Fix representation of spatial inversion.
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Marek Nečada
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\selectlanguage{finnish}%
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\section{Symmetries}\label{sec:Symmetries}
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If the system has nontrivial point group symmetries, group theory
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gives additional understanding of the system properties, and can be
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used to reduce the computational costs.
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As an example, if our system has a $D_{2h}$ symmetry and our truncated
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$\left(I-T\trops\right)$ matrix has size $N\times N$, it can be
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block-diagonalized into eight blocks of size about $N/8\times N/8$,
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each of which can be LU-factorised separately (this is due to the
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fact that $D_{2h}$ has eight different one-dimensional irreducible
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representations). This can reduce both memory and time requirements
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to solve the scattering problem (\ref{eq:Multiple-scattering problem block form})
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by a factor of 64.
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In periodic systems (problems (\ref{eq:Multiple-scattering problem unit cell block form}),
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(\ref{eq:lattice mode equation})) due to small number of particles
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per unit cell, the costliest part is usually the evaluation of the
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lattice sums in the $W\left(\omega,\vect k\right)$ matrix, not the
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linear algebra. However, the lattice modes can be searched for in
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each irrep separately, and the irrep dimension gives a priori information
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about mode degeneracy.
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\subsection{Excitation coefficients under point group operations}
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In order to use the point group symmetries, we first need to know
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how they affect our basis functions, i.e. the VSWFs.
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Let $g$ be a member of orthogonal group $O(3)$, i.e. a 3D point
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rotation or reflection operation that transforms vectors in $\reals^{3}$
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with an orthogonal matrix $R_{g}$:
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\[
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\vect r\mapsto R_{g}\vect r.
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\]
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Spherical harmonics $\ush lm$, being a basis the $l$-dimensional
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representation of $O(3)$, transform as \cite[???]{dresselhaus_group_2008}
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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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where $D_{m,m'}^{l}\left(g\right)$ denotes the elements of the \emph{Wigner
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matrix} representing the operation $g$. By their definition, vector
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spherical harmonics $\vsh 2lm,\vsh 3lm$ transform in the same way,
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\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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but the remaining set $\vsh 1lm$ transforms differently due to their
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pseudovector nature stemming from the cross product in their definition:
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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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where $\widetilde{D_{m,m'}^{l}}\left(g\right)=D_{m,m'}^{l}\left(g\right)$
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if $g$ is a proper rotation, but for spatial inversion operation
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$i:\vect r\mapsto-\vect r$ we have $\widetilde{D_{m,m'}^{l}}\left(i\right)=\left(-1\right)^{l+m}D_{m,m'}^{l}\left(i\right)$.
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The transformation behaviour of vector spherical harmonics directly
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propagates to the spherical vector waves, cf. (\ref{eq:VSWF regular}),
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(\ref{eq:VSWF outgoing}):
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\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*}
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(and analogously for the regular waves $\vswfrtlm{\tau}lm$). For
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convenience, we introduce the symbol $D_{m,m'}^{\tau l}$ that describes
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the transformation of both types (``magnetic'' and ``electric'')
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of waves at once:
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\[
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\vswfouttlm{\tau}lm\left(R_{g}\vect r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\vect r\right).
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\]
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Using these, we can express the VSWF expansion (\ref{eq:E field expansion})
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of the electric field around origin in a rotated/reflected system,
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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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which, together with the $T$-matrix definition, (\ref{eq:T-matrix definition})
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can be used to obtain a $T$-matrix of a rotated or mirror-reflected
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particle. Let $T$ be the $T$-matrix of an original particle; the
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$T$-matrix of a particle physically transformed by operation $g\in O(3)$
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is then
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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}
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\end{equation}
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If the particle is symmetric (so that $g$ produces a particle indistinguishable
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from the original one), the $T$-matrix must remain invariant under
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the transformation (\ref{eq:T-matrix of a transformed particle}),
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$T'_{\tau lm;\tau'l'm'}=T{}_{\tau lm;\tau'l'm'}$. Explicit forms
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of these invariance properties for the most imporant point group symmetries
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can be found in \cite{schulz_point-group_1999}.
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If the field expansion is done around a point $\vect r_{p}$ different
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from the global origin, as in \ref{eq:E field expansion multiparticle},
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we have\foreignlanguage{english}{
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\begin{align}
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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(\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)+\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}
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\end{align}
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}
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\begin{figure}
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\caption{Scatterer orbits under $D_{2}$ symmetry. Particles $A,B,C,D$ lie
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outside of origin or any mirror planes, and together constitute an
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orbit of the size equal to the order of the group, $\left|D_{2}\right|=4$.
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Particles $E,F$ lie on the $xz$ plane, hence the corresponding reflection
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maps each of them to itself, but the $yz$ reflection (or the $\pi$
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rotation around the $z$ axis) maps them to each other, forming a
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particle orbit of size 2. The particle $O$ in the very origin is
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always mapped to itself, constituting its own orbit.}\label{fig:D2-symmetric structure particle orbits}
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\end{figure}
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With these transformation properties in hand, we can proceed to the
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effects of point symmetries on the whole many-particle system. Let
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us have a many-particle system symmetric with respect to a point group
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$G$. A symmetry operation $g\in G$ determines a permutation of the
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particles: $p\mapsto\pi_{g}(p)$, $p\in\mathcal{P}$. For a given
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particle $p$, we will call the set of particles onto which any of
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the symmetries maps the particle $p$, i.e. the set $\left\{ \pi_{g}\left(p\right);g\in G\right\} $,
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as the \emph{orbit} of particle $p$. The whole set $\mathcal{P}$
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can therefore be divided into the different particle orbits; an example
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is in Fig. \ref{fig:D2-symmetric structure particle orbits}. The
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importance of the particle orbits stems from the following: in the
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multiple-scattering problem, outside of the scatterers one has \foreignlanguage{english}{
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\begin{align}
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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(\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)+\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)\label{eq:rotated E field expansion around outside origin-1}\\
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& =\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)+\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).
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\end{align}
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This means that the field expansion coefficients $\rcoeffp p,\outcoeffp p$
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transform as
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\begin{align}
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\rcoeffptlm p{\tau}lm & \mapsto\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right),\nonumber \\
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\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}
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\end{align}
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Obviously, the expansion coefficients belonging to particles in different
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orbits do not mix together. As before, we introduce a short-hand block-matrix
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notation for \ref{eq:excitation coefficient under symmetry operation}}
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\selectlanguage{english}%
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\begin{align}
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\rcoeff & \mapsto D\left(g\right)a,\nonumber \\
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\outcoeff & \mapsto D\left(g\right)\outcoeff.\label{eq:excitation coefficient under symmetry operation block form}
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\end{align}
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\selectlanguage{finnish}%
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\subsection{Irrep decomposition}
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\subsection{Periodic systems}
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\selectlanguage{english}%
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