1438 lines
29 KiB
Plaintext
1438 lines
29 KiB
Plaintext
#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\lyxformat 584
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\begin_document
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\begin_header
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\origin unavailable
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\textclass article
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\use_package amsmath 1
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\use_package amssymb 1
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\use_package cancel 1
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\use_package esint 1
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\use_package mathdots 1
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\use_package mathtools 1
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\use_package mhchem 1
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\use_package stackrel 1
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\use_package stmaryrd 1
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\use_package undertilde 1
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\cite_engine basic
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\index Index
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\shortcut idx
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\color #008000
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\end_index
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\secnumdepth 3
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\tocdepth 3
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\paragraph_separation indent
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\end_header
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\begin_body
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\begin_layout Section
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Symmetries
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:Symmetries"
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\end_inset
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\end_layout
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\begin_layout Standard
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If the system has nontrivial point group symmetries, group theory gives
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additional understanding of the system properties, and can be used to reduce
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the computational costs.
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\end_layout
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\begin_layout Standard
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As an example, if our system has a
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\begin_inset Formula $D_{2h}$
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\end_inset
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symmetry and our truncated
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\begin_inset Formula $\left(I-T\trops\right)$
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\end_inset
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matrix has size
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\begin_inset Formula $N\times N$
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\end_inset
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,
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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nepoužívám
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\begin_inset Formula $N$
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\end_inset
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už v jiném kontextu?
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\end_layout
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||
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\end_inset
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it can be block-diagonalized into eight blocks of size about
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\begin_inset Formula $N/8\times N/8$
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\end_inset
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, each of which can be LU-factorised separately (this is due to the fact
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that
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\begin_inset Formula $D_{2h}$
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\end_inset
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||
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has eight different one-dimensional irreducible representations).
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This can reduce both memory and time requirements to solve the scattering
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problem
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem block form"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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by a factor of 64.
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\end_layout
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\begin_layout Standard
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In periodic systems (problems
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem unit cell block form"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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||
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,
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:lattice mode equation"
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plural "false"
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||
caps "false"
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||
noprefix "false"
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||
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||
\end_inset
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||
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||
) due to small number of particles per unit cell, the costliest part is
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usually the evaluation of the lattice sums in the
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\begin_inset Formula $W\left(\omega,\vect k\right)$
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||
\end_inset
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||
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matrix, not the linear algebra.
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||
However, the lattice modes can be searched for in each irrep separately,
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||
and the irrep dimension gives a priori information about mode degeneracy.
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||
\end_layout
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||
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||
\begin_layout Subsection
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Excitation coefficients under point group operations
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||
\end_layout
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||
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||
\begin_layout Standard
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||
\begin_inset Note Note
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||
status open
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||
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\begin_layout Plain Layout
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||
TODO Zkontrolovat všechny vzorečky zde!!!
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||
\end_layout
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||
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||
\end_inset
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||
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||
In order to use the point group symmetries, we first need to know how they
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affect our basis functions, i.e.
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the VSWFs.
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||
\end_layout
|
||
|
||
\begin_layout Standard
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||
Let
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\begin_inset Formula $g$
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||
\end_inset
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||
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||
be a member of orthogonal group
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||
\begin_inset Formula $O(3)$
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||
\end_inset
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||
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||
, i.e.
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||
a 3D point rotation or reflection operation that transforms vectors in
|
||
|
||
\begin_inset Formula $\reals^{3}$
|
||
\end_inset
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||
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||
with an orthogonal matrix
|
||
\begin_inset Formula $R_{g}$
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||
\end_inset
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||
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||
:
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||
\begin_inset Formula
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||
\[
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||
\vect r\mapsto R_{g}\vect r.
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||
\]
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||
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||
\end_inset
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||
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||
Spherical harmonics
|
||
\begin_inset Formula $\ush lm$
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||
\end_inset
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||
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||
, being a basis the
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||
\begin_inset Formula $l$
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||
\end_inset
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||
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||
-dimensional representation of
|
||
\begin_inset Formula $O(3)$
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||
\end_inset
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||
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||
, transform as
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
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||
after "???"
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||
key "dresselhaus_group_2008"
|
||
literal "false"
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||
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||
\end_inset
|
||
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||
|
||
\begin_inset Formula
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||
\[
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||
\ush lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\ush l{m'}\left(\uvec r\right)
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||
\]
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||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $D_{m,m'}^{l}\left(g\right)$
|
||
\end_inset
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||
|
||
denotes the elements of the
|
||
\emph on
|
||
Wigner matrix
|
||
\emph default
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||
representing the operation
|
||
\begin_inset Formula $g$
|
||
\end_inset
|
||
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||
.
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||
By their definition, vector spherical harmonics
|
||
\begin_inset Formula $\vsh 2lm,\vsh 3lm$
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||
\end_inset
|
||
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||
transform in the same way,
|
||
\begin_inset Formula
|
||
\begin{align*}
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||
\vsh 2lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 2l{m'}\left(\uvec r\right),\\
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\vsh 3lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 3l{m'}\left(\uvec r\right),
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||
\end{align*}
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||
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||
\end_inset
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||
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||
but the remaining set
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||
\begin_inset Formula $\vsh 1lm$
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||
\end_inset
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||
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||
transforms differently due to their pseudovector nature stemming from the
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||
cross product in their definition:
|
||
\begin_inset Formula
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||
\[
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||
\vsh 3lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 3l{m'}\left(\uvec r\right),
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||
\]
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||
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||
\end_inset
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||
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||
where
|
||
\begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(g\right)=D_{m,m'}^{l}\left(g\right)$
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||
\end_inset
|
||
|
||
if
|
||
\begin_inset Formula $g$
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||
\end_inset
|
||
|
||
is a proper rotation, but for spatial inversion operation
|
||
\begin_inset Formula $i:\vect r\mapsto-\vect r$
|
||
\end_inset
|
||
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||
we have
|
||
\begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(i\right)=\left(-1\right)^{l+m}D_{m,m'}^{l}\left(i\right)$
|
||
\end_inset
|
||
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||
.
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||
The transformation behaviour of vector spherical harmonics directly propagates
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||
to the spherical vector waves, cf.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:VSWF regular"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
,
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||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:VSWF outgoing"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\begin{align*}
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||
\vswfouttlm 1lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vswfouttlm 1l{m'}\left(\vect r\right),\\
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||
\vswfouttlm 2lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vswfouttlm 2l{m'}\left(\vect r\right),
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||
\end{align*}
|
||
|
||
\end_inset
|
||
|
||
(and analogously for the regular waves
|
||
\begin_inset Formula $\vswfrtlm{\tau}lm$
|
||
\end_inset
|
||
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||
).
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
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||
TODO víc obdivu.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
For convenience, we introduce the symbol
|
||
\begin_inset Formula $D_{m,m'}^{\tau l}$
|
||
\end_inset
|
||
|
||
that describes the transformation of both types (
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
magnetic
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
electric
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
) of waves at once:
|
||
\begin_inset Formula
|
||
\[
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||
\vswfouttlm{\tau}lm\left(R_{g}\vect r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\vect r\right).
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||
\]
|
||
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||
\end_inset
|
||
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||
Using these, we can express the VSWF expansion
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
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||
reference "eq:E field expansion"
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||
plural "false"
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||
caps "false"
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||
noprefix "false"
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||
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||
\end_inset
|
||
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||
of the electric field around origin in a rotated/reflected system,
|
||
\begin_inset Formula
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||
\[
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||
\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\vect r\right)+\outcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\vect r\right)\right),
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||
\]
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||
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||
\end_inset
|
||
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||
which, together with the
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||
\begin_inset Formula $T$
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||
\end_inset
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||
|
||
-matrix definition,
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
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||
reference "eq:T-matrix definition"
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||
plural "false"
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||
caps "false"
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||
noprefix "false"
|
||
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||
\end_inset
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||
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||
can be used to obtain a
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||
\begin_inset Formula $T$
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||
\end_inset
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||
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||
-matrix of a rotated or mirror-reflected particle.
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||
Let
|
||
\begin_inset Formula $T$
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||
\end_inset
|
||
|
||
be the
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix of an original particle; the
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix of a particle physically transformed by operation
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||
\begin_inset Formula $g\in O(3)$
|
||
\end_inset
|
||
|
||
is then
|
||
\begin_inset Note Note
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||
status open
|
||
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||
\begin_layout Plain Layout
|
||
check sides
|
||
\end_layout
|
||
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||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
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||
\begin{equation}
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||
T'_{\tau lm;\tau'l'm'}=\sum_{\mu=-l}^{l}\sum_{\mu'=-l'}^{l'}\left(D_{\mu,m}^{\tau l}\left(g\right)\right)^{*}T_{\tau l\mu;\tau'l'm'}D_{m',\mu'}^{\tau l}\left(g\right).\label{eq:T-matrix of a transformed particle}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
If the particle is symmetric (so that
|
||
\begin_inset Formula $g$
|
||
\end_inset
|
||
|
||
produces a particle indistinguishable from the original one), the
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix must remain invariant under the transformation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:T-matrix of a transformed particle"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $T'_{\tau lm;\tau'l'm'}=T{}_{\tau lm;\tau'l'm'}$
|
||
\end_inset
|
||
|
||
.
|
||
Explicit forms of these invariance properties for the most imporant point
|
||
group symmetries can be found in
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "schulz_point-group_1999"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
If the field expansion is done around a point
|
||
\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
|
||
\begin_inset Formula
|
||
\begin{multline}
|
||
\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)\right.+\\
|
||
+\left.\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{multline}
|
||
|
||
\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
|
||
\align center
|
||
\begin_inset CommandInset include
|
||
LatexCommand input
|
||
filename "orbits.tex"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\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 $yz$
|
||
\end_inset
|
||
|
||
plane, hence the corresponding reflection maps each of them to itself,
|
||
but the
|
||
\begin_inset Formula $xz$
|
||
\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
|
||
\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)\right.+\label{eq:rotated E field expansion around outside origin-1}\\
|
||
& \quad+\left.\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)\\
|
||
& =\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)\right.+\\
|
||
& \quad+\left.\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
|
||
\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
|
||
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 unitary matrix, each block (also unitary) representing
|
||
the action of
|
||
\begin_inset Formula $g$
|
||
\end_inset
|
||
|
||
on one particle orbit.
|
||
All the
|
||
\begin_inset Formula $J\left(g\right)$
|
||
\end_inset
|
||
|
||
s make together a (reducible) linear representation of
|
||
\begin_inset Formula $G$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Irrep decomposition
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Knowledge of symmetry group actions
|
||
\begin_inset Formula $J\left(g\right)$
|
||
\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
|
||
|
||
.
|
||
Let
|
||
\begin_inset Formula $\Gamma_{n}$
|
||
\end_inset
|
||
|
||
be the
|
||
\begin_inset Formula $d_{n}$
|
||
\end_inset
|
||
|
||
-dimensional irreducible matrix representations of
|
||
\begin_inset Formula $G$
|
||
\end_inset
|
||
|
||
consisting of matrices
|
||
\begin_inset Formula $D^{\Gamma_{n}}\left(g\right)$
|
||
\end_inset
|
||
|
||
.
|
||
Then the projection operators
|
||
\begin_inset Formula
|
||
\[
|
||
P_{kl}^{\left(\Gamma_{n}\right)}\equiv\frac{d_{n}}{\left|G\right|}\sum_{g\in G}\left(D^{\Gamma_{n}}\left(g\right)\right)_{kl}^{*}J\left(g\right),\quad k,l=1,\dots,d_{n}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
project the full scattering system field expansion coefficient vectors
|
||
\begin_inset Formula $\rcoeff,\outcoeff$
|
||
\end_inset
|
||
|
||
onto a subspace corresponding to the irreducible representation
|
||
\begin_inset Formula $\Gamma_{n}$
|
||
\end_inset
|
||
|
||
.
|
||
The projectors can be used to construct a unitary transformation
|
||
\begin_inset Formula $U$
|
||
\end_inset
|
||
|
||
with components
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
U_{nri;p\tau lm}=\frac{d_{n}}{\left|G\right|}\sum_{g\in G}\left(D^{\Gamma_{n}}\left(g\right)\right)_{rr}^{*}J\left(g\right)_{p'\tau'l'm'(nri);p\tau lm}\label{eq:SAB unitary transformation operator}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $r$
|
||
\end_inset
|
||
|
||
goes from
|
||
\begin_inset Formula $1$
|
||
\end_inset
|
||
|
||
through
|
||
\begin_inset Formula $d_{n}$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $i$
|
||
\end_inset
|
||
|
||
goes from 1 through the multiplicity of irreducible representation
|
||
\begin_inset Formula $\Gamma_{n}$
|
||
\end_inset
|
||
|
||
in the (reducible) representation of
|
||
\begin_inset Formula $G$
|
||
\end_inset
|
||
|
||
spanned by the field expansion coefficients
|
||
\begin_inset Formula $\rcoeff$
|
||
\end_inset
|
||
|
||
or
|
||
\begin_inset Formula $\outcoeff$
|
||
\end_inset
|
||
|
||
.
|
||
The indices
|
||
\begin_inset Formula $p',\tau',l',m'$
|
||
\end_inset
|
||
|
||
are given by an arbitrary bijective mapping
|
||
\begin_inset Formula $\left(n,r,i\right)\mapsto\left(p',\tau',l',m'\right)$
|
||
\end_inset
|
||
|
||
with the constraint that for given
|
||
\begin_inset Formula $n,r,i$
|
||
\end_inset
|
||
|
||
there are at least some non-zero elements
|
||
\begin_inset Formula $U_{nri;p\tau lm}$
|
||
\end_inset
|
||
|
||
.
|
||
For details, we refer the reader to textbooks about group representation
|
||
theory
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
or linear representations?
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
, e.g.
|
||
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "Chapter 4"
|
||
key "dresselhaus_group_2008"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
or
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "???"
|
||
key "bradley_mathematical_1972"
|
||
literal "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
The transformation given by
|
||
\begin_inset Formula $U$
|
||
\end_inset
|
||
|
||
transforms the excitation coefficient vectors
|
||
\begin_inset Formula $\rcoeff,\outcoeff$
|
||
\end_inset
|
||
|
||
into a new,
|
||
\emph on
|
||
symmetry-adapted basis
|
||
\emph default
|
||
.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
One can show that if an operator
|
||
\begin_inset Formula $M$
|
||
\end_inset
|
||
|
||
acting on the excitation coefficient vectors is invariant under the operations
|
||
of group
|
||
\begin_inset Formula $G$
|
||
\end_inset
|
||
|
||
, meaning that
|
||
\begin_inset Formula
|
||
\[
|
||
\forall g\in G:J\left(g\right)MJ\left(g\right)^{\dagger}=M,
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
then in the symmetry-adapted basis,
|
||
\begin_inset Formula $M$
|
||
\end_inset
|
||
|
||
is block diagonal, or more specifically
|
||
\begin_inset Formula
|
||
\[
|
||
M_{\Gamma,r,i;\Gamma',r',j}^{\mathrm{s.a.b.}}=\frac{\delta_{\Gamma\Gamma'}\delta_{ij}}{d_{\Gamma}}\sum_{q}M{}_{\Gamma,r,q;\Gamma',r',q}^{\mathrm{s.a.b.}}.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
Both the
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\trops$
|
||
\end_inset
|
||
|
||
operators (and trivially also the identity
|
||
\begin_inset Formula $I$
|
||
\end_inset
|
||
|
||
) in
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Multiple-scattering problem block form"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
are invariant under the actions of whole system symmetry group, so
|
||
\begin_inset Formula $\left(I-T\trops\right)$
|
||
\end_inset
|
||
|
||
is also invariant, hence
|
||
\begin_inset Formula $U\left(I-T\trops\right)U^{\dagger}$
|
||
\end_inset
|
||
|
||
is a block-diagonal matrix, and the problem
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Multiple-scattering problem block form"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
can be solved for each block separately.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
From the computational perspective, it is important to note that
|
||
\begin_inset Formula $U$
|
||
\end_inset
|
||
|
||
is at least as sparse as
|
||
\begin_inset Formula $J\left(g\right)$
|
||
\end_inset
|
||
|
||
(which is
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
orbit-block
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
diagonal), hence the block-diagonalisation can be performed fast.
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Kvantifikovat!
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
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
|
||
\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
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
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.
|
||
in
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "chapters 10–11"
|
||
key "dresselhaus_group_2008"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
; here we use the same notation.
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
We analyse the symmetries of the system in the same VSWF representation
|
||
as used in the
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix formalism introduced above.
|
||
We are interested in the modes at the
|
||
\begin_inset Formula $\Kp$
|
||
\end_inset
|
||
|
||
-point of the hexagonal lattice, which has the
|
||
\begin_inset Formula $D_{3h}$
|
||
\end_inset
|
||
|
||
point symmetry.
|
||
The six irreducible representations (irreps) of the
|
||
\begin_inset Formula $D_{3h}$
|
||
\end_inset
|
||
|
||
group are known and are available in the literature in their explicit forms.
|
||
In order to find and classify the modes, we need to find a decomposition
|
||
of the lattice mode representation
|
||
\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}=\Gamma^{\mathrm{equiv.}}\otimes\Gamma_{\mathrm{vec.}}$
|
||
\end_inset
|
||
|
||
into the irreps of
|
||
\begin_inset Formula $D_{3h}$
|
||
\end_inset
|
||
|
||
.
|
||
The equivalence representation
|
||
\begin_inset Formula $\Gamma^{\mathrm{equiv.}}$
|
||
\end_inset
|
||
|
||
is the
|
||
\begin_inset Formula $E'$
|
||
\end_inset
|
||
|
||
representation as can be deduced from
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "eq. (11.19)"
|
||
key "dresselhaus_group_2008"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
, eq.
|
||
(11.19) and the character table for
|
||
\begin_inset Formula $D_{3h}$
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\begin_inset Formula $\Gamma_{\mathrm{vec.}}$
|
||
\end_inset
|
||
|
||
operates on a space spanned by the VSWFs around each nanoparticle in the
|
||
unit cell (the effects of point group operations on VSWFs are described
|
||
in
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "schulz_point-group_1999"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
).
|
||
This space can be then decomposed into invariant subspaces of the
|
||
\begin_inset Formula $D_{3h}$
|
||
\end_inset
|
||
|
||
using the projectors
|
||
\begin_inset Formula $\hat{P}_{ab}^{\left(\Gamma\right)}$
|
||
\end_inset
|
||
|
||
defined by
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "eq. (4.28)"
|
||
key "dresselhaus_group_2008"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
This way, we obtain a symmetry adapted basis
|
||
\begin_inset Formula $\left\{ \vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}\right\} $
|
||
\end_inset
|
||
|
||
as linear combinations of VSWFs
|
||
\begin_inset Formula $\vswfs lm{p,t}$
|
||
\end_inset
|
||
|
||
around the constituting nanoparticles (labeled
|
||
\begin_inset Formula $p$
|
||
\end_inset
|
||
|
||
),
|
||
\begin_inset Formula
|
||
\[
|
||
\vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}=\sum_{l,m,p,t}U_{\Gamma,r,i}^{p,t,l,m}\vswfs lm{p,t},
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\Gamma$
|
||
\end_inset
|
||
|
||
stands for one of the six different irreps of
|
||
\begin_inset Formula $D_{3h}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $r$
|
||
\end_inset
|
||
|
||
labels the different realisations of the same irrep, and the last index
|
||
|
||
\begin_inset Formula $i$
|
||
\end_inset
|
||
|
||
going from 1 to
|
||
\begin_inset Formula $d_{\Gamma}$
|
||
\end_inset
|
||
|
||
(the dimensionality of
|
||
\begin_inset Formula $\Gamma$
|
||
\end_inset
|
||
|
||
) labels the different partners of the same given irrep.
|
||
The number of how many times is each irrep contained in
|
||
\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}$
|
||
\end_inset
|
||
|
||
(i.e.
|
||
the range of index
|
||
\begin_inset Formula $r$
|
||
\end_inset
|
||
|
||
for given
|
||
\begin_inset Formula $\Gamma$
|
||
\end_inset
|
||
|
||
) depends on the multipole degree cutoff
|
||
\begin_inset Formula $l_{\mathrm{max}}$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
Each mode at the
|
||
\begin_inset Formula $\Kp$
|
||
\end_inset
|
||
|
||
-point shall lie in the irreducible spaces of only one of the six possible
|
||
irreps and it can be shown via
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "eq. (2.51)"
|
||
key "dresselhaus_group_2008"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
that, at the
|
||
\begin_inset Formula $\Kp$
|
||
\end_inset
|
||
|
||
-point, the matrix
|
||
\begin_inset Formula $M\left(\omega,\vect k\right)$
|
||
\end_inset
|
||
|
||
defined above takes a block-diagonal form in the symmetry-adapted basis,
|
||
|
||
\begin_inset Formula
|
||
\[
|
||
M\left(\omega,\vect K\right)_{\Gamma,r,i;\Gamma',r',j}^{\mathrm{s.a.b.}}=\frac{\delta_{\Gamma\Gamma'}\delta_{ij}}{d_{\Gamma}}\sum_{q}M\left(\omega,\vect K\right)_{\Gamma,r,q;\Gamma',r',q}^{\mathrm{s.a.b.}}.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
This enables us to decompose the matrix according to the irreps and to solve
|
||
the singular value problem in each irrep separately, as done in Fig.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "smfig:dispersions"
|
||
|
||
\end_inset
|
||
|
||
(a).
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|