begin_inset Formula $\left(I-TS\right)$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+
 \end_layout
 
 \begin_layout Subsection
@@ -501,6 +790,12 @@ Periodic systems
 
 \begin_layout Standard
 
+\lang english
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+
 \lang english
 A general overview of utilizing group theory to find lattice modes at high-symme
 try points of the Brillouin zone can be found e.g.
@@ -516,7 +811,7 @@ literal "true"
 ; here we use the same notation.
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \lang english
 We analyse the symmetries of the system in the same VSWF representation
@@ -671,7 +966,7 @@ where
 .
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \lang english
 Each mode at the 
@@ -717,5 +1012,10 @@ reference "smfig:dispersions"
 (a).
 \end_layout
 
+\end_inset
+
+
+\end_layout
+
 \end_body
 \end_document