We solve the 2D Gross-Pitaevskii equation with a magnetic field. This is similar to the previous example (Gross-Pitaevskii equation in one dimension), but with an extra term for the magnetic field. We reproduce here the results of https://arxiv.org/pdf/1611.02045.pdf Fig. 10
using DFTK
using StaticArrays
using Plots
Unit cell. Having one of the lattice vectors as zero means a 2D system
a = 15
lattice = a .* [[1 0 0.]; [0 1 0]; [0 0 0]];
Confining scalar potential, and magnetic vector potential
pot(x, y, z) = ((x - a/2)^2 + (y - a/2)^2)/2
ω = .6
Apot(x, y, z) = ω * @SVector [y - a/2, -(x - a/2), 0]
Apot(X) = Apot(X...);
Parameters
Ecut = 20 # Increase this for production
η = 500
C = η/2
α = 2
n_electrons = 1; # Increase this for fun
Collect all the terms, build and run the model
terms = [Kinetic(),
ExternalFromReal(X -> pot(X...)),
PowerNonlinearity(C, α),
Magnetic(Apot),
]
model = Model(lattice; n_electrons=n_electrons,
terms=terms, spin_polarization=:spinless) # "spinless electrons"
basis = PlaneWaveBasis(model, Ecut, kgrid=(1, 1, 1))
scfres = direct_minimization(basis, tol=1e-5) # Reduce tol for production
heatmap(scfres.ρ.real[:, :, 1], c=:blues)
Iter Function value Gradient norm 0 3.178874e+01 8.073477e+00 * time: 0.0019261837005615234 1 2.979741e+01 5.177095e+00 * time: 0.008530139923095703 2 2.136178e+01 5.751207e+00 * time: 0.01683521270751953 3 1.751289e+01 7.333431e+00 * time: 0.02464914321899414 4 1.192422e+01 2.676646e+00 * time: 0.03176307678222656 5 9.798932e+00 1.021552e+00 * time: 0.03886914253234863 6 9.152591e+00 1.245030e+00 * time: 0.04542708396911621 7 8.787339e+00 7.303126e-01 * time: 0.05136704444885254 8 8.488029e+00 5.894573e-01 * time: 0.05793118476867676 9 8.298257e+00 5.768405e-01 * time: 0.063751220703125 10 8.145146e+00 5.158911e-01 * time: 0.0696101188659668 11 8.033811e+00 4.742253e-01 * time: 0.0761561393737793 12 7.951600e+00 4.700619e-01 * time: 0.11073899269104004 13 7.923141e+00 6.973155e-01 * time: 0.11396503448486328 14 7.870060e+00 4.118104e-01 * time: 0.11706900596618652 15 7.857083e+00 5.350892e-01 * time: 0.12032914161682129 16 7.820183e+00 4.187180e-01 * time: 0.12432599067687988 17 7.781929e+00 5.020283e-01 * time: 0.12743902206420898 18 7.755426e+00 4.000637e-01 * time: 0.13132405281066895 19 7.741935e+00 4.450722e-01 * time: 0.13444304466247559 20 7.733792e+00 3.611611e-01 * time: 0.1376180648803711 21 7.722208e+00 2.701878e-01 * time: 0.1407790184020996 22 7.713792e+00 3.308856e-01 * time: 0.1438741683959961 23 7.707877e+00 3.095860e-01 * time: 0.147003173828125 24 7.702640e+00 2.850163e-01 * time: 0.15026307106018066 25 7.693755e+00 2.520440e-01 * time: 0.15374016761779785 26 7.686768e+00 2.699542e-01 * time: 0.15687012672424316 27 7.685180e+00 2.624248e-01 * time: 0.16009020805358887 28 7.681179e+00 1.951314e-01 * time: 0.16480302810668945 29 7.677673e+00 2.632465e-01 * time: 0.1702280044555664 30 7.672823e+00 2.599632e-01 * time: 0.1748790740966797 31 7.664361e+00 1.334921e-01 * time: 0.18097305297851562 32 7.662072e+00 2.750370e-01 * time: 0.18641018867492676 33 7.656434e+00 1.760239e-01 * time: 0.1929171085357666 34 7.656242e+00 2.819530e-01 * time: 0.19822907447814941 35 7.651541e+00 1.863965e-01 * time: 0.20467400550842285 36 7.648331e+00 2.164933e-01 * time: 0.20975208282470703 37 7.647984e+00 2.305950e-01 * time: 0.21582913398742676 38 7.644223e+00 1.377365e-01 * time: 0.22176098823547363 39 7.640991e+00 1.308886e-01 * time: 0.2278749942779541 40 7.638552e+00 1.094372e-01 * time: 0.25028109550476074 41 7.637247e+00 1.559888e-01 * time: 0.2533860206604004 42 7.634933e+00 1.335024e-01 * time: 0.2563471794128418 43 7.633332e+00 1.378269e-01 * time: 0.2592802047729492 44 7.631486e+00 1.471017e-01 * time: 0.262559175491333 45 7.629342e+00 7.429038e-02 * time: 0.26625514030456543 46 7.627941e+00 7.927926e-02 * time: 0.26926112174987793 47 7.627447e+00 1.294184e-01 * time: 0.2722771167755127 48 7.625968e+00 7.551238e-02 * time: 0.2768821716308594 49 7.625001e+00 6.862656e-02 * time: 0.27999305725097656 50 7.624053e+00 6.496034e-02 * time: 0.2830522060394287 51 7.623417e+00 6.819647e-02 * time: 0.2860841751098633 52 7.622818e+00 4.990528e-02 * time: 0.28994321823120117 53 7.622578e+00 6.896675e-02 * time: 0.2931480407714844 54 7.622075e+00 3.735801e-02 * time: 0.2970302104949951 55 7.621869e+00 6.357034e-02 * time: 0.30001115798950195 56 7.621503e+00 4.867156e-02 * time: 0.3030731678009033 57 7.621233e+00 5.502045e-02 * time: 0.30612802505493164 58 7.620985e+00 4.253322e-02 * time: 0.3092520236968994 59 7.620853e+00 5.839866e-02 * time: 0.31233716011047363 60 7.620371e+00 5.027390e-02 * time: 0.31530308723449707 61 7.620139e+00 6.203246e-02 * time: 0.31838321685791016 62 7.619735e+00 5.082239e-02 * time: 0.3214151859283447 63 7.619393e+00 4.064180e-02 * time: 0.3247561454772949 64 7.619333e+00 6.924952e-02 * time: 0.32776904106140137 65 7.619021e+00 4.010354e-02 * time: 0.3307371139526367 66 7.618750e+00 7.635480e-02 * time: 0.3337230682373047 67 7.618612e+00 4.561417e-02 * time: 0.33678722381591797 68 7.618541e+00 7.632950e-02 * time: 0.3405771255493164 69 7.618310e+00 7.969570e-02 * time: 0.34369611740112305 70 7.617821e+00 4.287992e-02 * time: 0.3630940914154053 71 7.617425e+00 6.870406e-02 * time: 0.3660430908203125 72 7.616875e+00 3.366292e-02 * time: 0.3699030876159668 73 7.616566e+00 5.532085e-02 * time: 0.372866153717041 74 7.616297e+00 3.974816e-02 * time: 0.3768000602722168 75 7.616243e+00 5.849349e-02 * time: 0.37979817390441895 76 7.616223e+00 4.705534e-02 * time: 0.3827931880950928 77 7.616054e+00 6.111076e-02 * time: 0.3867170810699463 78 7.615660e+00 4.233811e-02 * time: 0.39074206352233887 79 7.615379e+00 7.147692e-02 * time: 0.3938741683959961 80 7.614933e+00 6.416441e-02 * time: 0.3968780040740967 81 7.614879e+00 4.140046e-02 * time: 0.39997005462646484 82 7.614550e+00 3.417496e-02 * time: 0.4038112163543701 83 7.614260e+00 2.930962e-02 * time: 0.40680503845214844 84 7.614059e+00 6.696491e-02 * time: 0.4098021984100342 85 7.613832e+00 4.517744e-02 * time: 0.41358208656311035 86 7.613485e+00 3.696788e-02 * time: 0.4175751209259033 87 7.613430e+00 4.164094e-02 * time: 0.4206051826477051 88 7.613236e+00 2.950946e-02 * time: 0.4236412048339844 89 7.613060e+00 4.014367e-02 * time: 0.42660999298095703 90 7.612920e+00 4.530450e-02 * time: 0.4296431541442871 91 7.612666e+00 4.512535e-02 * time: 0.4337332248687744 92 7.612589e+00 4.888226e-02 * time: 0.43712615966796875 93 7.612517e+00 5.051830e-02 * time: 0.4408760070800781 94 7.612259e+00 4.780878e-02 * time: 0.4441821575164795 95 7.612153e+00 4.427455e-02 * time: 0.4472529888153076 96 7.612051e+00 5.523071e-02 * time: 0.4503471851348877 97 7.611797e+00 4.794785e-02 * time: 0.45337820053100586 98 7.611531e+00 4.616947e-02 * time: 0.4676392078399658 99 7.611316e+00 3.163070e-02 * time: 0.4714641571044922 100 7.611276e+00 4.972612e-02 * time: 0.47460412979125977 101 7.611120e+00 4.902382e-02 * time: 0.4784560203552246 102 7.610873e+00 4.003331e-02 * time: 0.48153209686279297 103 7.610599e+00 2.962464e-02 * time: 0.485288143157959 104 7.610364e+00 2.879457e-02 * time: 0.48934412002563477 105 7.610209e+00 2.745396e-02 * time: 0.49321413040161133 106 7.610100e+00 3.967563e-02 * time: 0.4966590404510498 107 7.609961e+00 3.071004e-02 * time: 0.5007731914520264 108 7.609774e+00 2.141246e-02 * time: 0.5046210289001465 109 7.609687e+00 1.699303e-02 * time: 0.5085041522979736 110 7.609656e+00 3.522981e-02 * time: 0.5128481388092041 111 7.609654e+00 3.987945e-02 * time: 0.5158631801605225 112 7.609501e+00 2.501769e-02 * time: 0.519017219543457 113 7.609378e+00 3.072030e-02 * time: 0.5220541954040527 114 7.609239e+00 3.377606e-02 * time: 0.5250520706176758 115 7.609011e+00 2.629497e-02 * time: 0.5283660888671875 116 7.608991e+00 4.683545e-02 * time: 0.5314991474151611 117 7.608933e+00 3.356111e-02 * time: 0.5345752239227295 118 7.608736e+00 2.505826e-02 * time: 0.5377521514892578 119 7.608555e+00 3.665424e-02 * time: 0.5407431125640869 120 7.608372e+00 2.363868e-02 * time: 0.5449399948120117 121 7.608194e+00 2.957404e-02 * time: 0.5493760108947754 122 7.608047e+00 1.836890e-02 * time: 0.5532190799713135 123 7.607988e+00 2.857453e-02 * time: 0.556779146194458 124 7.607896e+00 2.995345e-02 * time: 0.5611422061920166 125 7.607725e+00 1.992903e-02 * time: 0.5763092041015625 126 7.607663e+00 2.746698e-02 * time: 0.5792632102966309 127 7.607529e+00 2.106534e-02 * time: 0.5830950736999512 128 7.607421e+00 2.387112e-02 * time: 0.5860772132873535 129 7.607341e+00 2.323160e-02 * time: 0.5892210006713867 130 7.607216e+00 3.027004e-02 * time: 0.5922009944915771 131 7.607112e+00 2.959508e-02 * time: 0.5951700210571289 132 7.607094e+00 2.932824e-02 * time: 0.5982379913330078 133 7.607015e+00 2.870490e-02 * time: 0.6014730930328369 134 7.606992e+00 3.914016e-02 * time: 0.6047031879425049 135 7.606850e+00 2.764842e-02 * time: 0.6077940464019775 136 7.606713e+00 3.164189e-02 * time: 0.61080002784729 137 7.606542e+00 2.572833e-02 * time: 0.6146290302276611 138 7.606435e+00 1.820954e-02 * time: 0.6183362007141113 139 7.606255e+00 2.055410e-02 * time: 0.6223130226135254 140 7.606119e+00 1.972259e-02 * time: 0.6263031959533691 141 7.606002e+00 2.063439e-02 * time: 0.6293540000915527 142 7.605994e+00 2.788884e-02 * time: 0.6323561668395996 143 7.605975e+00 3.637467e-02 * time: 0.6356191635131836 144 7.605871e+00 2.875584e-02 * time: 0.6395890712738037 145 7.605787e+00 3.737618e-02 * time: 0.6427321434020996 146 7.605721e+00 4.622652e-02 * time: 0.6458430290222168 147 7.605668e+00 3.754051e-02 * time: 0.649224042892456 148 7.605544e+00 3.361410e-02 * time: 0.6524381637573242 149 7.605355e+00 2.880915e-02 * time: 0.656400203704834 150 7.605282e+00 3.323432e-02 * time: 0.6595320701599121 151 7.605104e+00 3.567696e-02 * time: 0.6632671356201172 152 7.605087e+00 4.277122e-02 * time: 0.6663320064544678 153 7.604903e+00 3.671312e-02 * time: 0.6702370643615723 154 7.604721e+00 3.423207e-02 * time: 0.6845030784606934 155 7.604599e+00 3.365080e-02 * time: 0.6875340938568115 156 7.604468e+00 2.139750e-02 * time: 0.690709114074707 157 7.604294e+00 2.916278e-02 * time: 0.6946451663970947 158 7.604144e+00 1.680242e-02 * time: 0.698479175567627 159 7.604071e+00 4.157167e-02 * time: 0.7015590667724609 160 7.603942e+00 3.413969e-02 * time: 0.7047510147094727 161 7.603774e+00 2.456552e-02 * time: 0.7085740566253662 162 7.603711e+00 2.305136e-02 * time: 0.712367057800293 163 7.603599e+00 2.211205e-02 * time: 0.7171900272369385 164 7.603512e+00 2.205009e-02 * time: 0.7213001251220703 165 7.603482e+00 2.602874e-02 * time: 0.7245612144470215 166 7.603381e+00 2.400506e-02 * time: 0.7282421588897705 167 7.603311e+00 1.492612e-02 * time: 0.732370138168335 168 7.603247e+00 2.066597e-02 * time: 0.7353332042694092 169 7.603235e+00 2.816003e-02 * time: 0.7382621765136719 170 7.603204e+00 2.462509e-02 * time: 0.7412362098693848 171 7.603131e+00 2.253180e-02 * time: 0.7443389892578125 172 7.602970e+00 2.212456e-02 * time: 0.747298002243042 173 7.602913e+00 2.262561e-02 * time: 0.7504680156707764 174 7.602869e+00 2.376628e-02 * time: 0.7536020278930664 175 7.602775e+00 2.924218e-02 * time: 0.7567241191864014 176 7.602663e+00 1.756108e-02 * time: 0.760897159576416 177 7.602573e+00 1.902948e-02 * time: 0.7639551162719727 178 7.602494e+00 1.892095e-02 * time: 0.7669742107391357 179 7.602451e+00 3.367653e-02 * time: 0.7715680599212646 180 7.602349e+00 2.319436e-02 * time: 0.775705099105835 181 7.602341e+00 2.888374e-02 * time: 0.7899911403656006 182 7.602234e+00 3.170918e-02 * time: 0.7933192253112793 183 7.602180e+00 4.313864e-02 * time: 0.7963712215423584 184 7.601940e+00 3.215992e-02 * time: 0.7999510765075684 185 7.601717e+00 3.566516e-02 * time: 0.8038690090179443 186 7.601505e+00 3.347708e-02 * time: 0.8069941997528076 187 7.601498e+00 4.710457e-02 * time: 0.8099150657653809 188 7.601187e+00 4.480785e-02 * time: 0.8130011558532715 189 7.601025e+00 6.264988e-02 * time: 0.8160312175750732 190 7.600704e+00 4.261948e-02 * time: 0.8197472095489502 191 7.600438e+00 4.055995e-02 * time: 0.826714038848877 192 7.600185e+00 4.764402e-02 * time: 0.8305211067199707 193 7.599901e+00 6.146472e-02 * time: 0.8335700035095215 194 7.599566e+00 3.747784e-02 * time: 0.8375611305236816 195 7.599287e+00 3.859206e-02 * time: 0.8407211303710938 196 7.599091e+00 3.549398e-02 * time: 0.8445870876312256 197 7.598845e+00 3.426347e-02 * time: 0.8476331233978271 198 7.598604e+00 2.526132e-02 * time: 0.8515481948852539 199 7.598401e+00 2.921411e-02 * time: 0.85465407371521 200 7.598266e+00 3.131272e-02 * time: 0.8577160835266113 201 7.598058e+00 3.919834e-02 * time: 0.8607370853424072 202 7.598027e+00 4.014128e-02 * time: 0.8637611865997314 203 7.597894e+00 2.835059e-02 * time: 0.8667440414428711 204 7.597665e+00 1.999775e-02 * time: 0.8702352046966553 205 7.597518e+00 3.115744e-02 * time: 0.8732750415802002 206 7.597366e+00 2.292758e-02 * time: 0.8762969970703125 207 7.597291e+00 2.748302e-02 * time: 0.8792481422424316 208 7.597222e+00 2.824212e-02 * time: 0.8825821876525879 209 7.597101e+00 1.707600e-02 * time: 0.8856470584869385 210 7.597000e+00 2.337972e-02 * time: 0.901576042175293 211 7.596877e+00 2.414328e-02 * time: 0.9053490161895752 212 7.596748e+00 3.249982e-02 * time: 0.9084970951080322 213 7.596599e+00 2.574114e-02 * time: 0.9126532077789307 214 7.596406e+00 2.198796e-02 * time: 0.9175870418548584 215 7.596308e+00 4.039008e-02 * time: 0.9206202030181885 216 7.596129e+00 2.585984e-02 * time: 0.9245471954345703 217 7.596015e+00 4.297080e-02 * time: 0.927666187286377 218 7.595744e+00 2.005980e-02 * time: 0.9334790706634521 219 7.595662e+00 3.387969e-02 * time: 0.9367051124572754 220 7.595448e+00 3.439935e-02 * time: 0.9401421546936035 221 7.595436e+00 4.093395e-02 * time: 0.9435520172119141 222 7.595265e+00 4.132066e-02 * time: 0.946641206741333 223 7.595023e+00 2.564528e-02 * time: 0.9515261650085449 224 7.594813e+00 3.123543e-02 * time: 0.9546411037445068 225 7.594670e+00 3.760023e-02 * time: 0.9577770233154297 226 7.594502e+00 3.632711e-02 * time: 0.9617199897766113 227 7.594477e+00 4.688546e-02 * time: 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