{
 "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     3.122979e+01     8.041349e+00\n",
      " * time: 0.0014331340789794922\n",
      "     1     2.962691e+01     5.194341e+00\n",
      " * time: 0.00603795051574707\n",
      "     2     2.182118e+01     6.600144e+00\n",
      " * time: 0.012025117874145508\n",
      "     3     1.563171e+01     4.917492e+00\n",
      " * time: 0.01752305030822754\n",
      "     4     1.186117e+01     2.149640e+00\n",
      " * time: 0.022284984588623047\n",
      "     5     1.137053e+01     2.929259e+00\n",
      " * time: 0.026194095611572266\n",
      "     6     1.053403e+01     1.750955e+00\n",
      " * time: 0.031018972396850586\n",
      "     7     9.747891e+00     1.552575e+00\n",
      " * time: 0.034928083419799805\n",
      "     8     9.090514e+00     1.679429e+00\n",
      " * time: 0.038886070251464844\n",
      "     9     8.779532e+00     1.127845e+00\n",
      " * time: 0.04311013221740723\n",
      "    10     8.582690e+00     9.808984e-01\n",
      " * time: 0.0481419563293457\n",
      "    11     8.381509e+00     8.715960e-01\n",
      " * time: 0.052304983139038086\n",
      "    12     8.223375e+00     6.743384e-01\n",
      " * time: 0.05624508857727051\n",
      "    13     8.197315e+00     8.439144e-01\n",
      " * time: 0.05989503860473633\n",
      "    14     8.139930e+00     7.023806e-01\n",
      " * time: 0.06377601623535156\n",
      "    15     8.074470e+00     6.752596e-01\n",
      " * time: 0.09373712539672852\n",
      "    16     8.036873e+00     5.538479e-01\n",
      " * time: 0.09615302085876465\n",
      "    17     7.995625e+00     5.610557e-01\n",
      " * time: 0.0984959602355957\n",
      "    18     7.942597e+00     4.213812e-01\n",
      " * time: 0.10098791122436523\n",
      "    19     7.923442e+00     8.019301e-01\n",
      " * time: 0.10300397872924805\n",
      "    20     7.907011e+00     8.261580e-01\n",
      " * time: 0.10527896881103516\n",
      "    21     7.870397e+00     4.219163e-01\n",
      " * time: 0.10834407806396484\n",
      "    22     7.855176e+00     5.505108e-01\n",
      " * time: 0.11079597473144531\n",
      "    23     7.815045e+00     4.594445e-01\n",
      " * time: 0.11294412612915039\n",
      "    24     7.799625e+00     4.662650e-01\n",
      " * time: 0.1157069206237793\n",
      "    25     7.777751e+00     3.029118e-01\n",
      " * time: 0.11850595474243164\n",
      "    26     7.756934e+00     3.360724e-01\n",
      " * time: 0.12151503562927246\n",
      "    27     7.749661e+00     3.981501e-01\n",
      " * time: 0.12383699417114258\n",
      "    28     7.748521e+00     3.674228e-01\n",
      " * time: 0.12640595436096191\n",
      "    29     7.743930e+00     4.164525e-01\n",
      " * time: 0.12885308265686035\n",
      "    30     7.724561e+00     2.842537e-01\n",
      " * time: 0.1319410800933838\n",
      "    31     7.721135e+00     5.095137e-01\n",
      " * time: 0.13440299034118652\n",
      "    32     7.708546e+00     4.804166e-01\n",
      " * time: 0.13692307472229004\n",
      "    33     7.691247e+00     2.350169e-01\n",
      " * time: 0.13994407653808594\n",
      "    34     7.682490e+00     3.041951e-01\n",
      " * time: 0.14243292808532715\n",
      "    35     7.679673e+00     2.418369e-01\n",
      " * time: 0.14464807510375977\n",
      "    36     7.674242e+00     1.786314e-01\n",
      " * time: 0.14735102653503418\n",
      "    37     7.664293e+00     1.583925e-01\n",
      " * time: 0.1502389907836914\n",
      "    38     7.661325e+00     2.861231e-01\n",
      " * time: 0.15380597114562988\n",
      "    39     7.652962e+00     2.294544e-01\n",
      " * time: 0.1591949462890625\n",
      "    40     7.647011e+00     1.843652e-01\n",
      " * time: 0.16402602195739746\n",
      "    41     7.640637e+00     1.971624e-01\n",
      " * time: 0.16797494888305664\n",
      "    42     7.636014e+00     1.372626e-01\n",
      " * time: 0.17357897758483887\n",
      "    43     7.633162e+00     1.062178e-01\n",
      " * time: 0.1783759593963623\n",
      "    44     7.631505e+00     1.006395e-01\n",
      " * time: 0.1817479133605957\n",
      "    45     7.631221e+00     1.342639e-01\n",
      " * time: 0.1857929229736328\n",
      "    46     7.629368e+00     8.469784e-02\n",
      " * time: 0.20447993278503418\n",
      "    47     7.628685e+00     9.651779e-02\n",
      " * time: 0.20651507377624512\n",
      "    48     7.627694e+00     8.109871e-02\n",
      " * time: 0.20859003067016602\n",
      "    49     7.626311e+00     4.839437e-02\n",
      " * time: 0.21058011054992676\n",
      "    50     7.625166e+00     5.113374e-02\n",
      " * time: 0.21263599395751953\n",
      "    51     7.624780e+00     7.262042e-02\n",
      " * time: 0.21499013900756836\n",
      "    52     7.624503e+00     6.358642e-02\n",
      " * time: 0.2175159454345703\n",
      "    53     7.624236e+00     8.465746e-02\n",
      " * time: 0.21996712684631348\n",
      "    54     7.624051e+00     7.048943e-02\n",
      " * time: 0.22196507453918457\n",
      "    55     7.623163e+00     7.060252e-02\n",
      " * time: 0.22400403022766113\n",
      "    56     7.622416e+00     3.661848e-02\n",
      " * time: 0.22653603553771973\n",
      "    57     7.621963e+00     7.270198e-02\n",
      " * time: 0.22852802276611328\n",
      "    58     7.621737e+00     8.689250e-02\n",
      " * time: 0.23093008995056152\n",
      "    59     7.621224e+00     5.613937e-02\n",
      " * time: 0.23328304290771484\n",
      "    60     7.620919e+00     6.026545e-02\n",
      " * time: 0.2357640266418457\n",
      "    61     7.620240e+00     5.781363e-02\n",
      " * time: 0.23807501792907715\n",
      "    62     7.619508e+00     4.033903e-02\n",
      " * time: 0.24078893661499023\n",
      "    63     7.619377e+00     6.979910e-02\n",
      " * time: 0.24297404289245605\n",
      "    64     7.618908e+00     6.716091e-02\n",
      " * time: 0.24570703506469727\n",
      "    65     7.618355e+00     3.383349e-02\n",
      " * time: 0.24873113632202148\n",
      "    66     7.617652e+00     3.791594e-02\n",
      " * time: 0.251068115234375\n",
      "    67     7.617143e+00     4.593312e-02\n",
      " * time: 0.253277063369751\n",
      "    68     7.616893e+00     8.083180e-02\n",
      " * time: 0.255648136138916\n",
      "    69     7.616190e+00     4.484075e-02\n",
      " * time: 0.2586350440979004\n",
      "    70     7.616032e+00     7.532133e-02\n",
      " * time: 0.2624959945678711\n",
      "    71     7.615566e+00     4.377557e-02\n",
      " * time: 0.2677431106567383\n",
      "    72     7.615047e+00     5.773214e-02\n",
      " * time: 0.27170705795288086\n",
      "    73     7.614853e+00     5.599879e-02\n",
      " * time: 0.275130033493042\n",
      "    74     7.614841e+00     7.548428e-02\n",
      " * time: 0.27919602394104004\n",
      "    75     7.614441e+00     7.990188e-02\n",
      " * time: 0.28328704833984375\n",
      "    76     7.614056e+00     7.432898e-02\n",
      " * time: 0.2869751453399658\n",
      "    77     7.613328e+00     6.679307e-02\n",
      " * time: 0.29041504859924316\n",
      "    78     7.612781e+00     5.976445e-02\n",
      " * time: 0.30577611923217773\n",
      "    79     7.612486e+00     5.926440e-02\n",
      " * time: 0.3080129623413086\n",
      "    80     7.612055e+00     6.092114e-02\n",
      " * time: 0.31112003326416016\n",
      "    81     7.611601e+00     4.362699e-02\n",
      " * time: 0.3140439987182617\n",
      "    82     7.611309e+00     3.585826e-02\n",
      " * time: 0.31603407859802246\n",
      "    83     7.610903e+00     5.130359e-02\n",
      " * time: 0.31803107261657715\n",
      "    84     7.610518e+00     3.544783e-02\n",
      " * time: 0.3205270767211914\n",
      "    85     7.610339e+00     3.879526e-02\n",
      " * time: 0.3227670192718506\n",
      "    86     7.609995e+00     3.580910e-02\n",
      " * time: 0.325747013092041\n",
      "    87     7.609913e+00     7.756345e-02\n",
      " * time: 0.3281710147857666\n",
      "    88     7.609824e+00     5.500128e-02\n",
      " * time: 0.3305840492248535\n",
      "    89     7.609462e+00     5.539547e-02\n",
      " * time: 0.33307504653930664\n",
      "    90     7.609008e+00     5.098483e-02\n",
      " * time: 0.3355889320373535\n",
      "    91     7.608538e+00     3.716341e-02\n",
      " * time: 0.33866405487060547\n",
      "    92     7.608480e+00     5.906393e-02\n",
      " * time: 0.3411281108856201\n",
      "    93     7.608180e+00     4.713079e-02\n",
      " * time: 0.3442881107330322\n",
      "    94     7.607972e+00     5.505148e-02\n",
      " * time: 0.3466460704803467\n",
      "    95     7.607676e+00     5.182515e-02\n",
      " * time: 0.3488340377807617\n",
      "    96     7.607194e+00     3.502500e-02\n",
      " * time: 0.35094308853149414\n",
      "    97     7.607184e+00     5.021192e-02\n",
      " * time: 0.35303807258605957\n",
      "    98     7.606871e+00     5.456627e-02\n",
      " * time: 0.3560910224914551\n",
      "    99     7.606568e+00     3.041682e-02\n",
      " * time: 0.3591940402984619\n",
      "   100     7.606377e+00     5.002653e-02\n",
      " * time: 0.3623509407043457\n",
      "   101     7.606121e+00     3.774674e-02\n",
      " * time: 0.3657550811767578\n",
      "   102     7.606108e+00     5.366885e-02\n",
      " * time: 0.3691270351409912\n",
      "   103     7.605826e+00     3.891657e-02\n",
      " * time: 0.3742949962615967\n",
      "   104     7.605723e+00     4.869222e-02\n",
      " * time: 0.37764906883239746\n",
      "   105     7.605583e+00     7.508928e-02\n",
      " * time: 0.38048505783081055\n",
      "   106     7.605452e+00     5.287392e-02\n",
      " * time: 0.3834221363067627\n",
      "   107     7.605135e+00     3.823169e-02\n",
      " * time: 0.38662004470825195\n",
      "   108     7.604837e+00     2.049159e-02\n",
      " * time: 0.39923095703125\n",
      "   109     7.604801e+00     5.590117e-02\n",
      " * time: 0.4014449119567871\n",
      "   110     7.604780e+00     4.867948e-02\n",
      " * time: 0.40383100509643555\n",
      "   111     7.604596e+00     4.126217e-02\n",
      " * time: 0.40619611740112305\n",
      "   112     7.604281e+00     4.214552e-02\n",
      " * time: 0.4082491397857666\n",
      "   113     7.604201e+00     5.422412e-02\n",
      " * time: 0.4102311134338379\n",
      "   114     7.604004e+00     6.177909e-02\n",
      " * time: 0.4121971130371094\n",
      "   115     7.603796e+00     6.492113e-02\n",
      " * time: 0.4141850471496582\n",
      "   116     7.603532e+00     4.775545e-02\n",
      " * time: 0.416748046875\n",
      "   117     7.603377e+00     5.655964e-02\n",
      " * time: 0.41919398307800293\n",
      "   118     7.603273e+00     5.289166e-02\n",
      " * time: 0.4216439723968506\n",
      "   119     7.603077e+00     5.213989e-02\n",
      " * time: 0.4237360954284668\n",
      "   120     7.603010e+00     6.035143e-02\n",
      " * time: 0.42572808265686035\n",
      "   121     7.602684e+00     4.559793e-02\n",
      " * time: 0.42824792861938477\n",
      "   122     7.602468e+00     4.612345e-02\n",
      " * time: 0.4308459758758545\n",
      "   123     7.602223e+00     4.130215e-02\n",
      " * time: 0.4330711364746094\n",
      "   124     7.601883e+00     4.218979e-02\n",
      " * time: 0.43625593185424805\n",
      "   125     7.601691e+00     2.352147e-02\n",
      " * time: 0.4395010471343994\n",
      "   126     7.601578e+00     2.887224e-02\n",
      " * time: 0.4419851303100586\n",
      "   127     7.601397e+00     2.530797e-02\n",
      " * time: 0.44446301460266113\n",
      "   128     7.601246e+00     2.756150e-02\n",
      " * time: 0.4475901126861572\n",
      "   129     7.601098e+00     3.283176e-02\n",
      " * time: 0.45073699951171875\n",
      "   130     7.600982e+00     3.638977e-02\n",
      " * time: 0.4540529251098633\n",
      "   131     7.600751e+00     2.043097e-02\n",
      " * time: 0.45861005783081055\n",
      "   132     7.600673e+00     3.205637e-02\n",
      " * time: 0.4622640609741211\n",
      "   133     7.600524e+00     3.257767e-02\n",
      " * time: 0.4670069217681885\n",
      "   134     7.600368e+00     4.152013e-02\n",
      " * time: 0.4708230495452881\n",
      "   135     7.600159e+00     3.068077e-02\n",
      " * time: 0.4757080078125\n",
      "   136     7.599995e+00     2.019210e-02\n",
      " * time: 0.47872400283813477\n",
      "   137     7.599949e+00     3.422529e-02\n",
      " * time: 0.48124098777770996\n",
      "   138     7.599757e+00     2.716672e-02\n",
      " * time: 0.48394107818603516\n",
      "   139     7.599620e+00     3.177704e-02\n",
      " * time: 0.49555110931396484\n",
      "   140     7.599512e+00     2.554000e-02\n",
      " * time: 0.4980599880218506\n",
      "   141     7.599491e+00     3.421710e-02\n",
      " * time: 0.5007600784301758\n",
      "   142     7.599362e+00     3.093265e-02\n",
      " * time: 0.5038719177246094\n",
      "   143     7.599191e+00     4.511819e-02\n",
      " * time: 0.506385087966919\n",
      "   144     7.599054e+00     3.398795e-02\n",
      " * time: 0.5094509124755859\n",
      "   145     7.598900e+00     2.125321e-02\n",
      " * time: 0.5125961303710938\n",
      "   146     7.598814e+00     4.055044e-02\n",
      " * time: 0.5152161121368408\n",
      "   147     7.598655e+00     3.750311e-02\n",
      " * time: 0.5175681114196777\n",
      "   148     7.598470e+00     2.800670e-02\n",
      " * time: 0.5205399990081787\n",
      "   149     7.598397e+00     3.532578e-02\n",
      " * time: 0.5229840278625488\n",
      "   150     7.598195e+00     2.672998e-02\n",
      " * time: 0.5253291130065918\n",
      "   151     7.598021e+00     3.399095e-02\n",
      " * time: 0.5278270244598389\n",
      "   152     7.597696e+00     2.404643e-02\n",
      " * time: 0.5311551094055176\n",
      "   153     7.597640e+00     4.313169e-02\n",
      " * time: 0.5336430072784424\n",
      "   154     7.597565e+00     3.324135e-02\n",
      " * time: 0.5361270904541016\n",
      "   155     7.597423e+00     3.367817e-02\n",
      " * time: 0.5386481285095215\n",
      "   156     7.597405e+00     4.493830e-02\n",
      " * time: 0.5411150455474854\n",
      "   157     7.597230e+00     4.822960e-02\n",
      " * time: 0.5436179637908936\n",
      "   158     7.597003e+00     3.339382e-02\n",
      " * time: 0.5469570159912109\n",
      "   159     7.596835e+00     3.311032e-02\n",
      " * time: 0.5498371124267578\n",
      "   160     7.596641e+00     3.784571e-02\n",
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      "   521     7.592305e+00     7.493153e-04\n",
      " * time: 1.6538851261138916\n",
      "   522     7.592305e+00     1.057233e-03\n",
      " * time: 1.6566050052642822\n",
      "   523     7.592305e+00     9.580617e-04\n",
      " * time: 1.659924030303955\n",
      "   524     7.592305e+00     6.010324e-04\n",
      " * time: 1.6632399559020996\n",
      "   525     7.592305e+00     1.082501e-03\n",
      " * time: 1.6658079624176025\n",
      "   526     7.592305e+00     1.143050e-03\n",
      " * time: 1.668442964553833\n",
      "   527     7.592305e+00     8.288707e-04\n",
      " * time: 1.6710829734802246\n",
      "   528     7.592304e+00     6.077063e-04\n",
      " * time: 1.6736769676208496\n",
      "   529     7.592304e+00     4.346339e-04\n",
      " * time: 1.6761720180511475\n",
      "   530     7.592304e+00     5.527290e-04\n",
      " * time: 1.6786811351776123\n",
      "   531     7.592304e+00     6.257253e-04\n",
      " * time: 1.6811821460723877\n",
      "   532     7.592304e+00     4.734995e-04\n",
      " * time: 1.6845099925994873\n",
      "   533     7.592304e+00     3.386831e-04\n",
      " * time: 1.6878421306610107\n",
      "   534     7.592304e+00     5.460467e-04\n",
      " * time: 1.6903760433197021\n",
      "   535     7.592304e+00     4.113025e-04\n",
      " * time: 1.6929590702056885\n",
      "   536     7.592304e+00     1.013812e-03\n",
      " * time: 1.6954660415649414\n",
      "   537     7.592304e+00     8.324561e-04\n",
      " * time: 1.6987159252166748\n",
      "   538     7.592304e+00     5.762140e-04\n",
      " * time: 1.7020759582519531\n",
      "   539     7.592304e+00     8.159976e-04\n",
      " * time: 1.7046599388122559\n",
      "   540     7.592304e+00     6.416366e-04\n",
      " * time: 1.7071759700775146\n",
      "   541     7.592304e+00     1.174705e-03\n",
      " * time: 1.7188971042633057\n",
      "   542     7.592304e+00     5.963562e-04\n",
      " * time: 1.7213780879974365\n",
      "   543     7.592304e+00     5.636157e-04\n",
      " * time: 1.7238540649414062\n",
      "   544     7.592304e+00     7.771219e-04\n",
      " * time: 1.726341962814331\n",
      "   545     7.592304e+00     1.097493e-03\n",
      " * time: 1.7288200855255127\n",
      "   546     7.592303e+00     1.174788e-03\n",
      " * time: 1.7314491271972656\n",
      "   547     7.592303e+00     6.213433e-04\n",
      " * time: 1.7347290515899658\n",
      "   548     7.592303e+00     7.352676e-04\n",
      " * time: 1.7371830940246582\n",
      "   549     7.592303e+00     7.122682e-04\n",
      " * time: 1.739673137664795\n",
      "   550     7.592303e+00     3.879021e-04\n",
      " * time: 1.742811918258667\n",
      "   551     7.592303e+00     6.751406e-04\n",
      " * time: 1.745361089706421\n",
      "   552     7.592303e+00     5.760892e-04\n",
      " * time: 1.7487330436706543\n",
      "   553     7.592303e+00     3.866982e-04\n",
      " * time: 1.75205397605896\n",
      "   554     7.592303e+00     3.531694e-04\n",
      " * time: 1.7552099227905273\n",
      "   555     7.592303e+00     4.800337e-04\n",
      " * time: 1.757699966430664\n",
      "   556     7.592303e+00     6.236399e-04\n",
      " * time: 1.7602031230926514\n",
      "   557     7.592303e+00     4.359017e-04\n",
      " * time: 1.7628891468048096\n",
      "   558     7.592303e+00     4.449010e-04\n",
      " * time: 1.7656080722808838\n",
      "   559     7.592303e+00     5.581119e-04\n",
      " * time: 1.768226146697998\n",
      "   560     7.592303e+00     4.148960e-04\n",
      " * time: 1.7707419395446777\n",
      "   561     7.592303e+00     3.383739e-04\n",
      " * time: 1.7732279300689697\n",
      "   562     7.592303e+00     4.381269e-04\n",
      " * time: 1.7757360935211182\n",
      "   563     7.592303e+00     3.415827e-04\n",
      " * time: 1.7790210247039795\n",
      "   564     7.592303e+00     5.836848e-04\n",
      " * time: 1.7816951274871826\n",
      "   565     7.592303e+00     3.883394e-04\n",
      " * time: 1.7842481136322021\n",
      "   566     7.592303e+00     5.997273e-04\n",
      " * time: 1.786790132522583\n",
      "   567     7.592303e+00     4.828633e-04\n",
      " * time: 1.7899670600891113\n",
      "   568     7.592303e+00     6.059820e-04\n",
      " * time: 1.7925410270690918\n",
      "   569     7.592303e+00     6.807460e-04\n",
      " * time: 1.7951021194458008\n",
      "   570     7.592302e+00     6.052967e-04\n",
      " * time: 1.7984230518341064\n",
      "   571     7.592302e+00     8.221753e-04\n",
      " * time: 1.8008971214294434\n",
      "   572     7.592302e+00     7.858349e-04\n",
      " * time: 1.812593936920166\n",
      "   573     7.592302e+00     7.776014e-04\n",
      " * time: 1.8151681423187256\n",
      "   574     7.592302e+00     7.448790e-04\n",
      " * time: 1.8176419734954834\n",
      "   575     7.592302e+00     5.515709e-04\n",
      " * time: 1.820786952972412\n",
      "   576     7.592302e+00     5.781691e-04\n",
      " * time: 1.8233790397644043\n",
      "   577     7.592302e+00     3.993094e-04\n",
      " * time: 1.826643943786621\n",
      "   578     7.592302e+00     6.794209e-04\n",
      " * time: 1.8292970657348633\n",
      "   579     7.592302e+00     8.990385e-04\n",
      " * time: 1.8318049907684326\n",
      "   580     7.592302e+00     8.154840e-04\n",
      " * time: 1.83430814743042\n",
      "   581     7.592302e+00     8.498985e-04\n",
      " * time: 1.8367979526519775\n",
      "   582     7.592302e+00     6.293984e-04\n",
      " * time: 1.840116024017334\n",
      "   583     7.592302e+00     3.552261e-04\n",
      " * time: 1.8434350490570068\n",
      "   584     7.592302e+00     5.479099e-04\n",
      " * time: 1.8459219932556152\n",
      "   585     7.592302e+00     3.402267e-04\n",
      " * time: 1.8484070301055908\n",
      "   586     7.592302e+00     3.828023e-04\n",
      " * time: 1.8509209156036377\n",
      "   587     7.592302e+00     2.221537e-04\n",
      " * time: 1.8540871143341064\n",
      "   588     7.592302e+00     6.386296e-04\n",
      " * time: 1.8566789627075195\n",
      "   589     7.592302e+00     4.635184e-04\n",
      " * time: 1.859377145767212\n",
      "   590     7.592301e+00     2.196817e-04\n",
      " * time: 1.8625340461730957\n",
      "   591     7.592301e+00     5.021191e-04\n",
      " * time: 1.8650119304656982\n",
      "   592     7.592301e+00     3.984708e-04\n",
      " * time: 1.8675010204315186\n",
      "   593     7.592301e+00     2.675558e-04\n",
      " * time: 1.8700709342956543\n",
      "   594     7.592301e+00     3.526317e-04\n",
      " * time: 1.872736930847168\n",
      "   595     7.592301e+00     6.571966e-04\n",
      " * time: 1.8755300045013428\n",
      "   596     7.592301e+00     5.496870e-04\n",
      " * time: 1.8781640529632568\n",
      "   597     7.592301e+00     4.318822e-04\n",
      " * time: 1.8807640075683594\n",
      "   598     7.592301e+00     3.808529e-04\n",
      " * time: 1.8834640979766846\n"
     ]
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "Plot{Plots.GRBackend() n=1}",
      "image/png": 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       "</svg>\n"
      ]
     },
     "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.real[:, :, 1], c=:blues)"
   ],
   "metadata": {},
   "execution_count": 5
  }
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