{ "cells": [ { "cell_type": "markdown", "source": [ "# Gross-Pitaevskii equation with magnetism" ], "metadata": {} }, { "cell_type": "markdown", "source": [ "We solve the 2D Gross-Pitaevskii equation with a magnetic field.\n", "This is similar to the\n", "previous example (Gross-Pitaevskii equation in one dimension),\n", "but with an extra term for the magnetic field.\n", "We reproduce here the results of https://arxiv.org/pdf/1611.02045.pdf Fig. 10" ], "metadata": {} }, { "outputs": [], "cell_type": "code", "source": [ "using DFTK\n", "using StaticArrays\n", "using Plots" ], "metadata": {}, "execution_count": 1 }, { "cell_type": "markdown", "source": [ "Unit cell. Having one of the lattice vectors as zero means a 2D system" ], "metadata": {} }, { "outputs": [], "cell_type": "code", "source": [ "a = 15\n", "lattice = a .* [[1 0 0.]; [0 1 0]; [0 0 0]];" ], "metadata": {}, "execution_count": 2 }, { "cell_type": "markdown", "source": [ "Confining scalar potential, and magnetic vector potential" ], "metadata": {} }, { "outputs": [], "cell_type": "code", "source": [ "pot(x, y, z) = ((x - a/2)^2 + (y - a/2)^2)/2\n", "ω = .6\n", "Apot(x, y, z) = ω * @SVector [y - a/2, -(x - a/2), 0]\n", "Apot(X) = Apot(X...);" ], "metadata": {}, "execution_count": 3 }, { "cell_type": "markdown", "source": [ "Parameters" ], "metadata": {} }, { "outputs": [], "cell_type": "code", "source": [ "Ecut = 20 # Increase this for production\n", "η = 500\n", "C = η/2\n", "α = 2\n", "n_electrons = 1; # Increase this for fun" ], "metadata": {}, "execution_count": 4 }, { "cell_type": "markdown", "source": [ "Collect all the terms, build and run the model" ], "metadata": {} }, { "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Iter Function value Gradient norm \n", " 0 3.076795e+01 7.503771e+00\n", " * time: 0.0020389556884765625\n", " 1 2.850236e+01 6.173472e+00\n", " * time: 0.008480072021484375\n", " 2 2.104104e+01 6.919747e+00\n", " * time: 0.015841007232666016\n", " 3 1.491597e+01 4.598344e+00\n", " * time: 0.023647069931030273\n", " 4 1.126953e+01 1.938035e+00\n", " * time: 0.030544042587280273\n", " 5 1.053462e+01 2.504776e+00\n", " * time: 0.03644394874572754\n", " 6 9.941404e+00 1.738964e+00\n", " * time: 0.0425260066986084\n", " 7 9.324278e+00 1.413210e+00\n", " * time: 0.048799991607666016\n", " 8 8.865733e+00 1.267031e+00\n", " * time: 0.05532193183898926\n", " 9 8.667788e+00 1.081462e+00\n", " * time: 0.061949968338012695\n", " 10 8.541607e+00 9.737555e-01\n", " * time: 0.06846785545349121\n", " 11 8.421264e+00 7.820865e-01\n", " * time: 0.07505106925964355\n", " 12 8.416931e+00 1.149503e+00\n", " * time: 0.11214494705200195\n", " 13 8.342651e+00 8.691841e-01\n", " * time: 0.11568093299865723\n", " 14 8.299733e+00 7.508414e-01\n", " * time: 0.11880302429199219\n", " 15 8.246963e+00 6.333104e-01\n", " * time: 0.12193489074707031\n", " 16 8.216101e+00 9.787482e-01\n", " * time: 0.12495803833007812\n", " 17 8.165582e+00 4.814081e-01\n", " * time: 0.12902498245239258\n", " 18 8.132928e+00 7.601908e-01\n", " * time: 0.13222789764404297\n", " 19 8.097300e+00 6.906450e-01\n", " * time: 0.1353929042816162\n", " 20 8.072718e+00 5.018843e-01\n", " * time: 0.13840985298156738\n", " 21 8.068286e+00 6.941872e-01\n", " * time: 0.14182806015014648\n", " 22 8.051294e+00 4.447927e-01\n", " * time: 0.14503693580627441\n", " 23 8.013217e+00 3.658953e-01\n", " * time: 0.1486189365386963\n", " 24 7.986649e+00 4.642554e-01\n", " * time: 0.1518239974975586\n", " 25 7.973411e+00 4.339732e-01\n", " * time: 0.15495800971984863\n", " 26 7.955554e+00 4.414018e-01\n", " * time: 0.15821003913879395\n", " 27 7.945662e+00 4.620689e-01\n", " * time: 0.16254901885986328\n", " 28 7.935328e+00 3.823380e-01\n", " * time: 0.1665658950805664\n", " 29 7.915427e+00 2.302696e-01\n", " * time: 0.17302489280700684\n", " 30 7.899154e+00 4.128918e-01\n", " * time: 0.17808103561401367\n", " 31 7.895051e+00 5.744434e-01\n", " * time: 0.1831209659576416\n", " 32 7.874350e+00 3.463292e-01\n", " * time: 0.18949103355407715\n", " 33 7.863409e+00 5.354438e-01\n", " * time: 0.1951148509979248\n", " 34 7.836191e+00 3.107415e-01\n", " * time: 0.20070600509643555\n", " 35 7.816445e+00 5.281816e-01\n", " * time: 0.20599102973937988\n", " 36 7.794423e+00 2.746795e-01\n", " * time: 0.21248602867126465\n", " 37 7.779161e+00 3.549642e-01\n", " * time: 0.21760201454162598\n", " 38 7.768131e+00 5.159801e-01\n", " * time: 0.22307085990905762\n", " 39 7.762015e+00 3.996403e-01\n", " * time: 0.22880196571350098\n", " 40 7.750701e+00 3.953044e-01\n", " * time: 0.2501869201660156\n", " 41 7.733739e+00 3.273122e-01\n", " * 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0.3733830451965332\n", " 71 7.621134e+00 7.483142e-02\n", " * time: 0.37730908393859863\n", " 72 7.620764e+00 6.928794e-02\n", " * time: 0.38079404830932617\n", " 73 7.620018e+00 7.769109e-02\n", " * time: 0.38504600524902344\n", " 74 7.619123e+00 3.989617e-02\n", " * time: 0.38902997970581055\n", " 75 7.618322e+00 4.010574e-02\n", " * time: 0.3929250240325928\n", " 76 7.617969e+00 6.326625e-02\n", " * time: 0.39818286895751953\n", " 77 7.617751e+00 1.106100e-01\n", " * time: 0.4016439914703369\n", " 78 7.616654e+00 5.653593e-02\n", " * time: 0.40582990646362305\n", " 79 7.615974e+00 9.941706e-02\n", " * time: 0.4090230464935303\n", " 80 7.614859e+00 6.690946e-02\n", " * time: 0.4134409427642822\n", " 81 7.613689e+00 6.148705e-02\n", " * time: 0.4168229103088379\n", " 82 7.612411e+00 5.938995e-02\n", " * time: 0.42017197608947754\n", " 83 7.611339e+00 4.599660e-02\n", " * time: 0.42407989501953125\n", " 84 7.610906e+00 9.858877e-02\n", " * time: 0.42763686180114746\n", " 85 7.610138e+00 7.441001e-02\n", " * time: 0.4307389259338379\n", " 86 7.608952e+00 1.010586e-01\n", " * time: 0.43398404121398926\n", " 87 7.607961e+00 8.436170e-02\n", " * time: 0.4378838539123535\n", " 88 7.607823e+00 1.221717e-01\n", " * time: 0.44114208221435547\n", " 89 7.607665e+00 1.406282e-01\n", " * time: 0.44571399688720703\n", " 90 7.606951e+00 9.656510e-02\n", " * time: 0.4487769603729248\n", " 91 7.605692e+00 9.165149e-02\n", " * time: 0.4527699947357178\n", " 92 7.605677e+00 1.172422e-01\n", " * time: 0.45705604553222656\n", " 93 7.604491e+00 7.218918e-02\n", " * time: 0.46202993392944336\n", " 94 7.604445e+00 8.763104e-02\n", " * time: 0.46704888343811035\n", " 95 7.603270e+00 7.334709e-02\n", " * time: 0.4817018508911133\n", " 96 7.602100e+00 6.818422e-02\n", " * time: 0.48502206802368164\n", " 97 7.601671e+00 8.317083e-02\n", " * time: 0.4884529113769531\n", " 98 7.600964e+00 5.071106e-02\n", " * time: 0.4915900230407715\n", " 99 7.600558e+00 3.375613e-02\n", " * time: 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n_electrons=n_electrons,\n", " terms=terms, spin_polarization=:spinless) # \"spinless electrons\"\n", "basis = PlaneWaveBasis(model, Ecut, kgrid=(1, 1, 1))\n", "scfres = direct_minimization(basis, tol=1e-5) # Reduce tol for production\n", "heatmap(scfres.ρ.real[:, :, 1], c=:blues)" ], "metadata": {}, "execution_count": 5 } ], "nbformat_minor": 3, "metadata": { "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.5.3" }, "kernelspec": { "name": "julia-1.5", "display_name": "Julia 1.5.3", "language": "julia" } }, "nbformat": 4 }