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\begin_body

\begin_layout Section
Symmetries
\begin_inset CommandInset label
LatexCommand label
name "sec:Symmetries"

\end_inset


\end_layout

\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.
 
\end_layout

\begin_layout Standard
As an example, if our system has a 
\begin_inset Formula $D_{2h}$
\end_inset

 symmetry and our truncated 
\begin_inset Formula $\left(I-T\trops\right)$
\end_inset

 matrix has size 
\begin_inset Formula $N\times N$
\end_inset

,
\begin_inset Note Note
status open

\begin_layout Plain Layout
nepoužívám 
\begin_inset Formula $N$
\end_inset

 už v jiném kontextu?
\end_layout

\end_inset

 it can be block-diagonalized into eight blocks of size about 
\begin_inset Formula $N/8\times N/8$
\end_inset

, each of which can be LU-factorised separately (this is due to the fact
 that 
\begin_inset Formula $D_{2h}$
\end_inset

 has eight different one-dimensional irreducible representations).
 This can reduce both memory and time requirements to solve the scattering
 problem 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem block form"
plural "false"
caps "false"
noprefix "false"

\end_inset

 by a factor of 64.
\end_layout

\begin_layout Standard
In periodic systems (problems 
\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

) due to small number of particles per unit cell, the costliest part is
 usually the evaluation of the lattice sums in the 
\begin_inset Formula $W\left(\omega,\vect k\right)$
\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.
\end_layout

\begin_layout Subsection
Finite systems
\end_layout

\begin_layout Subsection
Periodic systems
\end_layout

\begin_layout Standard

\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.
 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 Standard

\lang english
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 Standard

\lang english
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_body
\end_document