#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 our system has a \begin_inset Formula $D_{2h}$ \end_inset symmetry and our truncated \begin_inset Formula $\left(I-T\trops\right)$ \end_inset matrix has size \begin_inset Formula $N\times N$ \end_inset , \begin_inset Note Note status open \begin_layout Plain Layout nepoužívám \begin_inset Formula $N$ \end_inset už v jiném kontextu? \end_layout \end_inset it can be block-diagonalized into eight blocks of size about \begin_inset Formula $N/8\times N/8$ \end_inset , each of which can be LU-factorised separately (this is due to the fact that \begin_inset Formula $D_{2h}$ \end_inset has eight different one-dimensional irreducible representations). This can reduce both memory and time requirements to solve the scattering problem \begin_inset CommandInset ref LatexCommand eqref reference "eq:Multiple-scattering problem block form" plural "false" caps "false" noprefix "false" \end_inset by a factor of 64. \end_layout \begin_layout Standard In periodic systems (problems \begin_inset CommandInset ref LatexCommand eqref reference "eq:Multiple-scattering problem unit cell block form" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand eqref reference "eq:lattice mode equation" plural "false" caps "false" noprefix "false" \end_inset ) due to small number of particles per unit cell, the costliest part is usually the evaluation of the lattice sums in the \begin_inset Formula $W\left(\omega,\vect k\right)$ \end_inset matrix, not the linear algebra. However, the lattice modes can be searched for in each irrep separately, and the irrep dimension gives a priori information about mode degeneracy. \end_layout \begin_layout Subsection 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 orthogonal group \begin_inset Formula $O(3)$ \end_inset , i.e. a 3D point rotation or reflection operation that transforms vectors in \begin_inset Formula $\reals^{3}$ \end_inset with an orthogonal matrix \begin_inset Formula $R_{g}$ \end_inset : \begin_inset Formula \[ \vect r\mapsto R_{g}\vect r. \] \end_inset Spherical harmonics \begin_inset Formula $\ush lm$ \end_inset , being a basis the \begin_inset Formula $l$ \end_inset -dimensional representation of \begin_inset Formula $O(3)$ \end_inset , transform as \begin_inset CommandInset citation LatexCommand cite after "???" key "dresselhaus_group_2008" literal "false" \end_inset \begin_inset Formula \[ \ush lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\ush l{m'}\left(\uvec r\right) \] \end_inset where \begin_inset Formula $D_{m,m'}^{l}\left(g\right)$ \end_inset denotes the elements of the \emph on Wigner matrix \emph default representing the operation \begin_inset Formula $g$ \end_inset . By their definition, vector spherical harmonics \begin_inset Formula $\vsh 2lm,\vsh 3lm$ \end_inset transform in the same way, \begin_inset Formula \begin{align*} \vsh 2lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 2l{m'}\left(\uvec r\right),\\ \vsh 3lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 3l{m'}\left(\uvec r\right), \end{align*} \end_inset but the remaining set \begin_inset Formula $\vsh 1lm$ \end_inset transforms differently due to their pseudovector nature stemming from the cross product in their definition: \begin_inset Formula \[ \vsh 3lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 3l{m'}\left(\uvec r\right), \] \end_inset where \begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(g\right)=D_{m,m'}^{l}\left(g\right)$ \end_inset if \begin_inset Formula $g$ \end_inset is a proper rotation, but for spatial inversion operation \begin_inset Formula $i:\vect r\mapsto-\vect r$ \end_inset we have \begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(i\right)=\left(-1\right)^{l+m}D_{m,m'}^{l}\left(i\right)$ \end_inset . The transformation behaviour of vector spherical harmonics directly propagates to the spherical vector waves, cf. \begin_inset CommandInset ref LatexCommand eqref reference "eq:VSWF regular" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand eqref reference "eq:VSWF outgoing" plural "false" caps "false" noprefix "false" \end_inset : \begin_inset Formula \begin{align*} \vswfouttlm 1lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vswfouttlm 1l{m'}\left(\vect r\right),\\ \vswfouttlm 2lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vswfouttlm 2l{m'}\left(\vect r\right), \end{align*} \end_inset (and analogously for the regular waves \begin_inset Formula $\vswfrtlm{\tau}lm$ \end_inset ). \begin_inset Note Note status open \begin_layout Plain Layout TODO víc obdivu. \end_layout \end_inset For convenience, we introduce the symbol \begin_inset Formula $D_{m,m'}^{\tau l}$ \end_inset that describes the transformation of both types ( \begin_inset Quotes eld \end_inset magnetic \begin_inset Quotes erd \end_inset and \begin_inset Quotes eld \end_inset electric \begin_inset Quotes erd \end_inset ) of waves at once: \begin_inset Formula \[ \vswfouttlm{\tau}lm\left(R_{g}\vect r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\vect r\right). \] \end_inset Using these, we can express the VSWF expansion \begin_inset CommandInset ref LatexCommand eqref reference "eq:E field expansion" plural "false" caps "false" noprefix "false" \end_inset of the electric field around origin in a rotated/reflected system, \begin_inset Formula \[ \vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\vect r\right)+\outcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\vect r\right)\right), \] \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 \begin_inset Formula \begin{multline} \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)\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 . 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 the following: in the multiple-scattering problem, outside of the scatterers \begin_inset Note Note status open \begin_layout Plain Layout < FIXME \end_layout \end_inset one has \begin_inset Formula \begin{align} \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)\right.+\label{eq:rotated E field expansion around outside origin-1}\\ & \quad+\left.\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)\\ & =\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)\right.+\\ & \quad+\left.\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). \end{align} \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 & \mapsto\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right),\nonumber \\ \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} \end{align} \end_inset Obviously, the expansion coefficients belonging to particles in different orbits do not mix together. As before, we introduce a short-hand block-matrix notation for \begin_inset CommandInset ref LatexCommand ref 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!) \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \rcoeff & \mapsto J\left(g\right)a,\nonumber \\ \outcoeff & \mapsto J\left(g\right)\outcoeff.\label{eq:excitation coefficient under symmetry operation 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 through \begin_inset Formula $d_{n}$ \end_inset and \begin_inset Formula $i$ \end_inset goes from 1 through 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 $M\left(\omega,\vect k\right)=\left(I-W\left(\omega,\vect k\right)T\left(\omega\right)\right)$ \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 is illustrated in Fig. \begin_inset CommandInset ref LatexCommand ref reference "Phase factor illustration" plural "false" caps "false" noprefix "false" \end_inset . \begin_inset Float figure placement document alignment document wide false sideways false status open \begin_layout Plain Layout \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 \begin_layout Plain Layout \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