{
"cells": [
{
"cell_type": "markdown",
"source": [
"# Gross-Pitaevskii equation in one dimension\n",
"In this example we will use DFTK to solve\n",
"the Gross-Pitaevskii equation, and use this opportunity to explore a few internals."
],
"metadata": {}
},
{
"cell_type": "markdown",
"source": [
"## The model\n",
"The [Gross-Pitaevskii equation](https://en.wikipedia.org/wiki/Gross%E2%80%93Pitaevskii_equation) (GPE)\n",
"is a simple non-linear equation used to model bosonic systems\n",
"in a mean-field approach. Denoting by ``ψ`` the effective one-particle bosonic\n",
"wave function, the time-independent GPE reads in atomic units:\n",
"$$\n",
" H ψ = \\left(-\\frac12 Δ + V + 2 C |ψ|^2\\right) ψ = μ ψ \\qquad \\|ψ\\|_{L^2} = 1\n",
"$$\n",
"where ``C`` provides the strength of the boson-boson coupling.\n",
"It's in particular a favorite model of applied mathematicians because it\n",
"has a structure simpler than but similar to that of DFT, and displays\n",
"interesting behavior (especially in higher dimensions with magnetic fields, see\n",
"Gross-Pitaevskii equation with external magnetic field)."
],
"metadata": {}
},
{
"cell_type": "markdown",
"source": [
"We wish to model this equation in 1D using DFTK.\n",
"First we set up the lattice. For a 1D case we supply two zero lattice vectors,"
],
"metadata": {}
},
{
"outputs": [],
"cell_type": "code",
"source": [
"a = 10\n",
"lattice = a .* [[1 0 0.]; [0 0 0]; [0 0 0]];"
],
"metadata": {},
"execution_count": 1
},
{
"cell_type": "markdown",
"source": [
"which is special cased in DFTK to support 1D models.\n",
"\n",
"For the potential term `V` we just pick a harmonic\n",
"potential. The real-space grid is in ``[0,1)``\n",
"in fractional coordinates( see\n",
"Lattices and lattice vectors),\n",
"therefore:"
],
"metadata": {}
},
{
"outputs": [],
"cell_type": "code",
"source": [
"pot(x) = (x - a/2)^2;"
],
"metadata": {},
"execution_count": 2
},
{
"cell_type": "markdown",
"source": [
"We setup each energy term in sequence: kinetic, potential and nonlinear term.\n",
"For the non-linearity we use the `PowerNonlinearity(C, α)` term of DFTK.\n",
"This object introduces an energy term ``C ∫ ρ(r)^α dr``\n",
"to the total energy functional, thus a potential term ``α C ρ^{α-1}``.\n",
"In our case we thus need the parameters"
],
"metadata": {}
},
{
"outputs": [],
"cell_type": "code",
"source": [
"C = 1.0\n",
"α = 2;"
],
"metadata": {},
"execution_count": 3
},
{
"cell_type": "markdown",
"source": [
"... and with this build the model"
],
"metadata": {}
},
{
"outputs": [],
"cell_type": "code",
"source": [
"using DFTK\n",
"using LinearAlgebra\n",
"\n",
"n_electrons = 1 # Increase this for fun\n",
"terms = [Kinetic(),\n",
" ExternalFromReal(r -> pot(r[1])),\n",
" PowerNonlinearity(C, α),\n",
"]\n",
"model = Model(lattice; n_electrons=n_electrons, terms=terms,\n",
" spin_polarization=:spinless); # use \"spinless electrons\""
],
"metadata": {},
"execution_count": 4
},
{
"cell_type": "markdown",
"source": [
"We discretize using a moderate Ecut (For 1D values up to `5000` are completely fine)\n",
"and run a direct minimization algorithm:"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Iter Function value Gradient norm \n",
" 0 2.015684e+02 1.451541e+02\n",
" * time: 0.0006830692291259766\n",
" 1 1.842670e+02 1.117706e+02\n",
" * time: 0.0014410018920898438\n",
" 2 1.381893e+02 1.274119e+02\n",
" * time: 0.003370046615600586\n",
" 3 3.906053e+01 5.829388e+01\n",
" * time: 0.005837202072143555\n",
" 4 1.500294e+01 1.056158e+01\n",
" * time: 0.00799107551574707\n",
" 5 1.395365e+01 1.053121e+01\n",
" * time: 0.008944034576416016\n",
" 6 8.274507e+00 2.477097e+01\n",
" * time: 0.010503053665161133\n",
" 7 8.073489e+00 1.705653e+01\n",
" * time: 0.011342048645019531\n",
" 8 5.723847e+00 9.970609e+00\n",
" * time: 0.012673139572143555\n",
" 9 3.609341e+00 4.035644e+00\n",
" * time: 0.013773202896118164\n",
" 10 2.120041e+00 2.933524e+00\n",
" * time: 0.015034198760986328\n",
" 11 1.694492e+00 2.385546e+00\n",
" * time: 0.01613020896911621\n",
" 12 1.354819e+00 9.856240e-01\n",
" * time: 0.01720118522644043\n",
" 13 1.307600e+00 1.092034e+00\n",
" * time: 0.01797008514404297\n",
" 14 1.258360e+00 8.472756e-01\n",
" * time: 0.019118070602416992\n",
" 15 1.204909e+00 6.893778e-01\n",
" * time: 0.02001500129699707\n",
" 16 1.174138e+00 5.458289e-01\n",
" * time: 0.020847082138061523\n",
" 17 1.156438e+00 3.387400e-01\n",
" * time: 0.021685123443603516\n",
" 18 1.148475e+00 2.537470e-01\n",
" * time: 0.022644996643066406\n",
" 19 1.145530e+00 7.210905e-02\n",
" * time: 0.023485183715820312\n",
" 20 1.144664e+00 7.525648e-02\n",
" * time: 0.024079084396362305\n",
" 21 1.144361e+00 3.798331e-02\n",
" * time: 0.02500009536743164\n",
" 22 1.144250e+00 4.237083e-02\n",
" * time: 0.025930166244506836\n",
" 23 1.144154e+00 3.860363e-02\n",
" * time: 0.02681708335876465\n",
" 24 1.144097e+00 2.468123e-02\n",
" * time: 0.027675151824951172\n",
" 25 1.144055e+00 1.086695e-02\n",
" * time: 0.02866816520690918\n",
" 26 1.144046e+00 6.443505e-03\n",
" * time: 0.02957010269165039\n",
" 27 1.144040e+00 4.035874e-03\n",
" * time: 0.030518054962158203\n",
" 28 1.144038e+00 2.810783e-03\n",
" * time: 0.03145003318786621\n",
" 29 1.144037e+00 1.629475e-03\n",
" * time: 0.03235316276550293\n",
" 30 1.144037e+00 1.123801e-03\n",
" * time: 0.03327512741088867\n",
" 31 1.144037e+00 1.267942e-03\n",
" * time: 0.03391313552856445\n",
" 32 1.144037e+00 3.290930e-04\n",
" * time: 0.0345911979675293\n",
" 33 1.144037e+00 4.760476e-04\n",
" * time: 0.0352480411529541\n",
" 34 1.144037e+00 4.453289e-04\n",
" * time: 0.0362551212310791\n",
" 35 1.144037e+00 2.168422e-04\n",
" * time: 0.037188053131103516\n",
" 36 1.144037e+00 1.035559e-04\n",
" * time: 0.03815603256225586\n",
" 37 1.144037e+00 1.121138e-04\n",
" * time: 0.04375410079956055\n",
" 38 1.144037e+00 6.797317e-05\n",
" * time: 0.045060157775878906\n",
" 39 1.144037e+00 4.443128e-05\n",
" * time: 0.04605722427368164\n",
" 40 1.144037e+00 2.640349e-05\n",
" * time: 0.047057151794433594\n",
" 41 1.144037e+00 1.657018e-05\n",
" * time: 0.04814505577087402\n",
" 42 1.144037e+00 1.428427e-05\n",
" * time: 0.049221038818359375\n",
" 43 1.144037e+00 6.915570e-06\n",
" * time: 0.05090022087097168\n",
" 44 1.144037e+00 2.914267e-06\n",
" * time: 0.052781105041503906\n",
" 45 1.144037e+00 2.814019e-06\n",
" * time: 0.057882070541381836\n",
" 46 1.144037e+00 1.664617e-06\n",
" * time: 0.05901813507080078\n",
" 47 1.144037e+00 7.862169e-07\n",
" * time: 0.06042814254760742\n",
" 48 1.144037e+00 6.914599e-07\n",
" * time: 0.06145811080932617\n",
" 49 1.144037e+00 6.295958e-07\n",
" * time: 0.062177181243896484\n",
" 50 1.144037e+00 3.682016e-07\n",
" * time: 0.06310915946960449\n",
" 51 1.144037e+00 1.755326e-07\n",
" * time: 0.06403803825378418\n",
" 52 1.144037e+00 8.182556e-08\n",
" * time: 0.1143181324005127\n",
" 53 1.144037e+00 7.314198e-08\n",
" * time: 0.11549711227416992\n"
]
},
{
"output_type": "execute_result",
"data": {
"text/plain": "Energy breakdown (in Ha):\n Kinetic 0.2682057 \n ExternalFromReal 0.4707475 \n PowerNonlinearity 0.4050836 \n\n total 1.144036852755 "
},
"metadata": {},
"execution_count": 5
}
],
"cell_type": "code",
"source": [
"basis = PlaneWaveBasis(model, Ecut=500, kgrid=(1, 1, 1))\n",
"scfres = direct_minimization(basis, tol=1e-8) # This is a constrained preconditioned LBFGS\n",
"scfres.energies"
],
"metadata": {},
"execution_count": 5
},
{
"cell_type": "markdown",
"source": [
"## Internals\n",
"We use the opportunity to explore some of DFTK internals.\n",
"\n",
"Extract the converged density and the obtained wave function:"
],
"metadata": {}
},
{
"outputs": [],
"cell_type": "code",
"source": [
"ρ = real(scfres.ρ)[:, 1, 1, 1] # converged density, first spin component\n",
"ψ_fourier = scfres.ψ[1][:, 1]; # first k-point, all G components, first eigenvector"
],
"metadata": {},
"execution_count": 6
},
{
"cell_type": "markdown",
"source": [
"Transform the wave function to real space and fix the phase:"
],
"metadata": {}
},
{
"outputs": [],
"cell_type": "code",
"source": [
"ψ = G_to_r(basis, basis.kpoints[1], ψ_fourier)[:, 1, 1]\n",
"ψ /= (ψ[div(end, 2)] / abs(ψ[div(end, 2)]));"
],
"metadata": {},
"execution_count": 7
},
{
"cell_type": "markdown",
"source": [
"Check whether ``ψ`` is normalised:"
],
"metadata": {}
},
{
"outputs": [],
"cell_type": "code",
"source": [
"x = a * vec(first.(DFTK.r_vectors(basis)))\n",
"N = length(x)\n",
"dx = a / N # real-space grid spacing\n",
"@assert sum(abs2.(ψ)) * dx ≈ 1.0"
],
"metadata": {},
"execution_count": 8
},
{
"cell_type": "markdown",
"source": [
"The density is simply built from ψ:"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "1.1177931243345485e-15"
},
"metadata": {},
"execution_count": 9
}
],
"cell_type": "code",
"source": [
"norm(scfres.ρ - abs2.(ψ))"
],
"metadata": {},
"execution_count": 9
},
{
"cell_type": "markdown",
"source": [
"We summarize the ground state in a nice plot:"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "Plot{Plots.GRBackend() n=3}",
"image/png": 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]
},
"metadata": {},
"execution_count": 10
}
],
"cell_type": "code",
"source": [
"using Plots\n",
"\n",
"p = plot(x, real.(ψ), label=\"real(ψ)\")\n",
"plot!(p, x, imag.(ψ), label=\"imag(ψ)\")\n",
"plot!(p, x, ρ, label=\"ρ\")"
],
"metadata": {},
"execution_count": 10
},
{
"cell_type": "markdown",
"source": [
"The `energy_hamiltonian` function can be used to get the energy and\n",
"effective Hamiltonian (derivative of the energy with respect to the density matrix)\n",
"of a particular state (ψ, occupation).\n",
"The density ρ associated to this state is precomputed\n",
"and passed to the routine as an optimization."
],
"metadata": {}
},
{
"outputs": [],
"cell_type": "code",
"source": [
"E, ham = energy_hamiltonian(basis, scfres.ψ, scfres.occupation; ρ=scfres.ρ)\n",
"@assert E.total == scfres.energies.total"
],
"metadata": {},
"execution_count": 11
},
{
"cell_type": "markdown",
"source": [
"Now the Hamiltonian contains all the blocks corresponding to k-points. Here, we just have one k-point:"
],
"metadata": {}
},
{
"outputs": [],
"cell_type": "code",
"source": [
"H = ham.blocks[1];"
],
"metadata": {},
"execution_count": 12
},
{
"cell_type": "markdown",
"source": [
"`H` can be used as a linear operator (efficiently using FFTs), or converted to a dense matrix:"
],
"metadata": {}
},
{
"outputs": [],
"cell_type": "code",
"source": [
"ψ11 = scfres.ψ[1][:, 1] # first k-point, first eigenvector\n",
"Hmat = Array(H) # This is now just a plain Julia matrix,\n",
"# which we can compute and store in this simple 1D example\n",
"@assert norm(Hmat * ψ11 - H * ψ11) < 1e-10"
],
"metadata": {},
"execution_count": 13
},
{
"cell_type": "markdown",
"source": [
"Let's check that ψ11 is indeed an eigenstate:"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "1.5505053735266486e-7"
},
"metadata": {},
"execution_count": 14
}
],
"cell_type": "code",
"source": [
"norm(H * ψ11 - dot(ψ11, H * ψ11) * ψ11)"
],
"metadata": {},
"execution_count": 14
},
{
"cell_type": "markdown",
"source": [
"Build a finite-differences version of the GPE operator ``H``, as a sanity check:"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "0.0002234216204111667"
},
"metadata": {},
"execution_count": 15
}
],
"cell_type": "code",
"source": [
"A = Array(Tridiagonal(-ones(N - 1), 2ones(N), -ones(N - 1)))\n",
"A[1, end] = A[end, 1] = -1\n",
"K = A / dx^2 / 2\n",
"V = Diagonal(pot.(x) + C .* α .* (ρ.^(α-1)))\n",
"H_findiff = K + V;\n",
"maximum(abs.(H_findiff*ψ - (dot(ψ, H_findiff*ψ) / dot(ψ, ψ)) * ψ))"
],
"metadata": {},
"execution_count": 15
}
],
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"metadata": {
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.7.0"
},
"kernelspec": {
"name": "julia-1.7",
"display_name": "Julia 1.7.0",
"language": "julia"
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