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Improving the text on symmetries, WIP figs 

Former-commit-id: 49e6eabb7188f98308c1b6d1661dbd0b89a79a71
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Marek Nečada 2020-06-07 16:30:15 +03:00
parent 9bd876c273
commit d8d0efc1b3
3 changed files with 1067 additions and 167 deletions
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1055
lepaper/figs/hex/pokus.svg
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28
lepaper/finite.lyx
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@ -1541,6 +1541,34 @@ where
matrices as the off-diagonal blocks, whereas the diagonal blocks are set
to zeros.
\end_layout
\begin_layout Standard
We note that eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem block form"
plural "false"
caps "false"
noprefix "false"
\end_inset
with zero right-hand side describes the normal modes of the system; the
methods mentioned later in Section
\begin_inset CommandInset ref
LatexCommand eqref
reference "sec:Infinite"
plural "false"
caps "false"
noprefix "false"
\end_inset
for solving the band structure of a periodic system can be used as well
for finding the resonant frequencies of a finite system.
\end_layout
\begin_layout Standard

101
lepaper/symmetries.lyx
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@ -92,9 +92,8 @@ name "sec:Symmetries"
\begin_layout Standard
If the system has nontrivial point group symmetries, group theory gives
additional understanding of the system properties, and can be used to reduce
the computational costs.
additional understanding of the system properties, and can be used to substanti
ally reduce the computational costs.
\end_layout
\begin_layout Standard
@ -175,8 +174,21 @@ noprefix "false"
\end_inset
matrix, not the linear algebra.
However, the lattice modes can be searched for in each irrep separately,
and the irrep dimension gives a priori information about mode degeneracy.
However, decomposition of the lattice mode problem
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:lattice mode equation"
plural "false"
caps "false"
noprefix "false"
\end_inset
into the irreducible representations of the corresponding little co-groups
of the system's space group is nevertheless a useful tool in the mode analysis:
among other things, it enables separation of the lattice modes (which can
then be searched for each irrep separately), and the irrep dimension gives
a priori information about mode degeneracy.
\end_layout
\begin_layout Subsection
@ -193,8 +205,8 @@ TODO Zkontrolovat všechny vzorečky zde!!!
\end_inset
In order to use the point group symmetries, we first need to know how they
affect our basis functions, i.e.
In order to make use of the point group symmetries, we first need to know
how they affect our basis functions, i.e.
the VSWFs.
\end_layout
@ -1170,8 +1182,7 @@ Periodic systems
\end_layout
\begin_layout Standard
For periodic systems, we can in similar manner also block-diagonalise the
Also for periodic systems,
\begin_inset Formula $M\left(\omega,\vect k\right)=\left(I-T\left(\omega\right)W\left(\omega,\vect k\right)\right)$
\end_inset
@ -1196,7 +1207,7 @@ noprefix "false"
\end_inset
.
can be block-diagonalised in a similar manner.
Hovewer, in this case,
\begin_inset Formula $W\left(\omega,\vect k\right)$
\end_inset
@ -1218,25 +1229,29 @@ s happens unless
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
interest, for it is where the band edges are typically located.
The subsection does not aim for an exhaustive treatment of the topic of
space groups in physics (which can be found elsewhere
\begin_inset CommandInset citation
LatexCommand cite
key "dresselhaus_group_2008,bradley_mathematical_1972"
literal "false"
\end_inset
), here we rather demonstrate how the group action matrices are generated
on a specific example of a symmorphic space group.
\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
better formulation
\end_layout
\end_inset
are typically located.
\end_layout
\begin_layout Standard
@ -1468,6 +1483,36 @@ because in this case, the Bloch condition gives
\end_layout
\begin_layout Standard
Having the group action matrices, we can construct the projectors and decompose
the system into irreducible representations of the corresponding point
groups analogously to the finite case
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:SAB unitary transformation operator"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
This procedure can be repeated for any system with a symmorphic space group
symmetry, where the translation and point group operations are essentially
separable.
For systems with non-symmorphic space group symmetries (i.e.
those with glide reflection planes or screw rotation axes) a more refined
approach is required
\begin_inset CommandInset citation
LatexCommand cite
key "bradley_mathematical_1972,dresselhaus_group_2008"
literal "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float figure
placement document
@ -1537,20 +1582,6 @@ name "Phase factor illustration"
\end_layout
\begin_layout Standard
More rigorous analysis can be found e.g.
in
\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