From 737b02e6dabf01633088805a8b3ada02d0629be0 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= <marek@necada.org>
Date: Thu, 1 Aug 2019 10:48:29 +0300
Subject: [PATCH] Symmetries for infinite systems, sloppy version

Former-commit-id: 2bda4d6e361b274be9bf20beb1ae8a2e4e2c63ce
---
 lepaper/arrayscat.lyx  |   5 ++
 lepaper/symmetries.lyx | 165 +++++++++++++++++++++++++++++++++++++++++
 2 files changed, 170 insertions(+)

diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx
index 6ec1cc5..cc4442c 100644
--- a/lepaper/arrayscat.lyx
+++ b/lepaper/arrayscat.lyx
@@ -758,6 +758,11 @@ The text about symmetries is pretty dense.
  Make it more explanatory and human-readable.
 \end_layout
 
+\begin_layout Itemize
+Check whether everything written is correct also for non-symmetric space
+ groups.
+\end_layout
+
 \begin_layout Standard
 \begin_inset CommandInset include
 LatexCommand include
diff --git a/lepaper/symmetries.lyx b/lepaper/symmetries.lyx
index 935eb86..9016d58 100644
--- a/lepaper/symmetries.lyx
+++ b/lepaper/symmetries.lyx
@@ -1042,6 +1042,171 @@ Kvantifikovat!
 Periodic systems
 \end_layout
 
+\begin_layout Standard
+For periodic systems, we can in similar manner also block-diagonalise the
+ 
+\begin_inset Formula $M\left(\omega,\vect k\right)=\left(I-W\left(\omega,\vect k\right)T\left(\omega\right)\right)$
+\end_inset
+
+ from the left hand side of eqs.
+ 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "eq:Multiple-scattering problem unit cell block form"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "eq:lattice mode equation"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+ Hovewer, in this case, 
+\begin_inset Formula $W\left(\omega,\vect k\right)$
+\end_inset
+
+ is in general not invariant under the whole point group symmetry subgroup
+ of the system geometry due to the 
+\begin_inset Formula $\vect k$
+\end_inset
+
+ dependence.
+ In other words, only those point symmetries that the 
+\begin_inset Formula $e^{i\vect k\cdot\vect r}$
+\end_inset
+
+ modulation does not break are preserved, and no preservation of point symmetrie
+s happens unless 
+\begin_inset Formula $\vect k$
+\end_inset
+
+ lies somewhere in the high-symmetry parts of the Brillouin zone.
+ However, the high-symmetry points are usually the ones of the highest physical
+ interest, for it is where the band edges 
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+or 
+\begin_inset Quotes eld
+\end_inset
+
+dirac points
+\begin_inset Quotes erd
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+ are typically located.
+\end_layout
+
+\begin_layout Standard
+The transformation to the symmetry adapted basis 
+\begin_inset Formula $U$
+\end_inset
+
+ is constructed in a similar way as in the finite case, but because we do
+ not work with all the (infinite number of) scatterers but only with one
+ unit cell, additional phase factors 
+\begin_inset Formula $e^{i\vect k\cdot\vect r_{p}}$
+\end_inset
+
+ appear in the per-unit-cell group action 
+\begin_inset Formula $J(g)$
+\end_inset
+
+.
+ This is illustrated in Fig.
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "Phase factor illustration"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "Phase factor illustration"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+More rigorous analysis can be found e.g.
+ in 
+\lang english
+
+\begin_inset CommandInset citation
+LatexCommand cite
+after "chapters 10–11"
+key "dresselhaus_group_2008"
+literal "true"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+In the group-theoretical terminology, blablabla little groups blabla bla...
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 
 \lang english