{ "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 3.077016e+01 7.099970e+00\n", " * time: 0.0017120838165283203\n", " 1 2.776218e+01 4.512457e+00\n", " * time: 0.0064890384674072266\n", " 2 1.935778e+01 4.836689e+00\n", " * time: 0.01180410385131836\n", " 3 1.343812e+01 3.448865e+00\n", " * time: 0.017058134078979492\n", " 4 1.157216e+01 3.099052e+00\n", " * time: 0.02149510383605957\n", " 5 1.066942e+01 2.204533e+00\n", " * time: 0.05390596389770508\n", " 6 9.501471e+00 1.473333e+00\n", " * time: 0.05660605430603027\n", " 7 8.777708e+00 1.091840e+00\n", " * time: 0.059127092361450195\n", " 8 8.440348e+00 7.715083e-01\n", " * time: 0.06161093711853027\n", " 9 8.403502e+00 1.330032e+00\n", " * time: 0.06386399269104004\n", " 10 8.303916e+00 1.135657e+00\n", " * time: 0.06653308868408203\n", " 11 8.271293e+00 1.160964e+00\n", " * time: 0.06931710243225098\n", " 12 8.137761e+00 6.803516e-01\n", " * time: 0.07297110557556152\n", " 13 8.048212e+00 6.851147e-01\n", " * time: 0.07725000381469727\n", " 14 8.004141e+00 7.643399e-01\n", " * time: 0.08101511001586914\n", " 15 7.962402e+00 4.475359e-01\n", " * time: 0.0842750072479248\n", " 16 7.929738e+00 5.183875e-01\n", " * time: 0.08750295639038086\n", " 17 7.902781e+00 3.430982e-01\n", " * time: 0.09162712097167969\n", " 18 7.871907e+00 2.653334e-01\n", " * time: 0.09636998176574707\n", " 19 7.846709e+00 4.816162e-01\n", " * time: 0.0997779369354248\n", " 20 7.821301e+00 3.078363e-01\n", " * time: 0.10334396362304688\n", " 21 7.797091e+00 4.357181e-01\n", " * time: 0.10680794715881348\n", " 22 7.788851e+00 4.507833e-01\n", " * time: 0.11038804054260254\n", " 23 7.767902e+00 2.896440e-01\n", " * time: 0.11412310600280762\n", " 24 7.748439e+00 4.247666e-01\n", " * time: 0.11757612228393555\n", " 25 7.744732e+00 4.381962e-01\n", " * time: 0.1211240291595459\n", " 26 7.734628e+00 3.889163e-01\n", " * time: 0.12470698356628418\n", " 27 7.711182e+00 2.092718e-01\n", " * time: 0.12864112854003906\n", " 28 7.699710e+00 3.116059e-01\n", " * time: 0.13201093673706055\n", " 29 7.686328e+00 2.396813e-01\n", " * time: 0.13657903671264648\n", " 30 7.675046e+00 2.855162e-01\n", " * time: 0.14107799530029297\n", " 31 7.663566e+00 1.737080e-01\n", " * time: 0.1461191177368164\n", " 32 7.655602e+00 2.191812e-01\n", " * time: 0.14963793754577637\n", " 33 7.649349e+00 1.262314e-01\n", " * time: 0.1541740894317627\n", " 34 7.647846e+00 1.985115e-01\n", " * time: 0.15772700309753418\n", " 35 7.645812e+00 1.488947e-01\n", " * time: 0.176285982131958\n", " 36 7.643513e+00 1.658047e-01\n", " * time: 0.17835712432861328\n", " 37 7.641865e+00 1.402109e-01\n", " * time: 0.1803750991821289\n", " 38 7.638447e+00 1.417783e-01\n", " * time: 0.1824660301208496\n", " 39 7.637949e+00 1.295858e-01\n", " * time: 0.18458199501037598\n", " 40 7.635803e+00 7.828126e-02\n", " * time: 0.18709206581115723\n", " 41 7.634192e+00 1.751718e-01\n", " * time: 0.18947505950927734\n", " 42 7.633979e+00 1.848441e-01\n", " * time: 0.19156098365783691\n", " 43 7.633042e+00 1.365978e-01\n", " * time: 0.193648099899292\n", " 44 7.631218e+00 1.103198e-01\n", " * time: 0.19582104682922363\n", " 45 7.628926e+00 6.951779e-02\n", " * time: 0.19847702980041504\n", " 46 7.628550e+00 1.140155e-01\n", " * time: 0.2005629539489746\n", " 47 7.628082e+00 1.216560e-01\n", " * time: 0.20261907577514648\n", " 48 7.626822e+00 8.612412e-02\n", " * time: 0.20560693740844727\n", " 49 7.625805e+00 9.981885e-02\n", " * time: 0.20760512351989746\n", " 50 7.624409e+00 4.687745e-02\n", " * time: 0.21011710166931152\n", " 51 7.623660e+00 7.903797e-02\n", " * time: 0.21327805519104004\n", " 52 7.622912e+00 5.746376e-02\n", " * time: 0.2176680564880371\n", " 53 7.621963e+00 4.712308e-02\n", " * time: 0.22241592407226562\n", " 54 7.621056e+00 2.792622e-02\n", " * time: 0.22679710388183594\n", " 55 7.620473e+00 4.247146e-02\n", " * time: 0.23029112815856934\n", " 56 7.620007e+00 4.527969e-02\n", " * time: 0.23382210731506348\n", " 57 7.619345e+00 4.876799e-02\n", " * time: 0.23890399932861328\n", " 58 7.618601e+00 3.379615e-02\n", " * time: 0.24353694915771484\n", " 59 7.617984e+00 3.279501e-02\n", " * time: 0.24718308448791504\n", " 60 7.617432e+00 3.803242e-02\n", " * time: 0.2507619857788086\n", " 61 7.616830e+00 3.485162e-02\n", " * time: 0.2542901039123535\n", " 62 7.616441e+00 3.949164e-02\n", " * time: 0.25768303871154785\n", " 63 7.615949e+00 5.935234e-02\n", " * time: 0.2600820064544678\n", " 64 7.615870e+00 6.170036e-02\n", " * time: 0.2635040283203125\n", " 65 7.615670e+00 5.540708e-02\n", " * time: 0.2667360305786133\n", " 66 7.615193e+00 4.567557e-02\n", " * time: 0.27041196823120117\n", " 67 7.614811e+00 4.141398e-02\n", " * time: 0.2846550941467285\n", " 68 7.614615e+00 3.737054e-02\n", " * time: 0.2867860794067383\n", " 69 7.614409e+00 4.051927e-02\n", " * time: 0.2888491153717041\n", " 70 7.614267e+00 4.847423e-02\n", " * time: 0.29106807708740234\n", " 71 7.614074e+00 3.389768e-02\n", " * time: 0.29305100440979004\n", " 72 7.613861e+00 4.345371e-02\n", " * time: 0.2950451374053955\n", " 73 7.613659e+00 3.658832e-02\n", " * time: 0.29713892936706543\n", " 74 7.613401e+00 3.278005e-02\n", " * time: 0.29962992668151855\n", " 75 7.613152e+00 2.286754e-02\n", " * time: 0.30224013328552246\n", " 76 7.612922e+00 2.451509e-02\n", " * time: 0.3047349452972412\n", " 77 7.612815e+00 3.533614e-02\n", " * time: 0.3068680763244629\n", " 78 7.612645e+00 4.091742e-02\n", " * time: 0.30910801887512207\n", " 79 7.612290e+00 2.756706e-02\n", " * time: 0.3118460178375244\n", " 80 7.612222e+00 4.471561e-02\n", " * time: 0.31412696838378906\n", " 81 7.611944e+00 3.815673e-02\n", " * time: 0.31670212745666504\n", " 82 7.611844e+00 5.912756e-02\n", " * time: 0.3186929225921631\n", " 83 7.611519e+00 5.506328e-02\n", " * time: 0.32071995735168457\n", " 84 7.611479e+00 5.425979e-02\n", " * time: 0.32273411750793457\n", " 85 7.611125e+00 4.706506e-02\n", " * time: 0.32487010955810547\n", " 86 7.610800e+00 3.669273e-02\n", " * time: 0.3270409107208252\n", " 87 7.610594e+00 5.225045e-02\n", " * time: 0.3293769359588623\n", " 88 7.610233e+00 2.602166e-02\n", " * time: 0.3322739601135254\n", " 89 7.610010e+00 3.228904e-02\n", " * time: 0.33500194549560547\n", " 90 7.609850e+00 3.296205e-02\n", " * time: 0.33707404136657715\n", " 91 7.609641e+00 2.189554e-02\n", " * time: 0.3397400379180908\n", " 92 7.609456e+00 2.068626e-02\n", " * time: 0.34183192253112793\n", " 93 7.609412e+00 3.659626e-02\n", " * time: 0.34400510787963867\n", " 94 7.609405e+00 4.413358e-02\n", " * time: 0.3465549945831299\n", " 95 7.609303e+00 4.463400e-02\n", " * time: 0.3487880229949951\n", " 96 7.609146e+00 2.616280e-02\n", " * time: 0.3507699966430664\n", " 97 7.608947e+00 3.418183e-02\n", " * time: 0.3527860641479492\n", " 98 7.608737e+00 1.961115e-02\n", " * time: 0.3562641143798828\n", " 99 7.608591e+00 2.283681e-02\n", " * time: 0.36591100692749023\n", " 100 7.608438e+00 1.791892e-02\n", " * time: 0.3681190013885498\n", " 101 7.608436e+00 3.356030e-02\n", " * time: 0.3700859546661377\n", " 102 7.608348e+00 2.633680e-02\n", " * time: 0.37213706970214844\n", " 103 7.608270e+00 1.926483e-02\n", " * time: 0.3746640682220459\n", " 104 7.608202e+00 3.104673e-02\n", " * time: 0.37706995010375977\n", " 105 7.608122e+00 2.446319e-02\n", " * time: 0.3798251152038574\n", " 106 7.608011e+00 1.843032e-02\n", " * time: 0.3823869228363037\n", " 107 7.607988e+00 2.051295e-02\n", " * time: 0.38446712493896484\n", " 108 7.607907e+00 1.743878e-02\n", " * time: 0.38643693923950195\n", " 109 7.607831e+00 1.615467e-02\n", " * time: 0.38853001594543457\n", " 110 7.607745e+00 1.930346e-02\n", " * time: 0.3907029628753662\n", " 111 7.607686e+00 2.055955e-02\n", " * time: 0.39313411712646484\n", " 112 7.607671e+00 3.002613e-02\n", " * time: 0.39538097381591797\n", " 113 7.607616e+00 1.660431e-02\n", " * time: 0.39817214012145996\n", " 114 7.607552e+00 2.128334e-02\n", " * time: 0.40025806427001953\n", " 115 7.607517e+00 3.261214e-02\n", " * time: 0.40230393409729004\n", " 116 7.607402e+00 2.078757e-02\n", " * time: 0.4043569564819336\n", " 117 7.607360e+00 3.935413e-02\n", " * time: 0.4063839912414551\n", " 118 7.607244e+00 3.024583e-02\n", " * time: 0.4095640182495117\n", " 119 7.607225e+00 3.510511e-02\n", " * time: 0.4121730327606201\n", " 120 7.607151e+00 2.509252e-02\n", " * time: 0.4148139953613281\n", " 121 7.607045e+00 3.170160e-02\n", " * time: 0.4173390865325928\n", " 122 7.606970e+00 2.794080e-02\n", " * time: 0.4197361469268799\n", " 123 7.606821e+00 1.501891e-02\n", " * time: 0.4225120544433594\n", " 124 7.606755e+00 2.582995e-02\n", " * time: 0.42508411407470703\n", " 125 7.606684e+00 2.400049e-02\n", " * time: 0.4277489185333252\n", " 126 7.606675e+00 3.718530e-02\n", " * time: 0.42980408668518066\n", " 127 7.606649e+00 4.930170e-02\n", " * time: 0.4318821430206299\n", " 128 7.606530e+00 2.751900e-02\n", " * time: 0.4346311092376709\n", " 129 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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 }