410 lines
8.4 KiB
Plaintext
410 lines
8.4 KiB
Plaintext
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#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\lyxformat 583
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\begin_document
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\index Index
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\end_index
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\end_header
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\begin_body
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\begin_layout Section
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Symmetries
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:Symmetries"
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\end_inset
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\end_layout
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\begin_layout Standard
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If the system has nontrivial point group symmetries, group theory gives
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additional understanding of the system properties, and can be used to reduce
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the computational costs.
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\end_layout
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\begin_layout Standard
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As an example, if our system has a
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\begin_inset Formula $D_{2h}$
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\end_inset
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symmetry and our truncated
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\begin_inset Formula $\left(I-T\trops\right)$
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\end_inset
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matrix has size
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\begin_inset Formula $N\times N$
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\end_inset
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,
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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nepoužívám
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\begin_inset Formula $N$
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\end_inset
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už v jiném kontextu?
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\end_layout
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\end_inset
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it can be block-diagonalized into eight blocks of size about
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\begin_inset Formula $N/8\times N/8$
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\end_inset
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, each of which can be LU-factorised separately (this is due to the fact
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that
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\begin_inset Formula $D_{2h}$
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\end_inset
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has eight different one-dimensional irreducible representations).
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This can reduce both memory and time requirements to solve the scattering
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problem
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem block form"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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by a factor of 64.
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\end_layout
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\begin_layout Standard
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In periodic systems (problems
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem unit cell block form"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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,
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:lattice mode equation"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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) due to small number of particles per unit cell, the costliest part is
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usually the evaluation of the lattice sums in the
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\begin_inset Formula $W\left(\omega,\vect k\right)$
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\end_inset
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matrix, not the linear algebra.
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However, the lattice modes can be searched for in each irrep separately,
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and the irrep dimension gives a priori information about mode degeneracy.
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\end_layout
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\begin_layout Subsection
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Finite systems
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\end_layout
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\begin_layout Subsection
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Periodic systems
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\end_layout
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\begin_layout Standard
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\lang english
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A general overview of utilizing group theory to find lattice modes at high-symme
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try points of the Brillouin zone can be found e.g.
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in
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\begin_inset CommandInset citation
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LatexCommand cite
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after "chapters 10–11"
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key "dresselhaus_group_2008"
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literal "true"
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\end_inset
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; here we use the same notation.
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\end_layout
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\begin_layout Standard
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\lang english
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We analyse the symmetries of the system in the same VSWF representation
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as used in the
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\begin_inset Formula $T$
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\end_inset
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-matrix formalism introduced above.
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We are interested in the modes at the
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\begin_inset Formula $\Kp$
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\end_inset
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-point of the hexagonal lattice, which has the
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\begin_inset Formula $D_{3h}$
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\end_inset
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point symmetry.
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The six irreducible representations (irreps) of the
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\begin_inset Formula $D_{3h}$
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\end_inset
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group are known and are available in the literature in their explicit forms.
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In order to find and classify the modes, we need to find a decomposition
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of the lattice mode representation
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\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}=\Gamma^{\mathrm{equiv.}}\otimes\Gamma_{\mathrm{vec.}}$
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\end_inset
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into the irreps of
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\begin_inset Formula $D_{3h}$
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\end_inset
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.
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The equivalence representation
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\begin_inset Formula $\Gamma^{\mathrm{equiv.}}$
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\end_inset
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is the
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\begin_inset Formula $E'$
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\end_inset
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representation as can be deduced from
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\begin_inset CommandInset citation
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LatexCommand cite
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after "eq. (11.19)"
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key "dresselhaus_group_2008"
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literal "true"
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\end_inset
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, eq.
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(11.19) and the character table for
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\begin_inset Formula $D_{3h}$
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\end_inset
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.
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\begin_inset Formula $\Gamma_{\mathrm{vec.}}$
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\end_inset
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operates on a space spanned by the VSWFs around each nanoparticle in the
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unit cell (the effects of point group operations on VSWFs are described
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in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "schulz_point-group_1999"
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literal "true"
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\end_inset
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).
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This space can be then decomposed into invariant subspaces of the
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\begin_inset Formula $D_{3h}$
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\end_inset
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using the projectors
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\begin_inset Formula $\hat{P}_{ab}^{\left(\Gamma\right)}$
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\end_inset
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defined by
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\begin_inset CommandInset citation
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LatexCommand cite
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after "eq. (4.28)"
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key "dresselhaus_group_2008"
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literal "true"
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\end_inset
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.
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This way, we obtain a symmetry adapted basis
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\begin_inset Formula $\left\{ \vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}\right\} $
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\end_inset
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as linear combinations of VSWFs
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\begin_inset Formula $\vswfs lm{p,t}$
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\end_inset
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around the constituting nanoparticles (labeled
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\begin_inset Formula $p$
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\end_inset
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),
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\begin_inset Formula
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\[
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\vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}=\sum_{l,m,p,t}U_{\Gamma,r,i}^{p,t,l,m}\vswfs lm{p,t},
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\]
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\end_inset
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where
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\begin_inset Formula $\Gamma$
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\end_inset
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stands for one of the six different irreps of
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\begin_inset Formula $D_{3h}$
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\end_inset
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,
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\begin_inset Formula $r$
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\end_inset
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labels the different realisations of the same irrep, and the last index
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\begin_inset Formula $i$
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\end_inset
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going from 1 to
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\begin_inset Formula $d_{\Gamma}$
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\end_inset
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(the dimensionality of
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\begin_inset Formula $\Gamma$
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\end_inset
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) labels the different partners of the same given irrep.
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The number of how many times is each irrep contained in
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\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}$
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\end_inset
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(i.e.
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the range of index
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\begin_inset Formula $r$
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\end_inset
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for given
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\begin_inset Formula $\Gamma$
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\end_inset
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) depends on the multipole degree cutoff
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\begin_inset Formula $l_{\mathrm{max}}$
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\end_inset
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.
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\end_layout
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|||
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\begin_layout Standard
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\lang english
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Each mode at the
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\begin_inset Formula $\Kp$
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\end_inset
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-point shall lie in the irreducible spaces of only one of the six possible
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irreps and it can be shown via
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\begin_inset CommandInset citation
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LatexCommand cite
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after "eq. (2.51)"
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key "dresselhaus_group_2008"
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literal "true"
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\end_inset
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that, at the
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\begin_inset Formula $\Kp$
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\end_inset
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-point, the matrix
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\begin_inset Formula $M\left(\omega,\vect k\right)$
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\end_inset
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defined above takes a block-diagonal form in the symmetry-adapted basis,
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\begin_inset Formula
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\[
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M\left(\omega,\vect K\right)_{\Gamma,r,i;\Gamma',r',j}^{\mathrm{s.a.b.}}=\frac{\delta_{\Gamma\Gamma'}\delta_{ij}}{d_{\Gamma}}\sum_{q}M\left(\omega,\vect K\right)_{\Gamma,r,q;\Gamma',r',q}^{\mathrm{s.a.b.}}.
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\]
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\end_inset
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This enables us to decompose the matrix according to the irreps and to solve
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|||
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the singular value problem in each irrep separately, as done in Fig.
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|||
|
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|||
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\begin_inset CommandInset ref
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|||
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LatexCommand ref
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|||
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reference "smfig:dispersions"
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\end_inset
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(a).
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\end_layout
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\end_body
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\end_document
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