Irrep decomposition
Former-commit-id: aedcdb912573500bcba434ce088ce10887052bd1
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Marek Nečada
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@ -758,7 +758,21 @@ If the particle indices are ordered in a way that the particles belonging
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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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will be a block-diagonal unitary matrix, each block (also unitary) representing
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the action of
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\begin_inset Formula $g$
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\end_inset
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on one particle orbit.
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All the
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\begin_inset Formula $J\left(g\right)$
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\end_inset
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s make together a (reducible) linear representation of
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\begin_inset Formula $G$
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\end_inset
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.
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\end_layout
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\begin_layout Subsection
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@ -767,7 +781,7 @@ Irrep decomposition
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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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\begin_inset Formula $J\left(g\right)$
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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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@ -776,11 +790,251 @@ 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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Let
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\begin_inset Formula $\Gamma_{n}$
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\end_inset
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be the
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\begin_inset Formula $d_{n}$
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\end_inset
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-dimensional irreducible matrix representations of
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\begin_inset Formula $G$
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\end_inset
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consisting of matrices
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\begin_inset Formula $D^{\Gamma_{n}}\left(g\right)$
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\end_inset
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.
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Then the projection operators
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\begin_inset Formula
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\[
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P_{kl}^{\left(\Gamma_{n}\right)}\equiv\frac{d_{n}}{\left|G\right|}\sum_{g\in G}\left(D^{\Gamma_{n}}\left(g\right)\right)_{kl}^{*}J\left(g\right),\quad k,l=1,\dots,d_{n}
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\]
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\end_inset
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project the full scattering system field expansion coefficient vectors
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\begin_inset Formula $\rcoeff,\outcoeff$
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\end_inset
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onto a subspace corresponding to the irreducible representation
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\begin_inset Formula $\Gamma_{n}$
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\end_inset
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.
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The projectors can be used to construct a unitary transformation
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\begin_inset Formula $U$
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\end_inset
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with components
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\begin_inset Formula
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\begin{equation}
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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}
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\end{equation}
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\end_inset
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where
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\begin_inset Formula $r$
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\end_inset
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goes from
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\begin_inset Formula $1$
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\end_inset
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through
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\begin_inset Formula $d_{n}$
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\end_inset
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and
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\begin_inset Formula $i$
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\end_inset
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goes from 1 through the multiplicity of irreducible representation
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\begin_inset Formula $\Gamma_{n}$
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\end_inset
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in the (reducible) representation of
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\begin_inset Formula $G$
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\end_inset
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spanned by the field expansion coefficients
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\begin_inset Formula $\rcoeff$
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\end_inset
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or
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\begin_inset Formula $\outcoeff$
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\end_inset
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.
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The indices
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\begin_inset Formula $p',\tau',l',m'$
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\end_inset
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are given by an arbitrary bijective mapping
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\begin_inset Formula $\left(n,r,i\right)\mapsto\left(p',\tau',l',m'\right)$
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\end_inset
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with the constraint that for given
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\begin_inset Formula $n,r,i$
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\end_inset
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there are at least some non-zero elements
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\begin_inset Formula $U_{nri;p\tau lm}$
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\end_inset
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.
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For details, we refer the reader to textbooks about group representation
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theory
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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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or linear representations?
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\end_layout
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\end_inset
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, e.g.
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\begin_inset CommandInset citation
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LatexCommand cite
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after "Chapter 4"
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key "dresselhaus_group_2008"
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literal "false"
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\end_inset
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or
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\begin_inset CommandInset citation
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LatexCommand cite
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after "???"
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key "bradley_mathematical_1972"
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literal "false"
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\end_inset
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.
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The transformation given by
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\begin_inset Formula $U$
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\end_inset
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transforms the excitation coefficient vectors
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\begin_inset Formula $\rcoeff,\outcoeff$
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\end_inset
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into a new,
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\emph on
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symmetry-adapted basis
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\emph default
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.
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\end_layout
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\begin_layout Standard
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One can show that if an operator
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\begin_inset Formula $M$
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\end_inset
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acting on the excitation coefficient vectors is invariant under the operations
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of group
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\begin_inset Formula $G$
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\end_inset
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, meaning that
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\begin_inset Formula
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\[
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\forall g\in G:J\left(g\right)MJ\left(g\right)^{\dagger}=M,
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\]
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\end_inset
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then in the symmetry-adapted basis,
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\begin_inset Formula $M$
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\end_inset
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is block diagonal, or more specifically
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\begin_inset Formula
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\[
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M_{\Gamma,r,i;\Gamma',r',j}^{\mathrm{s.a.b.}}=\frac{\delta_{\Gamma\Gamma'}\delta_{ij}}{d_{\Gamma}}\sum_{q}M{}_{\Gamma,r,q;\Gamma',r',q}^{\mathrm{s.a.b.}}.
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\]
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\end_inset
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Both the
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\begin_inset Formula $T$
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\end_inset
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and
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\begin_inset Formula $\trops$
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\end_inset
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operators (and trivially also the identity
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\begin_inset Formula $I$
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\end_inset
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) in
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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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are invariant under the actions of whole system symmetry group, so
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\begin_inset Formula $\left(I-T\trops\right)$
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\end_inset
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is also invariant, hence
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\begin_inset Formula $U\left(I-T\trops\right)U^{\dagger}$
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\end_inset
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is a block-diagonal matrix, and the 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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can be solved for each block separately.
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\end_layout
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\begin_layout Standard
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From the computational perspective, it is important to note that
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\begin_inset Formula $U$
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\end_inset
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is at least as sparse as
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\begin_inset Formula $J\left(g\right)$
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\end_inset
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(which is
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\begin_inset Quotes eld
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\end_inset
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orbit-block
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\begin_inset Quotes erd
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\end_inset
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diagonal), hence the block-diagonalisation can be performed fast.
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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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Kvantifikovat!
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\end_layout
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\end_inset
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\end_layout
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