Kagome Lattice

Preamble

using LatticeTools
using LinearAlgebra
using Printf
using Plots

simplifyname(s::AbstractString) = (
    s
    |> (x-> replace(x, r"<sup>(.+?)</sup>" => s"\1"))
    |> (x-> replace(x, r"<sub>(.+?)</sub>" => s"[\1]"))
)
mkpath("example_kagome_symmetry")
extent = [-2, 2, -2, 2]
within(r) = (extent[1] <= r[1] <= extent[2] && extent[3] <= r[2] <= extent[4])

within (generic function with 1 method)

Generate Kagome Lattice

function make_kagome_lattice(size_matrix ::AbstractMatrix{<:Integer})
    latticevectors = [1 -0.5; 0 0.5*sqrt(3.0)];
    unitcell = makeunitcell(latticevectors, SiteType=String)
    addsite!(unitcell, "A", carte2fract(unitcell, [0.5, 0.0]))
    addsite!(unitcell, "B", carte2fract(unitcell, [0.25, 0.25*sqrt(3.0)]))
    addsite!(unitcell, "C", carte2fract(unitcell, [0.5+0.25, 0.25*sqrt(3.0)]))

    nnbondtypes = [
        ([ 0, 0], "A", [ 0, 0], "B", 1),
        ([ 0, 0], "A", [ 0, 0], "C", 1),
        ([ 0, 0], "B", [ 0, 0], "C", 1),
        ([ 1, 1], "A", [ 1, 0], "B",-1),
        ([ 1, 0], "B", [ 0, 0], "C",-1),
        ([ 0, 0], "C", [ 1, 1], "A",-1),
    ]

    nnnbondtypes = [
        ([ 0, 0], "A", [ 1, 0], "B", 1), # ◁
        ([ 1, 0], "B", [ 0,-1], "C", 1),
        ([ 0,-1], "C", [ 0, 0], "A", 1),
        ([ 0, 0], "C", [ 1, 0], "A",-1),
        ([ 1, 0], "A", [ 0,-1], "B",-1), # ▷
        ([ 0,-1], "B", [ 0, 0], "C",-1),
    ]

    lattice = makelattice(unitcell, size_matrix)
    hypercube = lattice.hypercube
    supercell = lattice.supercell
    tsym = TranslationSymmetry(lattice)
    psym = little_symmetry(tsym, project(PointSymmetryDatabase.get(25), [1 0 0; 0 1 0]))
    tsymbed = embed(lattice, tsym)
    psymbed = embed(lattice, psym)
    ssymbed = tsymbed ⋊ psymbed

    nnbonds = []
    nnnbonds = []

    for r in lattice.bravais_coordinates
        for (rowvec, roworb, colvec, colorb, bondsign) in nnbondtypes
            R_row, r_row = hypercube.wrap(r .+ rowvec)
            R_col, r_col = hypercube.wrap(r .+ colvec)
            roworb_super = (roworb, r_row)
            colorb_super = (colorb, r_col)
            irow = get(supercell.siteindices, roworb_super, -1)
            icol = get(supercell.siteindices, colorb_super, -1)
            push!(nnbonds, ((irow, icol), R_col-R_row, bondsign))
        end
        for (rowvec, roworb, colvec, colorb, bondsign) in nnnbondtypes
            R_row, r_row = hypercube.wrap(r .+ rowvec)
            R_col, r_col = hypercube.wrap(r .+ colvec)
            roworb_super = (roworb, r_row)
            colorb_super = (colorb, r_col)
            irow = get(supercell.siteindices, roworb_super, -1)
            icol = get(supercell.siteindices, colorb_super, -1)
            push!(nnnbonds, ((irow, icol), R_col-R_row, bondsign))
        end
    end

    nn_triangles = []
    for r in lattice.bravais_coordinates
      triangle = []
      for (rowvec, roworb, colvec, colorb, bondsign) in nnbondtypes[1:3]
        R_row, r_row = hypercube.wrap(r .+ rowvec)
        R_col, r_col = hypercube.wrap(r .+ colvec)
        roworb_super = (roworb, r_row)
        colorb_super = (colorb, r_col)
        irow = get(supercell.siteindices, roworb_super, -1)
        icol = get(supercell.siteindices, colorb_super, -1)
        push!(triangle, ((irow, icol), R_col-R_row))
      end
      push!(nn_triangles, (triangle, 1))

      triangle = []
      for (rowvec, roworb, colvec, colorb, bondsign) in nnbondtypes[4:6]
        R_row, r_row = hypercube.wrap(r .+ rowvec)
        R_col, r_col = hypercube.wrap(r .+ colvec)
        roworb_super = (roworb, r_row)
        colorb_super = (colorb, r_col)
        irow = get(supercell.siteindices, roworb_super, -1)
        icol = get(supercell.siteindices, colorb_super, -1)
        push!(triangle, ((irow, icol), R_col-R_row))
      end
      push!(nn_triangles, (triangle, -1))
    end

    return (unitcell=unitcell,
            lattice=lattice,
            space_symmetry_embedding=ssymbed,
            nearest_neighbor_bonds=nnbonds,
            next_nearest_neighbor_bonds=nnnbonds,
            nearest_neighbor_triangles=nn_triangles)
end

make_kagome_lattice (generic function with 1 method)

Symmetry Analysis

kagome = make_kagome_lattice([4 -2; 2 2])

tsymbed = kagome.space_symmetry_embedding.normal
psymbed = kagome.space_symmetry_embedding.rest
tsym = symmetry(tsymbed)
psym = symmetry(psymbed)

println("Point Symmetry")
println("==============")
println()
println("Point group: ", psym.hermann_mauguin)
println("Number of elements: ", group_order(psym))
println("Number of irreps: ", num_irreps(psym))

Point Symmetry
==============

Point group: 6mm
Number of elements: 12
Number of irreps: 6

Site Map

site_map = findsitemap(kagome.lattice.unitcell, psym)

println("Site map")
println("-----------")
println()

for (n, map) in zip(element_names(psym), site_map)
    @printf("%32s:", n)
    for (i_elem, (j_elem, R)) in enumerate(map)
        @printf("  %d ↦ %d, %-8s", i_elem, j_elem, string(R))
    end
    println()
end
println()

Site map
-----------

                               1:  1 ↦ 1, [0, 0]    2 ↦ 2, [0, 0]    3 ↦ 3, [0, 0]
     3<sup>+</sup><sub>001</sub>:  1 ↦ 3, [-1, 0]   2 ↦ 1, [-1, 0]   3 ↦ 2, [-1, 0]
     3<sup>-</sup><sub>001</sub>:  1 ↦ 2, [-1, -1]  2 ↦ 3, [-1, -1]  3 ↦ 1, [-1, -1]
                 2<sub>001</sub>:  1 ↦ 1, [-1, 0]   2 ↦ 2, [-1, -1]  3 ↦ 3, [-2, -1]
     6<sup>-</sup><sub>001</sub>:  1 ↦ 3, [-1, -1]  2 ↦ 1, [0, 0]    3 ↦ 2, [0, -1]
     6<sup>+</sup><sub>001</sub>:  1 ↦ 2, [0, 0]    2 ↦ 3, [-1, 0]   3 ↦ 1, [0, 1]
                 m<sub>110</sub>:  1 ↦ 3, [-1, -1]  2 ↦ 2, [-1, -1]  3 ↦ 1, [-1, -1]
                 m<sub>100</sub>:  1 ↦ 1, [-1, 0]   2 ↦ 3, [-1, 0]   3 ↦ 2, [-1, 0]
                 m<sub>010</sub>:  1 ↦ 2, [0, 0]    2 ↦ 1, [0, 0]    3 ↦ 3, [0, 0]
                m<sub>1-10</sub>:  1 ↦ 3, [-1, 0]   2 ↦ 2, [0, 0]    3 ↦ 1, [0, 1]
                 m<sub>120</sub>:  1 ↦ 1, [0, 0]    2 ↦ 3, [-1, -1]  3 ↦ 2, [0, -1]
                 m<sub>210</sub>:  1 ↦ 2, [-1, -1]  2 ↦ 1, [-1, 0]   3 ↦ 3, [-2, -1]

Plot Translation Symmetry

for (i_elem, perm) in enumerate(elements(tsymbed))
    elname = element_name(tsym, i_elem)
    fig = plot(title=elname, aspect=1, size=(200, 200), grid=false, titlefont=Plots.font("sans-serif", pointsize=8))
    orbcoords = []
    orbnames = []

    for iorb in eachindex(kagome.lattice.supercell.sites)
        orbfc = getsitecoord(kagome.lattice.supercell, perm(iorb))
        orbcc = fract2carte(kagome.lattice.supercell, orbfc)
        push!(orbnames, "$iorb")
        push!(orbcoords, orbcc)
    end
    orbcoords = hcat(orbcoords...)
    L = kagome.lattice.supercell.latticevectors
    for R1 in -2:2, R2 in -2:2
        R = [R1, R2]
        LR = L * R
        idx_filt = [i for (i, r) in enumerate(eachcol(orbcoords)) if within(r .+ LR)]
        scatter!(orbcoords[1,idx_filt] .+ LR[1],
                 orbcoords[2,idx_filt] .+ LR[2],
                 color="blue",
                 markerstrokecolor="blue",
                 series_annotations=[Plots.text(x, 6, :left, :bottom) for x in orbnames[idx_filt]], label=nothing)
    end
    xlims!(extent[1], extent[2])
    ylims!(extent[3], extent[4])
    savefig(fig, "example_kagome_symmetry/translation_symmetry-$i_elem.svg")
end

/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS

Plot Point Symmetry

for (i_elem, perm) in enumerate(elements(psymbed))
    elname = element_name(psym, i_elem)
    fig = plot(title=simplifyname(elname), aspect=1, size=(200, 200), grid=false, titlefont=Plots.font("sans-serif", pointsize=8))
    orbcoords = []
    orbnames = []

    for iorb in eachindex(kagome.lattice.supercell.sites)
        orbfc = getsitecoord(kagome.lattice.supercell, perm(iorb))
        orbcc = fract2carte(kagome.lattice.supercell, orbfc)
        push!(orbnames, "$iorb")
        push!(orbcoords, orbcc)
    end
    orbcoords = hcat(orbcoords...)
    L = kagome.lattice.supercell.latticevectors
    for R1 in -2:2, R2 in -2:2
        R = [R1, R2]
        LR = L * R
        idx_filt = [i for (i, r) in enumerate(eachcol(orbcoords)) if within(r .+ LR)]
        scatter!(orbcoords[1,idx_filt] .+ LR[1],
                 orbcoords[2,idx_filt] .+ LR[2],
                 color="blue",
                 markerstrokecolor="blue",
                 series_annotations=[Plots.text(x, 6, :left, :bottom) for x in orbnames[idx_filt]], label=nothing)
    end
    xlims!(extent[1], extent[2])
    ylims!(extent[3], extent[4])
    savefig(fig, "example_kagome_symmetry/point_symmetry-$i_elem.svg")
end

/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/RlE5Y/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS

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