#LyX 2.4 created this file. For more info see https://www.lyx.org/ \lyxformat 584 \begin_document \begin_header \save_transient_properties true \origin unavailable \textclass article \use_default_options true \maintain_unincluded_children false \language english \language_package default \inputencoding utf8 \fontencoding auto \font_roman "default" "default" \font_sans "default" "default" \font_typewriter "default" "default" \font_math "auto" "auto" \font_default_family default \use_non_tex_fonts false \font_sc false \font_roman_osf false \font_sans_osf false \font_typewriter_osf false \font_sf_scale 100 100 \font_tt_scale 100 100 \use_microtype false \use_dash_ligatures true \graphics default \default_output_format default \output_sync 0 \bibtex_command default \index_command default \paperfontsize default \use_hyperref false \papersize default \use_geometry false \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 1 \use_package mhchem 1 \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 \cite_engine basic \cite_engine_type default \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false \justification true \use_refstyle 1 \use_minted 0 \use_lineno 0 \index Index \shortcut idx \color #008000 \end_index \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \is_math_indent 0 \math_numbering_side default \quotes_style english \dynamic_quotes 0 \papercolumns 1 \papersides 1 \paperpagestyle default \tablestyle default \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \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 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, 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 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 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 the orthogonal group \begin_inset Formula $\mathrm{O}(3)$ \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 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 "???" key "dresselhaus_group_2008" literal "false" \end_inset \begin_inset Formula \[ \left(\groupop g\ush lm\right)\left(\uvec r\right)=\ush lm\left(R_{g}^{-1}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\ush l{m'}\left(\uvec r\right) \] \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 \[ \left(\groupop g\vect E\right)\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\vect r\right)+\outcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\vect r\right)\right), \] \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 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 (CHECK THIS CAREFULLY AND EXPLAIN) \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(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 ; 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(k\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(k\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(k\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(k\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(k\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(k\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 , 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 (TODO avoid notation clash here in a more consistent and readable way!) \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 "???" 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 $M\left(\omega,\vect k\right)=\left(I-T\left(\omega\right)W\left(\omega,\vect k\right)\right)$ \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 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 CommandInset ref LatexCommand ref reference "Phase factor illustration" plural "false" caps "false" noprefix "false" \end_inset . Fig. \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 . If 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{\vect0A} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect0E},\\ \outcoeff_{\vect0C} & \overset{\sigma_{xz}}{\longmapsto}\tilde{J}\left(\sigma_{xz}\right)\outcoeffp{\vect0C}, \end{align*} \end_inset as in eq. \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{\vect0A} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\left(0,1\right)E},\\ \outcoeff_{\vect0C} & \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 $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$ \end_inset : in this case, we have \begin_inset Formula $\outcoeffp{\left(0,1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect M_{1}\cdot\vect a_{2}}=\outcoeffp{\vect0\alpha}e^{i0}=\outcoeffp{\vect0\alpha}$ \end_inset , \begin_inset Formula $\outcoeffp{\left(1,0\right)\alpha}=e^{i\vect M_{1}\cdot\vect a_{2}}\outcoeffp{\vect0\alpha}=e^{i\pi}\outcoeffp{\vect0\alpha}=-\outcoeffp{\vect0\alpha},$ \end_inset so \begin_inset Formula \begin{align*} \outcoeffp{\vect0A} & \overset{\sigma_{yz}}{\longmapsto}-\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect0E},\\ \outcoeff_{\vect0C} & \overset{\sigma_{yz}}{\longmapsto}\tilde{J}\left(\sigma_{yz}\right)\outcoeffp{\vect0C}. \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 \begin_inset Formula \begin{align*} \outcoeffp{\vect0A} & \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{\vect0E},\\ \outcoeff_{\vect0C} & \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{\vect0A},\\ \outcoeff_{\vect0B} & \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{\vect0B}, \end{align*} \end_inset because in this case, the Bloch condition gives \begin_inset Formula $\outcoeffp{\left(0,-1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect K\cdot\left(-\vect a_{2}\right)}=\outcoeffp{\vect0\alpha}e^{-4\pi i/3}=\outcoeffp{\vect0\alpha}e^{2\pi i/3}=\outcoeffp{\vect0\alpha}$ \end_inset , \begin_inset Formula $\outcoeffp{\left(1,-1\right)\alpha}=\outcoeffp{\vect0\alpha}e^{i\vect K\cdot\left(\vect a_{1}-\vect a_{2}\right)}=e^{-2\pi i/3}\outcoeffp{\vect0\alpha}.$ \end_inset \end_layout \begin_layout Standard \begin_inset Float figure placement document alignment document wide false sideways false status collapsed \begin_layout Plain Layout \align center \begin_inset Graphics filename p6m_mpoint.png width 100col% \end_inset \end_layout \begin_layout Plain Layout \begin_inset Graphics filename p6m_kpoint.png width 100col% \end_inset \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 \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