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shinaoka/semicircular.ipynb

Created January 23, 2023 09:07
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semicircular.ipynb
{
"cells": [
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"# Semicircular DOS\n",
"\n",
"Semicircular DOS ($t=1$):\n",
"\n",
"$$\n",
"D(\\epsilon) = \\frac{1}{2\\pi} \\sqrt{4 - \\epsilon^2}\n",
"$$\n",
"\n",
"Hilbert transform:\n",
"\n",
"$$\n",
"\\tilde D(\\xi) = \\int_{-\\infty}^\\infty \\frac{D(\\epsilon)}{\\xi - \\epsilon} = \\frac{\\xi - s \\sqrt{\\xi^2 - 4}}{2},\n",
"$$\n",
"\n",
"where $s$ is the sign of $\\mathrm{Im}~\\xi$."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"using Plots\n",
"using SparseIR\n",
"import SparseIR: valueim\n",
"import LinearAlgebra as LA"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\u001b[32m\u001b[1mStatus\u001b[22m\u001b[39m `~/.julia/environments/v1.8/Project.toml`\n",
" \u001b[90m [4fe2279e] \u001b[39mSparseIR v1.0.3\n"
]
}
],
"source": [
"import Pkg\n",
"Pkg.status(\"SparseIR\")"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"N = 100\n",
"β = 10.0\n",
"\n",
"ωs = FermionicFreq.(2 .* collect(-N:(N-1)) .+ 1)\n",
"zs = valueim.(ωs, β)\n",
";"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"hilbert (generic function with 1 method)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"hilbert(ξ) = 0.5 * (ξ - sign(imag(ξ)) * sqrt(ξ^2 - 4))"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
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},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Giω = hilbert.(zs)\n",
"plot(imag.(zs), imag.(Giω), marker=:x)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"## Fourier transform to τ (through IR basis)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
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},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ωmax = 10.0\n",
"basis = FiniteTempBasis(Fermionic(), β, ωmax)\n",
"smpl_matsu = MatsubaraSampling(basis)\n",
"\n",
"ωsmpl = smpl_matsu.sampling_points\n",
"gl = fit(smpl_matsu, hilbert.(valueim.(ωsmpl, β)))\n",
"\n",
"plot(abs.(gl), yaxis=:log)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
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},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"τs = collect(LinRange(0, β, 100))\n",
"smpl_τ = TauSampling(basis, τs)\n",
"\n",
"gτ = evaluate(smpl_τ, gl)\n",
"\n",
"plot(τs, real.(gτ))"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"# Evaluate G(τ) at a given τ\n",
"evalτ(basis, τ, gl) = LA.dot(basis.u(τ), gl)\n",
"\n",
"gτ2 = [evalτ(basis, τ, gl) for τ in τs]\n",
"\n",
"@assert gτ ≈ gτ2"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.8.4",
"language": "julia",
"name": "julia-1.8"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.8.4"
},
"orig_nbformat": 4
},
"nbformat": 4,
"nbformat_minor": 2
}
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