Symmetries text in progress.
Former-commit-id: ecf71fe4af190cb6a3c1ed2c876183d01fbe6448
This commit is contained in:
Marek Nečada
parent
36cc152166
commit
526e108ec0
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@ -723,7 +723,11 @@ Truncation notation.
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\end_layout
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\begin_layout Itemize
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Example results.
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Example results!
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\end_layout
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\begin_layout Itemize
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Figures.
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\end_layout
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\begin_layout Itemize
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@ -738,7 +742,7 @@ Fix and unify notation (mainly indices) in infinite lattices section.
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Carefully check the transformation directions in sec.
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\begin_inset CommandInset ref
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LatexCommand eqref
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LatexCommand ref
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reference "sec:Symmetries"
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plural "false"
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caps "false"
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@ -749,6 +753,11 @@ noprefix "false"
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\end_layout
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\begin_layout Itemize
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The text about symmetries is pretty dense.
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Make it more explanatory and human-readable.
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset include
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LatexCommand include
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@ -180,7 +180,7 @@ noprefix "false"
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\end_layout
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\begin_layout Subsection
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Finite systems
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Excitation coefficients under point group operations
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\end_layout
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\begin_layout Standard
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@ -397,7 +397,7 @@ noprefix "false"
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of the electric field around origin in a rotated/reflected system,
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\begin_inset Formula
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\[
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\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}lm\left(k\vect r\right)+D_{m,m'}^{\tau l}\left(g\right)\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\left(k\vect r\right)\right),
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\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\vect r\right)+\outcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\vect r\right)\right),
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\]
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\end_inset
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@ -491,8 +491,297 @@ literal "false"
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\end_layout
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\begin_layout Standard
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With these point group transformation properties in hand, we can proceed
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to rotating (or mirror-reflecting) the whole many-particle system.
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If the field expansion is done around a point
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\begin_inset Formula $\vect r_{p}$
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\end_inset
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different from the global origin, as in
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:E field expansion multiparticle"
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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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, we have
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\lang english
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\begin_inset Formula
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\begin{align}
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\vect E\left(\omega,R_{g}\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)+\outcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right).\label{eq:rotated E field expansion around outside origin}
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\end{align}
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Float figure
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placement document
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alignment document
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wide false
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sideways false
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status open
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\begin_layout Plain Layout
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Scatterer orbits under
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\begin_inset Formula $D_{2}$
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\end_inset
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symmetry.
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Particles
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\begin_inset Formula $A,B,C,D$
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\end_inset
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lie outside of origin or any mirror planes, and together constitute an
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orbit of the size equal to the order of the group,
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\begin_inset Formula $\left|D_{2}\right|=4$
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\end_inset
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.
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Particles
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\begin_inset Formula $E,F$
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\end_inset
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lie on the
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\begin_inset Formula $xz$
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\end_inset
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plane, hence the corresponding reflection maps each of them to itself,
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but the
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\begin_inset Formula $yz$
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\end_inset
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reflection (or the
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\begin_inset Formula $\pi$
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\end_inset
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rotation around the
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\begin_inset Formula $z$
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\end_inset
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axis) maps them to each other, forming a particle orbit of size 2
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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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=???
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\end_layout
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\end_inset
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.
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The particle
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\begin_inset Formula $O$
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\end_inset
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in the very origin is always mapped to itself, constituting its own orbit.
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:D2-symmetric structure particle orbits"
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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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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TODO restructure this
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\end_layout
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\end_inset
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With these transformation properties in hand, we can proceed to the effects
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of point symmetries on the whole many-particle system.
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Let us have a many-particle system symmetric with respect to a point group
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\begin_inset Formula $G$
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\end_inset
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.
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A symmetry operation
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\begin_inset Formula $g\in G$
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\end_inset
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determines a permutation of the particles:
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\begin_inset Formula $p\mapsto\pi_{g}(p)$
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\end_inset
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,
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\begin_inset Formula $p\in\mathcal{P}$
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\end_inset
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.
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For a given particle
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\begin_inset Formula $p$
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\end_inset
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, we will call the set of particles onto which any of the symmetries maps
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the particle
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\begin_inset Formula $p$
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\end_inset
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, i.e.
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the set
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\begin_inset Formula $\left\{ \pi_{g}\left(p\right);g\in G\right\} $
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\end_inset
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, as the
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\emph on
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orbit
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\emph default
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of particle
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\begin_inset Formula $p$
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\end_inset
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.
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The whole set
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\begin_inset Formula $\mathcal{P}$
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\end_inset
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can therefore be divided into the different particle orbits; an example
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is in Fig.
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:D2-symmetric structure particle orbits"
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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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The importance of the particle orbits stems from the following: in the
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multiple-scattering problem, outside of the scatterers
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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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< FIXME
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\end_layout
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\end_inset
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one has
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\lang english
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\begin_inset Formula
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\begin{align}
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\vect E\left(\omega,R_{g}\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{\pi_{g}(p)}\right)\right)+\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right)\label{eq:rotated E field expansion around outside origin-1}\\
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& =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)+\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right).
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\end{align}
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\end_inset
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This means that the field expansion coefficients
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\begin_inset Formula $\rcoeffp p,\outcoeffp p$
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\end_inset
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transform as
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\begin_inset Formula
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\begin{align}
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\rcoeffptlm p{\tau}lm & \mapsto\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right),\nonumber \\
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\outcoeffptlm p{\tau}lm & \mapsto\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right).\label{eq:excitation coefficient under symmetry operation}
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\end{align}
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\end_inset
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Obviously, the expansion coefficients belonging to particles in different
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orbits do not mix together.
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As before, we introduce a short-hand block-matrix notation for
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:excitation coefficient under symmetry operation"
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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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(TODO avoid notation clash here in a more consistent and readable way!)
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\end_layout
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\begin_layout Standard
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\lang english
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\begin_inset Formula
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\begin{align}
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\rcoeff & \mapsto J\left(g\right)a,\nonumber \\
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\outcoeff & \mapsto J\left(g\right)\outcoeff.\label{eq:excitation coefficient under symmetry operation block form}
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\end{align}
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\end_inset
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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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\lang english
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The matrices
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\begin_inset Formula $D\left(g\right)$
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\end_inset
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,
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\begin_inset Formula $g\in G$
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\end_inset
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will play a crucial role blablabla
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\end_layout
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\end_inset
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If the particle indices are ordered in a way that the particles belonging
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to the same orbit are grouped together,
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\begin_inset Formula $J\left(g\right)$
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\end_inset
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will be a block-diagonal matrix, each block representing one particle orbit.
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\end_layout
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\begin_layout Subsection
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Irrep decomposition
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\end_layout
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\begin_layout Standard
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Knowledge of symmetry group actions
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\begin_inset Formula $D\left(g\right)$
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\end_inset
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on the field expansion coefficients give us the possibility to construct
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a symmetry adapted basis in which we can block-diagonalise the multiple-scatter
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ing problem matrix
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\begin_inset Formula $\left(I-TS\right)$
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Subsection
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@ -501,6 +790,12 @@ Periodic systems
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\begin_layout Standard
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\lang english
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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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\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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@ -516,7 +811,7 @@ literal "true"
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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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\begin_layout Plain Layout
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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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@ -671,7 +966,7 @@ where
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.
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\lang english
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Each mode at the
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@ -717,5 +1012,10 @@ reference "smfig:dispersions"
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(a).
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\end_layout
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\end_inset
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\end_layout
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\end_body
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\end_document
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