{
 "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     3.102290e+01     7.041416e+00\n",
      " * time: 0.0018489360809326172\n",
      "     1     2.840777e+01     4.949200e+00\n",
      " * time: 0.006920814514160156\n",
      "     2     2.097527e+01     5.716790e+00\n",
      " * time: 0.012439966201782227\n",
      "     3     1.370089e+01     2.886502e+00\n",
      " * time: 0.016038894653320312\n",
      "     4     1.116671e+01     1.133748e+00\n",
      " * time: 0.01960587501525879\n",
      "     5     1.040005e+01     2.048981e+00\n",
      " * time: 0.022694826126098633\n",
      "     6     9.888087e+00     1.255660e+00\n",
      " * time: 0.025656938552856445\n",
      "     7     9.461884e+00     1.011384e+00\n",
      " * time: 0.028729915618896484\n",
      "     8     9.000984e+00     8.456923e-01\n",
      " * time: 0.03168487548828125\n",
      "     9     8.655796e+00     9.045388e-01\n",
      " * time: 0.03452897071838379\n",
      "    10     8.390322e+00     5.943918e-01\n",
      " * time: 0.03753781318664551\n",
      "    11     8.350848e+00     1.030523e+00\n",
      " * time: 0.040007829666137695\n",
      "    12     8.294046e+00     8.644571e-01\n",
      " * time: 0.04257392883300781\n",
      "    13     8.198117e+00     6.974578e-01\n",
      " * time: 0.04592585563659668\n",
      "    14     8.048967e+00     4.116745e-01\n",
      " * time: 0.04960989952087402\n",
      "    15     7.957076e+00     3.845398e-01\n",
      " * time: 0.05331587791442871\n",
      "    16     7.943642e+00     6.021342e-01\n",
      " * time: 0.092681884765625\n",
      "    17     7.911156e+00     5.087298e-01\n",
      " * time: 0.09547591209411621\n",
      "    18     7.879390e+00     5.412756e-01\n",
      " * time: 0.09806990623474121\n",
      "    19     7.842516e+00     3.481325e-01\n",
      " * time: 0.10102200508117676\n",
      "    20     7.812414e+00     2.676037e-01\n",
      " * time: 0.10408997535705566\n",
      "    21     7.782916e+00     3.394597e-01\n",
      " * time: 0.10641288757324219\n",
      "    22     7.765237e+00     3.985629e-01\n",
      " * time: 0.10876202583312988\n",
      "    23     7.746721e+00     1.605332e-01\n",
      " * time: 0.11177182197570801\n",
      "    24     7.732907e+00     2.088353e-01\n",
      " * time: 0.1141350269317627\n",
      "    25     7.728162e+00     2.831760e-01\n",
      " * time: 0.11638784408569336\n",
      "    26     7.725938e+00     2.903319e-01\n",
      " * time: 0.11867284774780273\n",
      "    27     7.715058e+00     2.002263e-01\n",
      " * time: 0.12154793739318848\n",
      "    28     7.705152e+00     2.600964e-01\n",
      " * time: 0.12373900413513184\n",
      "    29     7.699972e+00     2.184188e-01\n",
      " * time: 0.12601494789123535\n",
      "    30     7.692585e+00     2.279659e-01\n",
      " * time: 0.12842392921447754\n",
      "    31     7.689894e+00     2.321909e-01\n",
      " * time: 0.1306748390197754\n",
      "    32     7.683332e+00     2.787574e-01\n",
      " * time: 0.13326787948608398\n",
      "    33     7.680395e+00     1.890848e-01\n",
      " * time: 0.13609600067138672\n",
      "    34     7.672437e+00     1.547245e-01\n",
      " * time: 0.13898992538452148\n",
      "    35     7.669930e+00     2.164050e-01\n",
      " * time: 0.14186501502990723\n",
      "    36     7.665492e+00     1.855944e-01\n",
      " * time: 0.14486002922058105\n",
      "    37     7.661820e+00     1.272048e-01\n",
      " * time: 0.14773893356323242\n",
      "    38     7.657584e+00     1.362699e-01\n",
      " * time: 0.15016889572143555\n",
      "    39     7.654934e+00     8.766086e-02\n",
      " * time: 0.15271401405334473\n",
      "    40     7.654480e+00     1.598830e-01\n",
      " * time: 0.15536999702453613\n",
      "    41     7.652239e+00     9.934691e-02\n",
      " * time: 0.1589670181274414\n",
      "    42     7.650453e+00     7.527233e-02\n",
      " * time: 0.16277003288269043\n",
      "    43     7.649083e+00     1.150946e-01\n",
      " * time: 0.165618896484375\n",
      "    44     7.648177e+00     8.135321e-02\n",
      " * time: 0.16861701011657715\n",
      "    45     7.646946e+00     7.720579e-02\n",
      " * time: 0.17166686058044434\n",
      "    46     7.645058e+00     5.406924e-02\n",
      " * time: 0.17465686798095703\n",
      "    47     7.643652e+00     5.052017e-02\n",
      " * time: 0.17757081985473633\n",
      "    48     7.642202e+00     6.189246e-02\n",
      " * time: 0.18045401573181152\n",
      "    49     7.641855e+00     9.368634e-02\n",
      " * time: 0.18334603309631348\n",
      "    50     7.641054e+00     6.412553e-02\n",
      " * time: 0.18594789505004883\n",
      "    51     7.639683e+00     3.917063e-02\n",
      " * time: 0.20866894721984863\n",
      "    52     7.638715e+00     5.194167e-02\n",
      " * time: 0.21099591255187988\n",
      "    53     7.638035e+00     6.588134e-02\n",
      " * time: 0.21354293823242188\n",
      "    54     7.636989e+00     6.142308e-02\n",
      " * time: 0.2164309024810791\n",
      "    55     7.636072e+00     4.057438e-02\n",
      " * time: 0.21939587593078613\n",
      "    56     7.635475e+00     4.872530e-02\n",
      " * time: 0.222398042678833\n",
      "    57     7.634980e+00     3.900013e-02\n",
      " * time: 0.22528791427612305\n",
      "    58     7.634404e+00     4.278678e-02\n",
      " * time: 0.2277059555053711\n",
      "    59     7.634009e+00     5.571754e-02\n",
      " * time: 0.2299790382385254\n",
      "    60     7.633680e+00     5.079089e-02\n",
      " * time: 0.23236584663391113\n",
      "    61     7.633375e+00     5.332203e-02\n",
      " * time: 0.23460984230041504\n",
      "    62     7.632928e+00     7.453823e-02\n",
      " * time: 0.2369098663330078\n",
      "    63     7.632317e+00     5.953921e-02\n",
      " * time: 0.2391979694366455\n",
      "    64     7.631522e+00     4.226178e-02\n",
      " * time: 0.24220800399780273\n",
      "    65     7.631328e+00     7.342402e-02\n",
      " * time: 0.24451398849487305\n",
      "    66     7.630578e+00     4.370234e-02\n",
      " * time: 0.2468578815460205\n",
      "    67     7.629892e+00     6.013812e-02\n",
      " * time: 0.24912190437316895\n",
      "    68     7.629020e+00     3.830866e-02\n",
      " * time: 0.2514188289642334\n",
      "    69     7.628330e+00     4.472795e-02\n",
      " * time: 0.25385403633117676\n",
      "    70     7.627872e+00     5.994983e-02\n",
      " * time: 0.25613903999328613\n",
      "    71     7.627804e+00     7.089002e-02\n",
      " * time: 0.25854992866516113\n",
      "    72     7.627268e+00     6.465028e-02\n",
      " * time: 0.261538028717041\n",
      "    73     7.627131e+00     9.168751e-02\n",
      " * time: 0.2640049457550049\n",
      "    74     7.627032e+00     8.133178e-02\n",
      " * time: 0.2662949562072754\n",
      "    75     7.626524e+00     6.939501e-02\n",
      " * time: 0.26857900619506836\n",
      "    76     7.625646e+00     8.837095e-02\n",
      " * time: 0.2708468437194824\n",
      "    77     7.624706e+00     5.111281e-02\n",
      " * time: 0.27394986152648926\n",
      "    78     7.624605e+00     1.144389e-01\n",
      " * time: 0.27618885040283203\n",
      "    79     7.623676e+00     6.356848e-02\n",
      " * time: 0.27858996391296387\n",
      "    80     7.622880e+00     6.605025e-02\n",
      " * time: 0.2809908390045166\n",
      "    81     7.622372e+00     6.413931e-02\n",
      " * time: 0.2839798927307129\n",
      "    82     7.621806e+00     4.222750e-02\n",
      " * time: 0.286984920501709\n",
      "    83     7.621357e+00     5.332367e-02\n",
      " * time: 0.2892758846282959\n",
      "    84     7.620831e+00     4.491713e-02\n",
      " * time: 0.30832791328430176\n",
      "    85     7.620395e+00     3.744157e-02\n",
      " * time: 0.311262845993042\n",
      "    86     7.619932e+00     2.750860e-02\n",
      " * time: 0.314162015914917\n",
      "    87     7.619625e+00     4.952384e-02\n",
      " * time: 0.31640100479125977\n",
      "    88     7.619325e+00     4.109088e-02\n",
      " * time: 0.3193199634552002\n",
      "    89     7.618805e+00     4.077561e-02\n",
      " * time: 0.3223879337310791\n",
      "    90     7.618440e+00     6.256811e-02\n",
      " * time: 0.32460999488830566\n",
      "    91     7.618277e+00     5.624196e-02\n",
      " * time: 0.3268589973449707\n",
      "    92     7.617923e+00     4.926166e-02\n",
      " * time: 0.3297419548034668\n",
      "    93     7.617529e+00     3.595190e-02\n",
      " * time: 0.33199286460876465\n",
      "    94     7.617030e+00     4.915827e-02\n",
      " * time: 0.33424997329711914\n",
      "    95     7.616617e+00     4.009074e-02\n",
      " * time: 0.3372659683227539\n",
      "    96     7.616325e+00     3.765083e-02\n",
      " * time: 0.3400909900665283\n",
      "    97     7.615827e+00     4.831876e-02\n",
      " * time: 0.3425748348236084\n",
      "    98     7.615529e+00     4.149599e-02\n",
      " * time: 0.3454148769378662\n",
      "    99     7.615277e+00     5.030796e-02\n",
      " * time: 0.3477199077606201\n",
      "   100     7.614993e+00     2.840701e-02\n",
      " * time: 0.35056400299072266\n",
      "   101     7.614957e+00     6.053424e-02\n",
      " * time: 0.3530080318450928\n",
      "   102     7.614873e+00     4.829080e-02\n",
      " * time: 0.3552408218383789\n",
      "   103     7.614364e+00     4.870874e-02\n",
      " * time: 0.3575890064239502\n",
      "   104     7.614015e+00     6.855373e-02\n",
      " * time: 0.3598289489746094\n",
      "   105     7.613472e+00     5.783060e-02\n",
      " * time: 0.36226892471313477\n",
      "   106     7.613176e+00     5.116245e-02\n",
      " * time: 0.36455202102661133\n",
      "   107     7.612838e+00     3.983134e-02\n",
      " * time: 0.36757683753967285\n",
      "   108     7.612509e+00     3.729807e-02\n",
      " * time: 0.36981201171875\n",
      "   109     7.612240e+00     3.433138e-02\n",
      " * time: 0.37215685844421387\n",
      "   110     7.611957e+00     4.622372e-02\n",
      " * time: 0.37439489364624023\n",
      "   111     7.611711e+00     3.217555e-02\n",
      " * time: 0.3774528503417969\n",
      "   112     7.611399e+00     3.286435e-02\n",
      " * time: 0.380511999130249\n",
      "   113     7.611322e+00     4.109409e-02\n",
      " * time: 0.3829958438873291\n",
      "   114     7.611141e+00     3.494357e-02\n",
      " * time: 0.3852348327636719\n",
      "   115     7.610745e+00     5.965660e-02\n",
      " * time: 0.38759398460388184\n",
      "   116     7.610352e+00     4.204375e-02\n",
      " * time: 0.3898189067840576\n",
      "   117     7.610031e+00     3.620723e-02\n",
      " * time: 0.3923499584197998\n",
      "   118     7.609874e+00     4.097104e-02\n",
      " * time: 0.4032559394836426\n",
      "   119     7.609354e+00     4.943510e-02\n",
      " * time: 0.4055008888244629\n",
      "   120     7.609015e+00     2.944610e-02\n",
      " * time: 0.40834784507751465\n",
      "   121     7.608768e+00     4.623617e-02\n",
      " * time: 0.41059398651123047\n",
      "   122     7.608608e+00     3.723652e-02\n",
      " * time: 0.41300296783447266\n",
      "   123     7.608435e+00     2.723995e-02\n",
      " * time: 0.4158759117126465\n",
      "   124     7.608404e+00     5.236463e-02\n",
      " * time: 0.41828489303588867\n",
      "   125     7.608051e+00     3.931512e-02\n",
      " * time: 0.421205997467041\n",
      "   126     7.607862e+00     3.253523e-02\n",
      " * time: 0.4241750240325928\n",
      "   127     7.607687e+00     3.829422e-02\n",
      " * time: 0.4265158176422119\n",
      "   128     7.607525e+00     3.936380e-02\n",
      " * time: 0.4288468360900879\n",
      "   129     7.607300e+00     2.277224e-02\n",
      " * time: 0.43172693252563477\n",
      "   130     7.607037e+00     2.225021e-02\n",
      " * time: 0.43400001525878906\n",
      "   131     7.606898e+00     2.747788e-02\n",
      " * time: 0.43624091148376465\n",
      "   132     7.606719e+00     2.867698e-02\n",
      " * time: 0.43854784965515137\n",
      "   133     7.606591e+00     1.936495e-02\n",
      " * time: 0.44152283668518066\n",
      "   134     7.606474e+00     2.327721e-02\n",
      " * time: 0.4444398880004883\n",
      "   135     7.606350e+00     1.397533e-02\n",
      " * time: 0.44738292694091797\n",
      "   136     7.606283e+00     2.037536e-02\n",
      " * time: 0.4496469497680664\n",
      "   137     7.606139e+00     2.166394e-02\n",
      " * time: 0.45197582244873047\n",
      "   138     7.606028e+00     1.691680e-02\n",
      " * time: 0.4548468589782715\n",
      "   139     7.605931e+00     3.363934e-02\n",
      " * time: 0.4571249485015869\n",
      "   140     7.605885e+00     3.210235e-02\n",
      " * time: 0.4593830108642578\n",
      "   141     7.605735e+00     3.743706e-02\n",
      " * time: 0.46168994903564453\n",
      "   142     7.605672e+00     2.553263e-02\n",
      " * time: 0.4640069007873535\n",
      "   143     7.605569e+00     3.283540e-02\n",
      " * time: 0.46627283096313477\n",
      "   144     7.605387e+00     2.316794e-02\n",
      " * time: 0.468552827835083\n",
      "   145     7.605260e+00     3.224783e-02\n",
      " * time: 0.4708118438720703\n",
      "   146     7.605209e+00     3.186753e-02\n",
      " * time: 0.47321295738220215\n",
      "   147     7.605165e+00     3.512431e-02\n",
      " * time: 0.4754478931427002\n",
      "   148     7.605161e+00     5.130535e-02\n",
      " * time: 0.4778409004211426\n",
      "   149     7.605003e+00     2.895404e-02\n",
      " * time: 0.4800739288330078\n",
      "   150     7.604861e+00     4.363684e-02\n",
      " * time: 0.482496976852417\n",
      "   151     7.604709e+00     1.558589e-02\n",
      " * time: 0.4855179786682129\n",
      "   152     7.604545e+00     2.500268e-02\n",
      " * time: 0.49652791023254395\n",
      "   153     7.604422e+00     2.204940e-02\n",
      " * time: 0.4995279312133789\n",
      "   154     7.604363e+00     3.075254e-02\n",
      " * time: 0.5018858909606934\n",
      "   155     7.604280e+00     3.219137e-02\n",
      " * time: 0.5042359828948975\n",
      "   156     7.604230e+00     3.849645e-02\n",
      " * time: 0.5066089630126953\n",
      "   157     7.604108e+00     2.877284e-02\n",
      " * time: 0.5096139907836914\n",
      "   158     7.603934e+00     2.888318e-02\n",
      " * time: 0.512617826461792\n",
      "   159     7.603731e+00     1.754829e-02\n",
      " * time: 0.5156219005584717\n",
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      "   520     7.592305e+00     5.829484e-04\n",
      " * time: 1.5217769145965576\n",
      "   521     7.592305e+00     7.709669e-04\n",
      " * time: 1.5240809917449951\n",
      "   522     7.592305e+00     9.223135e-04\n",
      " * time: 1.5269789695739746\n",
      "   523     7.592304e+00     4.144980e-04\n",
      " * time: 1.5299839973449707\n",
      "   524     7.592304e+00     6.367513e-04\n",
      " * time: 1.5323128700256348\n",
      "   525     7.592304e+00     7.192259e-04\n",
      " * time: 1.5345749855041504\n",
      "   526     7.592304e+00     5.958315e-04\n",
      " * time: 1.5460779666900635\n",
      "   527     7.592304e+00     6.007328e-04\n",
      " * time: 1.5484249591827393\n",
      "   528     7.592304e+00     5.538829e-04\n",
      " * time: 1.5507190227508545\n",
      "   529     7.592304e+00     6.352578e-04\n",
      " * time: 1.5531208515167236\n",
      "   530     7.592304e+00     9.156473e-04\n",
      " * time: 1.5553369522094727\n",
      "   531     7.592304e+00     1.113372e-03\n",
      " * time: 1.5577068328857422\n",
      "   532     7.592304e+00     1.270301e-03\n",
      " * time: 1.5599958896636963\n",
      "   533     7.592303e+00     8.576146e-04\n",
      " * time: 1.5623078346252441\n",
      "   534     7.592303e+00     8.850594e-04\n",
      " * time: 1.5646400451660156\n",
      "   535     7.592303e+00     1.627696e-03\n",
      " * time: 1.5669519901275635\n",
      "   536     7.592303e+00     9.423361e-04\n",
      " * time: 1.5693318843841553\n",
      "   537     7.592303e+00     1.034615e-03\n",
      " * time: 1.5715668201446533\n",
      "   538     7.592303e+00     8.449986e-04\n",
      " * time: 1.5749030113220215\n",
      "   539     7.592303e+00     6.295347e-04\n",
      " * time: 1.577193021774292\n",
      "   540     7.592302e+00     7.176453e-04\n",
      " * time: 1.579530954360962\n",
      "   541     7.592302e+00     6.586048e-04\n",
      " * time: 1.5824990272521973\n",
      "   542     7.592302e+00     1.942404e-03\n",
      " * time: 1.584751844406128\n",
      "   543     7.592302e+00     1.070989e-03\n",
      " * time: 1.5877599716186523\n",
      "   544     7.592302e+00     9.657404e-04\n",
      " * time: 1.5900559425354004\n",
      "   545     7.592302e+00     9.402287e-04\n",
      " * time: 1.5923488140106201\n",
      "   546     7.592302e+00     1.020338e-03\n",
      " * time: 1.594667911529541\n",
      "   547     7.592302e+00     8.866450e-04\n",
      " * time: 1.5969128608703613\n",
      "   548     7.592302e+00     6.839219e-04\n",
      " * time: 1.5999388694763184\n",
      "   549     7.592302e+00     9.526252e-04\n",
      " * time: 1.6022298336029053\n",
      "   550     7.592302e+00     1.080698e-03\n",
      " * time: 1.6045429706573486\n",
      "   551     7.592301e+00     8.966271e-04\n",
      " * time: 1.6068480014801025\n",
      "   552     7.592301e+00     7.540693e-04\n",
      " * time: 1.6091740131378174\n",
      "   553     7.592301e+00     1.231184e-03\n",
      " * time: 1.6114189624786377\n",
      "   554     7.592301e+00     1.019640e-03\n",
      " * time: 1.6138370037078857\n",
      "   555     7.592301e+00     9.502074e-04\n",
      " * time: 1.6161129474639893\n",
      "   556     7.592301e+00     7.177609e-04\n",
      " * time: 1.619182825088501\n",
      "   557     7.592301e+00     6.939495e-04\n",
      " * time: 1.6221868991851807\n",
      "   558     7.592301e+00     8.901981e-04\n",
      " * time: 1.6245129108428955\n",
      "   559     7.592301e+00     8.014725e-04\n",
      " * time: 1.62680983543396\n",
      "   560     7.592301e+00     6.701744e-04\n",
      " * time: 1.6298069953918457\n",
      "   561     7.592301e+00     4.573906e-04\n",
      " * time: 1.6411418914794922\n",
      "   562     7.592301e+00     5.266435e-04\n",
      " * time: 1.6434438228607178\n",
      "   563     7.592300e+00     8.559942e-04\n",
      " * time: 1.6456730365753174\n",
      "   564     7.592300e+00     4.540772e-04\n",
      " * time: 1.6487088203430176\n",
      "   565     7.592300e+00     5.476328e-04\n",
      " * time: 1.6509678363800049\n",
      "   566     7.592300e+00     6.307882e-04\n",
      " * time: 1.653291940689087\n",
      "   567     7.592300e+00     3.943299e-04\n",
      " * time: 1.6555328369140625\n",
      "   568     7.592300e+00     5.727576e-04\n",
      " * time: 1.6577799320220947\n",
      "   569     7.592300e+00     3.645821e-04\n",
      " * time: 1.6606390476226807\n",
      "   570     7.592300e+00     5.189390e-04\n",
      " * time: 1.6630849838256836\n",
      "   571     7.592300e+00     3.436121e-04\n",
      " * time: 1.6653549671173096\n",
      "   572     7.592300e+00     5.047516e-04\n",
      " * time: 1.6675758361816406\n",
      "   573     7.592300e+00     3.588241e-04\n",
      " * time: 1.6697778701782227\n",
      "   574     7.592300e+00     3.213661e-04\n",
      " * time: 1.6720948219299316\n",
      "   575     7.592300e+00     3.363915e-04\n",
      " * time: 1.6745679378509521\n",
      "   576     7.592300e+00     4.599944e-04\n",
      " * time: 1.6768219470977783\n",
      "   577     7.592300e+00     3.321465e-04\n",
      " * time: 1.6796610355377197\n",
      "   578     7.592300e+00     3.555291e-04\n",
      " * time: 1.6819629669189453\n",
      "   579     7.592300e+00     2.982073e-04\n",
      " * time: 1.6850378513336182\n",
      "   580     7.592300e+00     3.049252e-04\n",
      " * time: 1.6872608661651611\n",
      "   581     7.592300e+00     2.045465e-04\n",
      " * time: 1.690194845199585\n",
      "   582     7.592300e+00     1.968502e-04\n",
      " * time: 1.6924188137054443\n",
      "   583     7.592300e+00     1.599554e-04\n",
      " * time: 1.6948168277740479\n",
      "   584     7.592300e+00     1.835117e-04\n",
      " * time: 1.697026014328003\n",
      "   585     7.592300e+00     1.217371e-04\n",
      " * time: 1.6999719142913818\n",
      "   586     7.592300e+00     1.793300e-04\n",
      " * time: 1.7022149562835693\n",
      "   587     7.592300e+00     2.289451e-04\n",
      " * time: 1.7046480178833008\n",
      "   588     7.592300e+00     1.842698e-04\n",
      " * time: 1.706885814666748\n",
      "   589     7.592300e+00     1.771351e-04\n",
      " * time: 1.709928035736084\n",
      "   590     7.592300e+00     1.610654e-04\n",
      " * time: 1.7121479511260986\n",
      "   591     7.592300e+00     1.221196e-04\n",
      " * time: 1.7152299880981445\n",
      "   592     7.592300e+00     1.481587e-04\n",
      " * time: 1.7174479961395264\n",
      "   593     7.592300e+00     1.301925e-04\n",
      " * time: 1.7198619842529297\n"
     ]
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "Plot{Plots.GRBackend() n=1}",
      "image/png": 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      ]
     },
     "metadata": {},
     "execution_count": 5
    }
   ],
   "cell_type": "code",
   "source": [
    "terms = [Kinetic(),\n",
    "         ExternalFromReal(X -> pot(X...)),\n",
    "         PowerNonlinearity(C, α),\n",
    "         Magnetic(Apot),\n",
    "]\n",
    "model = Model(lattice; 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[:, :, 1, 1], c=:blues)"
   ],
   "metadata": {},
   "execution_count": 5
  }
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