qpms/lepaper/symmetries.lyx

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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 Standard
\begin_inset Note Note
status open

\begin_layout Plain Layout
TODO Zkontrolovat všechny vzorečky zde!!!
\end_layout

\end_inset

In order to use the point group symmetries, we first need to know how they
 affect our basis functions, i.e.
 the VSWFs.
\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $g$
\end_inset

 be a member of orthogonal group 
\begin_inset Formula $O(3)$
\end_inset

, i.e.
 a 3D point rotation or reflection operation that transforms vectors in
 
\begin_inset Formula $\reals^{3}$
\end_inset

 with an orthogonal matrix 
\begin_inset Formula $R_{g}$
\end_inset

:
\begin_inset Formula 
\[
\vect r\mapsto R_{g}\vect r.
\]

\end_inset

Spherical harmonics 
\begin_inset Formula $\ush lm$
\end_inset

, being a basis the 
\begin_inset Formula $l$
\end_inset

-dimensional representation of 
\begin_inset Formula $O(3)$
\end_inset

, transform as 
\begin_inset CommandInset citation
LatexCommand cite
after "???"
key "dresselhaus_group_2008"
literal "false"

\end_inset


\begin_inset Formula 
\[
\ush lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\ush l{m'}\left(\uvec r\right)
\]

\end_inset

where 
\begin_inset Formula $D_{m,m'}^{l}\left(g\right)$
\end_inset

 denotes the elements of the 
\emph on
Wigner matrix
\emph default
 representing the operation 
\begin_inset Formula $g$
\end_inset

.
 By their definition, vector spherical harmonics 
\begin_inset Formula $\vsh 2lm,\vsh 3lm$
\end_inset

 transform in the same way,
\begin_inset Formula 
\begin{align*}
\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),\\
\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),
\end{align*}

\end_inset

but the remaining set 
\begin_inset Formula $\vsh 1lm$
\end_inset

 transforms differently due to their pseudovector nature stemming from the
 cross product in their definition:
\begin_inset Formula 
\[
\vsh 3lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 3l{m'}\left(\uvec r\right),
\]

\end_inset

where 
\begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(g\right)=D_{m,m'}^{l}\left(g\right)$
\end_inset

 if 
\begin_inset Formula $g$
\end_inset

 is a proper rotation, but for spatial inversion operation 
\begin_inset Formula $i:\vect r\mapsto-\vect r$
\end_inset

 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

.
 The transformation behaviour of vector spherical harmonics directly propagates
 to the spherical vector waves, cf.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:VSWF regular"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:VSWF outgoing"
plural "false"
caps "false"
noprefix "false"

\end_inset

:
\begin_inset Formula 
\begin{align*}
\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),\\
\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),
\end{align*}

\end_inset

(and analogously for the regular waves 
\begin_inset Formula $\vswfrtlm{\tau}lm$
\end_inset

).
 
\begin_inset Note Note
status open

\begin_layout Plain Layout
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 
\[
\vswfouttlm{\tau}lm\left(R_{g}\vect r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\vect r\right).
\]

\end_inset

Using these, we can express the VSWF expansion 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:E field expansion"
plural "false"
caps "false"
noprefix "false"

\end_inset

 of the electric field around origin in a rotated/reflected system,
\begin_inset Formula 
\[
\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}lm\left(k\vect r\right)+D_{m,m'}^{\tau l}\left(g\right)\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\left(k\vect r\right)\right),
\]

\end_inset

which, together with the 
\begin_inset Formula $T$
\end_inset

-matrix definition, 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:T-matrix definition"
plural "false"
caps "false"
noprefix "false"

\end_inset

 can be used to obtain a 
\begin_inset Formula $T$
\end_inset

-matrix of a rotated or mirror-reflected particle.
 Let 
\begin_inset Formula $T$
\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 
\begin_inset Formula $g\in O(3)$
\end_inset

 is then 
\begin_inset Note Note
status open

\begin_layout Plain Layout
check sides
\end_layout

\end_inset


\begin_inset Formula 
\begin{equation}
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
With these point group transformation properties in hand, we can proceed
 to rotating (or mirror-reflecting) the whole many-particle system.
\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