{ "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 3.148120e+01 7.976782e+00\n", " * time: 0.0021741390228271484\n", " 1 2.860931e+01 4.834721e+00\n", " * time: 0.006232023239135742\n", " 2 1.878532e+01 4.231944e+00\n", " * time: 0.010436058044433594\n", " 3 1.305985e+01 3.290575e+00\n", " * time: 0.014642000198364258\n", " 4 1.154349e+01 3.492442e+00\n", " * time: 0.018026113510131836\n", " 5 1.133521e+01 3.110641e+00\n", " * time: 0.021433115005493164\n", " 6 9.973911e+00 1.461407e+00\n", " * time: 0.025832176208496094\n", " 7 8.825201e+00 1.696388e+00\n", " * time: 0.029942035675048828\n", " 8 8.340630e+00 1.257247e+00\n", " * time: 0.0341341495513916\n", " 9 8.188388e+00 7.369264e-01\n", " * time: 0.08735418319702148\n", " 10 8.181798e+00 1.369906e+00\n", " * time: 0.08998298645019531\n", " 11 8.143823e+00 1.008025e+00\n", " * time: 0.09273409843444824\n", " 12 8.070573e+00 5.562346e-01\n", " * time: 0.09541201591491699\n", " 13 8.019478e+00 5.871768e-01\n", " * time: 0.09789013862609863\n", " 14 7.987270e+00 5.179631e-01\n", " * time: 0.10050106048583984\n", " 15 7.949139e+00 4.413989e-01\n", " * time: 0.10306119918823242\n", " 16 7.921160e+00 2.473208e-01\n", " * time: 0.10559701919555664\n", " 17 7.903649e+00 4.275275e-01\n", " * time: 0.1083061695098877\n", " 18 7.887734e+00 2.483476e-01\n", " * time: 0.11151003837585449\n", " 19 7.877275e+00 2.716477e-01\n", " * time: 0.11405205726623535\n", " 20 7.863731e+00 3.190057e-01\n", " * time: 0.11655211448669434\n", " 21 7.851747e+00 1.992245e-01\n", " * time: 0.12076020240783691\n", " 22 7.845183e+00 2.990783e-01\n", " * time: 0.1255180835723877\n", " 23 7.831563e+00 2.079099e-01\n", " * time: 0.12980914115905762\n", " 24 7.819596e+00 2.182718e-01\n", " * time: 0.13413715362548828\n", " 25 7.813778e+00 2.517873e-01\n", " * time: 0.13860416412353516\n", " 26 7.803472e+00 2.572012e-01\n", " * time: 0.14226698875427246\n", " 27 7.791507e+00 1.526725e-01\n", " * time: 0.14589715003967285\n", " 28 7.783590e+00 2.070923e-01\n", " * time: 0.14956998825073242\n", " 29 7.774926e+00 1.744098e-01\n", " * time: 0.1532430648803711\n", " 30 7.771718e+00 2.111901e-01\n", " * time: 0.1575150489807129\n", " 31 7.764113e+00 2.157635e-01\n", " * time: 0.16212916374206543\n", " 32 7.757313e+00 1.946915e-01\n", " * time: 0.16663503646850586\n", " 33 7.749997e+00 2.537610e-01\n", " * time: 0.17126011848449707\n", " 34 7.746078e+00 2.994609e-01\n", " * time: 0.17576217651367188\n", " 35 7.736647e+00 1.872230e-01\n", " * time: 0.18019604682922363\n", " 36 7.733542e+00 2.962243e-01\n", " * time: 0.18457603454589844\n", " 37 7.723686e+00 1.600278e-01\n", " * time: 0.18925213813781738\n", " 38 7.716906e+00 1.818841e-01\n", " * time: 0.19369006156921387\n", " 39 7.709998e+00 1.974226e-01\n", " * time: 0.19814014434814453\n", " 40 7.703846e+00 2.382428e-01\n", " * time: 0.20284318923950195\n", " 41 7.696816e+00 1.051032e-01\n", " * time: 0.20855212211608887\n", " 42 7.694500e+00 2.571276e-01\n", " * time: 0.2129380702972412\n", " 43 7.690573e+00 2.034471e-01\n", " * time: 0.23691201210021973\n", " 44 7.685807e+00 2.182316e-01\n", " * time: 0.23952603340148926\n", " 45 7.681004e+00 9.912048e-02\n", " * time: 0.24279117584228516\n", " 46 7.676336e+00 1.797902e-01\n", " * time: 0.24538016319274902\n", " 47 7.671407e+00 1.006924e-01\n", " * time: 0.24869799613952637\n", " 48 7.666702e+00 1.548438e-01\n", " * time: 0.25131702423095703\n", " 49 7.663790e+00 1.384494e-01\n", " * time: 0.2539050579071045\n", " 50 7.660523e+00 1.402333e-01\n", " * time: 0.2564401626586914\n", " 51 7.658122e+00 1.531875e-01\n", " * time: 0.258991003036499\n", " 52 7.653058e+00 1.387979e-01\n", " * time: 0.2615470886230469\n", " 53 7.649242e+00 1.176318e-01\n", " * time: 0.26500821113586426\n", " 54 7.646228e+00 7.116394e-02\n", " * time: 0.268233060836792\n", " 55 7.642844e+00 9.009411e-02\n", " * time: 0.2708010673522949\n", " 56 7.641380e+00 1.094158e-01\n", " * time: 0.27336812019348145\n", " 57 7.641047e+00 1.180147e-01\n", " * time: 0.2758660316467285\n", " 58 7.640899e+00 8.176400e-02\n", " * time: 0.27842020988464355\n", " 59 7.640045e+00 1.683780e-01\n", " * time: 0.28124117851257324\n", " 60 7.638125e+00 8.808581e-02\n", " * time: 0.28443312644958496\n", " 61 7.636426e+00 1.047422e-01\n", " * time: 0.2869300842285156\n", " 62 7.635526e+00 8.817420e-02\n", " * time: 0.28956103324890137\n", " 63 7.633994e+00 5.655445e-02\n", " * time: 0.29207301139831543\n", " 64 7.633591e+00 8.076289e-02\n", " * time: 0.2963531017303467\n", " 65 7.632499e+00 5.568526e-02\n", " * time: 0.30078601837158203\n", " 66 7.631722e+00 4.675207e-02\n", " * time: 0.30649304389953613\n", " 67 7.631436e+00 9.169248e-02\n", " * time: 0.31104016304016113\n", " 68 7.630670e+00 7.257136e-02\n", " * time: 0.31676602363586426\n", " 69 7.629986e+00 6.682596e-02\n", " * time: 0.32233405113220215\n", " 70 7.629326e+00 5.659019e-02\n", " * time: 0.3282191753387451\n", " 71 7.629080e+00 5.506472e-02\n", " * time: 0.3325660228729248\n", " 72 7.628601e+00 6.706456e-02\n", " * time: 0.3370530605316162\n", " 73 7.627928e+00 3.153190e-02\n", " * time: 0.34305405616760254\n", " 74 7.627421e+00 4.915726e-02\n", " * time: 0.3477320671081543\n", " 75 7.627369e+00 5.874008e-02\n", " * time: 0.3653261661529541\n", " 76 7.626745e+00 5.288012e-02\n", " * time: 0.3679921627044678\n", " 77 7.626244e+00 6.666101e-02\n", " * time: 0.37061214447021484\n", " 78 7.625692e+00 3.658142e-02\n", " * time: 0.3741281032562256\n", " 79 7.625420e+00 5.861060e-02\n", " * time: 0.37681007385253906\n", " 80 7.625175e+00 4.771474e-02\n", " * time: 0.37949514389038086\n", " 81 7.624876e+00 3.696046e-02\n", " * time: 0.38205718994140625\n", " 82 7.624821e+00 6.071859e-02\n", " * time: 0.38471198081970215\n", " 83 7.624447e+00 3.546631e-02\n", " * time: 0.38793206214904785\n", " 84 7.624115e+00 5.440242e-02\n", " * time: 0.3907740116119385\n", " 85 7.624091e+00 7.003944e-02\n", " * time: 0.39345407485961914\n", " 86 7.624034e+00 5.958998e-02\n", " * time: 0.3959810733795166\n", " 87 7.623693e+00 5.046968e-02\n", " * time: 0.39850616455078125\n", " 88 7.623268e+00 4.214160e-02\n", " * time: 0.4010441303253174\n", " 89 7.622815e+00 4.709536e-02\n", " * time: 0.4035520553588867\n", " 90 7.622685e+00 6.505086e-02\n", " * time: 0.40639305114746094\n", " 91 7.622286e+00 4.334952e-02\n", " * time: 0.40963315963745117\n", " 92 7.622043e+00 5.810219e-02\n", " * time: 0.41219210624694824\n", " 93 7.621672e+00 4.556278e-02\n", " * time: 0.41547513008117676\n", " 94 7.621499e+00 4.837781e-02\n", " * time: 0.4181091785430908\n", " 95 7.621024e+00 4.616776e-02\n", " * time: 0.42082810401916504\n", " 96 7.620702e+00 3.447455e-02\n", " * time: 0.42513203620910645\n", " 97 7.620201e+00 2.457585e-02\n", " * time: 0.43077898025512695\n", " 98 7.620025e+00 3.474768e-02\n", " * time: 0.4352850914001465\n", " 99 7.619919e+00 3.446584e-02\n", " * time: 0.4399731159210205\n", " 100 7.619796e+00 4.066059e-02\n", " * time: 0.44463419914245605\n", " 101 7.619711e+00 3.385564e-02\n", " * time: 0.4492671489715576\n", " 102 7.619422e+00 3.535886e-02\n", " * time: 0.4538850784301758\n", " 103 7.619145e+00 2.939781e-02\n", " * time: 0.4594600200653076\n", " 104 7.618927e+00 5.123870e-02\n", " * time: 0.4640212059020996\n", " 105 7.618819e+00 3.919593e-02\n", " * time: 0.4678230285644531\n", " 106 7.618702e+00 2.448087e-02\n", " * time: 0.47057008743286133\n", " 107 7.618486e+00 2.572167e-02\n", " * time: 0.4736900329589844\n", " 108 7.618325e+00 2.927185e-02\n", " * time: 0.48492908477783203\n", " 109 7.618110e+00 4.595612e-02\n", " * time: 0.4875810146331787\n", " 110 7.617931e+00 2.446269e-02\n", " * time: 0.49077320098876953\n", " 111 7.617770e+00 2.808868e-02\n", " * time: 0.49405908584594727\n", " 112 7.617557e+00 1.996933e-02\n", " * time: 0.4972701072692871\n", " 113 7.617396e+00 1.993439e-02\n", " * time: 0.5000290870666504\n", " 114 7.617339e+00 3.330451e-02\n", " * time: 0.5026090145111084\n", " 115 7.617299e+00 4.919919e-02\n", " * time: 0.5051090717315674\n", " 116 7.617074e+00 3.974564e-02\n", " * time: 0.5086400508880615\n", " 117 7.616943e+00 5.094035e-02\n", " * time: 0.5113182067871094\n", " 118 7.616784e+00 5.996913e-02\n", " * time: 0.5139870643615723\n", " 119 7.616474e+00 3.445397e-02\n", " * time: 0.5172979831695557\n", " 120 7.616265e+00 3.202310e-02\n", " * time: 0.5205211639404297\n", " 121 7.616112e+00 2.351918e-02\n", " * time: 0.5237431526184082\n", " 122 7.615987e+00 1.983246e-02\n", " * time: 0.5270512104034424\n", " 123 7.615848e+00 2.571351e-02\n", " * time: 0.5297491550445557\n", " 124 7.615661e+00 2.813978e-02\n", " * time: 0.5325310230255127\n", " 125 7.615485e+00 2.614831e-02\n", " * time: 0.5357010364532471\n", " 126 7.615394e+00 3.707523e-02\n", " * time: 0.5383419990539551\n", " 127 7.615213e+00 2.412467e-02\n", " * time: 0.5418450832366943\n", " 128 7.615074e+00 2.228796e-02\n", " * 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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": "1.5.4" }, "kernelspec": { "name": "julia-1.5", "display_name": "Julia 1.5.4", "language": "julia" } }, "nbformat": 4 }