From d702b11bc16d183fc0b38da67fa2fcb71a9fae9b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Wed, 31 Jul 2019 13:02:10 +0300 Subject: [PATCH] symmetries global intro and copypasta from Rui's paper Former-commit-id: fce7a1273471bd31c964e9e9072accc866aada81 --- lepaper/arrayscat.lyx | 2 +- lepaper/finite.lyx | 2 +- lepaper/infinite.lyx | 19 ++ lepaper/symmetries.lyx | 409 +++++++++++++++++++++++++++++++++++++++++ 4 files changed, 430 insertions(+), 2 deletions(-) create mode 100644 lepaper/symmetries.lyx diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index 71d17e5..8fa5dea 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -731,7 +731,7 @@ Concrete comparison with other methods. \end_layout \begin_layout Itemize -Fix notation (mainly index) clashes in infinite lattices. +Fix and unify notation (mainly indices) in infinite lattices section. \end_layout \begin_layout Standard diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index 4490db2..55de130 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -1565,7 +1565,7 @@ m & -m' & m'-m \end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ m & -m' & m'-m \end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\ -\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}}.\label{eq:translation operator} +\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\qquad\tau\ne\tau'.\label{eq:translation operator} \end{multline} \end_inset diff --git a/lepaper/infinite.lyx b/lepaper/infinite.lyx index 296ba20..8da4454 100644 --- a/lepaper/infinite.lyx +++ b/lepaper/infinite.lyx @@ -382,6 +382,25 @@ dispersion relation \end_inset will acquire complex values. + The solution +\begin_inset Formula $\outcoeffp{\vect 0}\left(\vect k\right)$ +\end_inset + + is then obtained as the right +\begin_inset Note Note +status open + +\begin_layout Plain Layout +CHECK! +\end_layout + +\end_inset + + singular vector of +\begin_inset Formula $M\left(\omega,\vect k\right)$ +\end_inset + + corresponding to the zero singular value. \end_layout \begin_layout Subsection diff --git a/lepaper/symmetries.lyx b/lepaper/symmetries.lyx new file mode 100644 index 0000000..27527c1 --- /dev/null +++ b/lepaper/symmetries.lyx @@ -0,0 +1,409 @@ +#LyX 2.4 created this file. For more info see https://www.lyx.org/ +\lyxformat 583 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass article +\use_default_options true +\maintain_unincluded_children false +\language finnish +\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 +Finite systems +\end_layout + +\begin_layout Subsection +Periodic systems +\end_layout + +\begin_layout Standard + +\lang english +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 Standard + +\lang english +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 Standard + +\lang english +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_body +\end_document