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 substanti
ally reduce the computational costs.
\end_layout

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

 symmetry and the corresponding 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, 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
Excitation coefficients under point group operations
\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 make use of the point group symmetries, we first need to know
 how they affect our basis functions, i.e.
\begin_inset space \space{}
\end_inset

the VSWFs.
 Let 
\begin_inset Formula $g$
\end_inset

 be a member of the orthogonal group 
\begin_inset Formula $\mathrm{O}(3)$
\end_inset

, i.e.
\begin_inset space \space{}
\end_inset

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

With 
\begin_inset Formula $\groupop g$
\end_inset

 we shall denote the action of 
\begin_inset Formula $g$
\end_inset

 on a field in real space.
 For a scalar field 
\begin_inset Formula $w$
\end_inset

 we have 
\begin_inset Formula $\left(\groupop gw\right)\left(\vect r\right)=w\left(R_{g}^{-1}\vect r\right)$
\end_inset

, whereas for a vector field 
\begin_inset Formula $\vect w$
\end_inset

, 
\begin_inset Formula $\left(\groupop g\vect w\right)\left(\vect r\right)=R_{g}\vect w\left(R_{g}^{-1}\vect r\right)$
\end_inset

.
\end_layout

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

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

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

, transform as 
\begin_inset CommandInset citation
LatexCommand cite
after "Chapter 15"
key "wigner_group_1959"
literal "false"

\end_inset


\begin_inset Formula 
\begin{equation}
\left(\groupop g\ush lm\right)\left(\uvec r\right)=\ush lm\left(R_{g}^{-1}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\ush l{m'}\left(\uvec r\right)\label{eq:Wigner matrices}
\end{equation}

\end_inset

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

 denotes the elements of the 
\emph on
Wigner matrix 
\begin_inset Note Note
status open

\begin_layout Plain Layout
TODO explicit formulation
\end_layout

\end_inset


\emph default
 representing the operation 
\begin_inset Formula $g$
\end_inset

.
 From their definitions 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:vector spherical harmonics definition"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and the properties of the gradient operator under coordinate transforms,
 vector spherical harmonics 
\begin_inset Formula $\vsh 2lm,\vsh 3lm$
\end_inset

 transform in the same way,
\begin_inset Formula 
\begin{align*}
\left(\groupop g\vsh 2lm\right)\left(\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 2l{m'}\left(\uvec r\right),\\
\left(\groupop g\vsh 3lm\right)\left(\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


\begin_inset Note Note
status open

\begin_layout Plain Layout
\begin_inset Formula 
\begin{align*}
\left(\groupop g\vsh 2lm\right)\left(\uvec r\right) & =R_{g}\vsh 2lm\left(R_{g}^{-1}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 2l{m'}\left(\uvec r\right),\\
\left(\groupop g\vsh 3lm\right)\left(\uvec r\right) & =R_{g}\vsh 2lm\left(R_{g}^{-1}\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


\end_layout

\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 
\[
\left(\groupop g\vsh 1lm\right)\left(\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 1l{m'}\left(\uvec r\right),
\]

\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, 
\begin_inset Formula $g\in\mathrm{SO(3)}$
\end_inset

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

 we have 
\begin_inset Formula $D_{m,m'}^{l}\left(i\right)=\left(-1\right)^{l}$
\end_inset

 but 
\begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(i\right)=\left(-1\right)^{l+1}$
\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*}
\left(\groupop g\vswfouttlm 1lm\right)\left(\vect r\right) & =\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vswfouttlm 1l{m'}\left(\vect r\right),\\
\left(\groupop g\vswfouttlm 2lm\right)\left(\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 (
\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

) types of waves at once:
\begin_inset Formula 
\[
\groupop g\vswfouttlm{\tau}lm\left(\vect r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\vect r\right).
\]

\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 
\[
\left(\groupop g\vect E\right)\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(\kappa\vect r\right)+\outcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\kappa\vect r\right)\right),
\]

\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
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 Note Note
status open

\begin_layout Plain Layout
\begin_inset Marginal
status open

\begin_layout Plain Layout
Check this carefully.
 Maybe explain in more detail?
\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset Formula 
\begin{multline}
\left(\groupop g\vect E\right)\left(\omega,\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(\kappa\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(\kappa\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

; their positions transform as 
\begin_inset Formula $\vect r_{\pi_{g}p}=R_{g}\vect r_{p}$
\end_inset

, 
\begin_inset Formula $\vect r_{\pi_{g}^{-1}p}=R_{g}^{-1}\vect r_{p}$
\end_inset

.
 In the symmetric multiple-scattering problem, transforming the whole field
 according to 
\begin_inset Formula $g$
\end_inset

, in terms of field expansion around a particle originally labelled as 
\begin_inset Formula $p$
\end_inset


\begin_inset Formula 
\begin{align*}
\left(\groupop g\vect E\right)\left(\omega,\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(\kappa\left(\vect r-R_{g}\vect r_{p}\right)\right)\right.+\\
 & \quad+\left.\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\kappa\left(\vect r-R_{g}\vect r_{p}\right)\right)\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(\kappa\left(\vect r-\vect r_{\pi_{g}p}\right)\right)\right.\\
 & \quad+\left.\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\kappa\left(\vect r-\vect r_{\pi_{g}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}q}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(\kappa\left(\vect r-\vect r_{q}\right)\right)\right.\\
 & \quad+\left.\outcoeffptlm{\pi_{g}^{-1}q}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\kappa\left(\vect r-\vect r_{q}\right)\right)\right).
\end{align*}

\end_inset

In the last step, we relabeled 
\begin_inset Formula $q=\pi_{g}p$
\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 & \overset{g}{\longmapsto}\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right),\nonumber \\
\outcoeffptlm p{\tau}lm & \overset{g}{\longmapsto}\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

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

,
\begin_inset space \space{}
\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 fact that the expansion
 coefficients belonging to particles in different orbits are not related
 together under the group action in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:excitation coefficient under symmetry operation"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 As before, we introduce a short-hand pairwise matrix notation for 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:excitation coefficient under symmetry operation"
plural "false"
caps "false"
noprefix "false"

\end_inset

 
\begin_inset Note Note
status open

\begin_layout Plain Layout
(TODO avoid notation clash here in a more consistent and readable way!)
\end_layout

\end_inset


\begin_inset Formula 
\begin{align}
\rcoeffp p & \overset{g}{\longmapsto}\tilde{J}\left(g\right)\rcoeffp{\pi_{g}^{-1}(p)},\nonumber \\
\outcoeffp p & \overset{g}{\longmapsto}\tilde{J}\left(g\right)\outcoeffp{\pi_{g}^{-1}(p)},\label{eq:excitation coefficient under symmetry operation matrix form}
\end{align}

\end_inset

and also a global block-matrix form
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\rcoeff & \overset{g}{\longmapsto}J\left(g\right)a,\nonumber \\
\outcoeff & \overset{g}{\longmapsto}J\left(g\right)\outcoeff.\label{eq:excitation coefficient under symmetry operation global 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

 to 
\begin_inset Formula $d_{n}$
\end_inset

 and 
\begin_inset Formula $i$
\end_inset

 goes from 1 to 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 "Chapter 2"
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
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

 from the left hand side of eqs.
\begin_inset space \space{}
\end_inset


\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

 can be block-diagonalised in a similar manner.
 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 are typically located.
 This 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
better formulation
\end_layout

\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 Graphics
	filename figs/hex/p6m_group_actions.pdf
	width 95text%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Representing symmetry action on electromagnetic Bloch waves in a lattice
 with 
\begin_inset Formula $p6m$
\end_inset

 wallpaper group symmetry.
 In a hexagonal array with five particles (labeled 
\begin_inset Formula $A$
\end_inset

–
\begin_inset Formula $E$
\end_inset

) per unit cell, we first choose into which unit cells do the particles
 on unit cell boundaries belong.
 a) At 
\begin_inset Formula $M$
\end_inset

 point, the little co-group contains a 
\begin_inset Formula $D_{2}$
\end_inset

 point group; the unit cells can be divided into two groups (alternating
 horizontal rows) with opposite sign.
 The horizontal mirror operation 
\begin_inset Formula $\sigma_{xz}$
\end_inset

 maps the particles from a single unit cell to each other.
 However, the vertical mirror operation 
\begin_inset Formula $\sigma_{yz}$
\end_inset

 maps them onto particles belonging to different unit cells, introducing
 possible phase factors in the point group action: 
\begin_inset Formula $B,C,D$
\end_inset

 map onto 
\begin_inset Formula $B,C,D$
\end_inset

 belonging to different unit cell from same phase group, so no additional
 phase is needed; however, 
\begin_inset Formula $A,E$
\end_inset

 map onto 
\begin_inset Formula $E,A$
\end_inset

 belonging to unitcells with relative phases 
\begin_inset Formula $\pm\pi$
\end_inset

, therefore the corresponding action matrix blocks will carry a factor 
\begin_inset Formula $-1$
\end_inset

.
 b) At 
\begin_inset Formula $K$
\end_inset

 point point, little co-group contains a 
\begin_inset Formula $D_{3}$
\end_inset

 point group, and the unit cells divide into three groups with relative
 phase shift 
\begin_inset Formula $e^{2\pi i/3}$
\end_inset

.
 The horizontal mirroring 
\begin_inset Formula $\sigma_{xz}$
\end_inset

 again does not introduce any additional phase.
 However, the 
\begin_inset Formula $C_{3}$
\end_inset

 rotation mixes particles belonging to different unit cells, so for example
 particle 
\begin_inset Formula $D$
\end_inset

 maps onto particle 
\begin_inset Formula $D$
\end_inset

, but with additional phase factor 
\begin_inset Formula $e^{-2\pi i/3}$
\end_inset

.
 
\begin_inset CommandInset label
LatexCommand label
name "Phase factor illustration"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\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 can happen if the point group symmetry maps some of the scatterers
 from the reference unit cell to scatterers belonging to other unit cells.
 This is illustrated in Fig.
\begin_inset space \space{}
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Phase factor illustration"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Fig.
\begin_inset space \space{}
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Phase factor illustration"
plural "false"
caps "false"
noprefix "false"

\end_inset

a shows a hexagonal periodic array with 
\begin_inset Formula $p6m$
\end_inset

 wallpaper group symmetry, with lattice vectors 
\begin_inset Formula $\vect a_{1}=\left(a,0\right)$
\end_inset

 and 
\begin_inset Formula $\vect a_{2}=\left(a/2,\sqrt{3}a/2\right)$
\end_inset

.
 We delimit our representative unit cell as the Wigner-Seitz cell with origin
 in a 
\begin_inset Formula $D_{6}$
\end_inset

 point group symmetry center (there is one per each unit cell); per unit
 cell, there are five different particles placed on the unit cell boundary,
 and we need to make a choice to which unit cell the particles on the boundary
 belong; in our case, we choose that a unit cell includes the particles
 on the left as denoted by different colors.
 If the Bloch vector is at the upper 
\begin_inset Formula $M$
\end_inset

 point, 
\begin_inset Formula $\vect k=\vect M_{1}=\left(0,2\pi/\sqrt{3}a\right)$
\end_inset

, it creates a relative phase of 
\begin_inset Formula $\pi$
\end_inset

 between the unit cell rows, and the original 
\begin_inset Formula $D_{6}$
\end_inset

 symmetry is reduced to 
\begin_inset Formula $D_{2}$
\end_inset

.
 The 
\begin_inset Quotes eld
\end_inset

horizontal
\begin_inset Quotes erd
\end_inset

 mirror operation 
\begin_inset Formula $\sigma_{xz}$
\end_inset

 maps, acording to our boundary division, all the particles only inside
 the same unit cell, e.g.
\begin_inset Formula 
\begin{align*}
\outcoeffp{\vect 0A} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect 0E},\\
\outcoeff_{\vect 0C} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect 0C},
\end{align*}

\end_inset

as in eq.
\begin_inset space \space{}
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:excitation coefficient under symmetry operation"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 However, both the 
\begin_inset Quotes eld
\end_inset

vertical
\begin_inset Quotes erd
\end_inset

 mirroring 
\begin_inset Formula $\sigma_{yz}$
\end_inset

 and the 
\begin_inset Formula $C_{2}$
\end_inset

 rotation map the boundary particles onto the boundaries that do not belong
 to the reference unit cell with 
\begin_inset Formula $\vect n=\left(0,0\right)$
\end_inset

, so we have, explicitly writing down also the lattice point indices 
\begin_inset Formula $\vect n$
\end_inset

,
\begin_inset Formula 
\begin{align*}
\outcoeffp{\vect 0A} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(0,1\right)E},\\
\outcoeff_{\vect 0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(1,0\right)C},
\end{align*}

\end_inset

but we want 
\begin_inset Formula $J(g)$
\end_inset

 to operate only inside one unit cell, so we use the Bloch condition 
\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
\end_inset

: in this case, we have 
\begin_inset Formula $\outcoeffp{\left(0,1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect M_{1}\cdot\vect a_{2}}=\outcoeffp{\vect 0\alpha}e^{i0}=\outcoeffp{\vect 0\alpha}$
\end_inset

, 
\begin_inset Formula $\outcoeffp{\left(1,0\right)\alpha}=e^{i\vect M_{1}\cdot\vect a_{2}}\outcoeffp{\vect 0\alpha}=e^{i\pi}\outcoeffp{\vect 0\alpha}=-\outcoeffp{\vect 0\alpha},$
\end_inset

so 
\begin_inset Formula 
\begin{align*}
\outcoeffp{\vect 0A} & \overset{\sigma_{yz}}{\longmapsto}-\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect 0E},\\
\outcoeff_{\vect 0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect 0C}.
\end{align*}

\end_inset

If we set instead 
\begin_inset Formula $\vect k=\vect K=\left(4\pi/3a,0\right),$
\end_inset

 the original 
\begin_inset Formula $D_{6}$
\end_inset

 point group symmetry reduces to 
\begin_inset Formula $D_{3}$
\end_inset

 and the unit cells can obtain a relative phase factor of 
\begin_inset Formula $e^{-2\pi i/3}$
\end_inset

 (blue) or 
\begin_inset Formula $e^{2\pi i/3}$
\end_inset

 (red).
 The 
\begin_inset Formula $\sigma_{xz}$
\end_inset

 mirror symmetry, as in the previous case, acts purely inside the reference
 unit cell with our boundary division.
 However, for a counterclockwise 
\begin_inset Formula $C_{3}$
\end_inset

 rotation as an example we have (see Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "Phase factor illustration"
plural "false"
caps "false"
noprefix "false"

\end_inset

b)
\begin_inset Formula 
\begin{align*}
\outcoeffp{\vect 0A} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(0,-1\right)E}=e^{2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect 0E},\\
\outcoeff_{\vect 0C} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)A}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect 0A},\\
\outcoeff_{\vect 0B} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(1,-1\right)B}=e^{-2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect 0B},
\end{align*}

\end_inset

because in this case, the Bloch condition gives 
\begin_inset Formula $\outcoeffp{\left(0,-1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect K\cdot\left(-\vect a_{2}\right)}=\outcoeffp{\vect 0\alpha}e^{-4\pi i/3}=\outcoeffp{\vect 0\alpha}e^{2\pi i/3}=\outcoeffp{\vect 0\alpha}$
\end_inset

, 
\begin_inset Formula $\outcoeffp{\left(1,-1\right)\alpha}=\outcoeffp{\vect 0\alpha}e^{i\vect K\cdot\left(\vect a_{1}-\vect a_{2}\right)}=e^{-2\pi i/3}\outcoeffp{\vect 0\alpha}.$
\end_inset


\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 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