Symmetries in periodic systems.
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Marek Nečada
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89b8943cf0f4c08a1ad487d561eebd971c2a721f
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@ -1252,6 +1252,148 @@ better formulation
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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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\align center
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\begin_inset Graphics
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filename figs/hex/p6m_group_actions.pdf
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width 95text%
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\end_inset
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\end_layout
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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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Representing symmetry action on electromagnetic Bloch waves in a lattice
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with
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\begin_inset Formula $p6m$
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\end_inset
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wallpaper group symmetry.
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In a hexagonal array with five particles (labeled
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\begin_inset Formula $A$
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\end_inset
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–
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\begin_inset Formula $E$
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\end_inset
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) per unit cell, we first choose into which unit cells do the particles
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on unit cell boundaries belong.
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a) At
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\begin_inset Formula $M$
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\end_inset
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point, the little co-group contains a
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\begin_inset Formula $D_{2}$
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\end_inset
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point group; the unit cells can be divided into two groups (alternating
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horizontal rows) with opposite sign.
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The horizontal mirror operation
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\begin_inset Formula $\sigma_{xz}$
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\end_inset
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maps the particles from a single unit cell to each other.
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However, the vertical mirror operation
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\begin_inset Formula $\sigma_{yz}$
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\end_inset
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maps them onto particles belonging to different unit cells, introducing
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possible phase factors in the point group action:
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\begin_inset Formula $B,C,D$
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\end_inset
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map onto
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\begin_inset Formula $B,C,D$
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\end_inset
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belonging to different unit cell from same phase group, so no additional
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phase is needed; however,
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\begin_inset Formula $A,E$
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\end_inset
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map onto
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\begin_inset Formula $E,A$
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\end_inset
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belonging to unitcells with relative phases
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\begin_inset Formula $\pm\pi$
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\end_inset
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, therefore the corresponding action matrix blocks will carry a factor
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\begin_inset Formula $-1$
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\end_inset
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.
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b) At
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\begin_inset Formula $K$
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\end_inset
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point point, little co-group contains a
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\begin_inset Formula $D_{3}$
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\end_inset
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point group, and the unit cells divide into three groups with relative
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phase shift
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\begin_inset Formula $e^{2\pi i/3}$
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\end_inset
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.
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The horizontal mirroring
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\begin_inset Formula $\sigma_{xz}$
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\end_inset
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again does not introduce any additional phase.
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However, the
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\begin_inset Formula $C_{3}$
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\end_inset
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rotation mixes particles belonging to different unit cells, so for example
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particle
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\begin_inset Formula $D$
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\end_inset
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maps onto particle
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\begin_inset Formula $D$
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\end_inset
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, but with additional phase factor
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\begin_inset Formula $e^{-2\pi i/3}$
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\end_inset
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.
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\begin_inset CommandInset label
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LatexCommand label
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name "Phase factor illustration"
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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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@ -1462,7 +1604,18 @@ the original
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\begin_inset Formula $C_{3}$
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\end_inset
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rotation, as an example we have
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rotation as an example we have (see Fig.
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "Phase factor illustration"
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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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b)
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\begin_inset Formula
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\begin{align*}
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\outcoeffp{\vect 0A} & \overset{C_{3}}{\longmapsto}\tilde{J}\left(C_{3}\right)\outcoeffp{\left(0,-1\right)E}=e^{2\pi i/3}\tilde{J}\left(C_{3}\right)\outcoeffp{\vect 0E},\\
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@ -1513,75 +1666,6 @@ literal "false"
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.
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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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\align center
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\begin_inset Graphics
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filename p6m_mpoint.png
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lyxscale 20
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width 100col%
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\align center
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\begin_inset Graphics
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filename p6m_kpoint.png
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lyxscale 20
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width 100col%
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\end_inset
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\end_layout
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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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Representing symmetry action on electromagnetic Bloch waves in a lattice
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with
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\begin_inset Formula $p6m$
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\end_inset
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wallpaper group symmetry for
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\begin_inset Formula $M$
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\end_inset
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(top) and
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\begin_inset Formula $K$
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\end_inset
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(bottom) points.
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\begin_inset CommandInset label
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LatexCommand label
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name "Phase factor illustration"
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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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