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[:, :, 1, 1], c=:blues)
Iter Function value Gradient norm 0 3.182901e+01 7.795833e+00 * time: 0.0015950202941894531 1 2.798890e+01 4.956841e+00 * time: 0.0071070194244384766 2 1.969671e+01 5.009689e+00 * time: 0.012188196182250977 3 1.282973e+01 3.257868e+00 * time: 0.016198158264160156 4 1.058469e+01 1.491212e+00 * time: 0.02001810073852539 5 9.702154e+00 1.997822e+00 * time: 0.023205041885375977 6 9.254038e+00 1.151561e+00 * time: 0.026545047760009766 7 8.963358e+00 7.520703e-01 * time: 0.029702186584472656 8 8.650594e+00 7.655641e-01 * time: 0.03285408020019531 9 8.408267e+00 6.017257e-01 * time: 0.036518096923828125 10 8.271961e+00 6.117114e-01 * time: 0.03993105888366699 11 8.188621e+00 5.625566e-01 * time: 0.04308509826660156 12 8.135284e+00 4.317111e-01 * time: 0.04674100875854492 13 8.102130e+00 6.246954e-01 * time: 0.04923415184020996 14 8.068053e+00 5.492248e-01 * time: 0.0516970157623291 15 8.025554e+00 4.863287e-01 * time: 0.10075712203979492 16 7.993303e+00 6.007184e-01 * time: 0.10330700874328613 17 7.985327e+00 6.639576e-01 * time: 0.10592508316040039 18 7.933416e+00 5.297995e-01 * time: 0.10845518112182617 19 7.912533e+00 6.157943e-01 * time: 0.11095619201660156 20 7.905912e+00 6.547221e-01 * time: 0.1134340763092041 21 7.871480e+00 4.694006e-01 * time: 0.11611008644104004 22 7.843980e+00 5.159290e-01 * time: 0.11857318878173828 23 7.840577e+00 7.385688e-01 * time: 0.12112212181091309 24 7.796935e+00 3.897199e-01 * time: 0.1244041919708252 25 7.763582e+00 4.413576e-01 * time: 0.12701010704040527 26 7.751453e+00 4.666119e-01 * time: 0.12949013710021973 27 7.730208e+00 3.057094e-01 * time: 0.13274502754211426 28 7.708048e+00 2.099653e-01 * time: 0.13609910011291504 29 7.693347e+00 1.816581e-01 * time: 0.1392500400543213 30 7.688766e+00 3.647389e-01 * time: 0.14177799224853516 31 7.676902e+00 2.190269e-01 * time: 0.1449570655822754 32 7.672153e+00 2.567602e-01 * time: 0.14786911010742188 33 7.669442e+00 2.991194e-01 * time: 0.15035510063171387 34 7.666644e+00 1.768711e-01 * time: 0.15282607078552246 35 7.654275e+00 1.469596e-01 * time: 0.1555311679840088 36 7.648596e+00 3.127774e-01 * time: 0.1580820083618164 37 7.639257e+00 1.720795e-01 * time: 0.16156911849975586 38 7.637666e+00 2.462429e-01 * time: 0.16434597969055176 39 7.633834e+00 1.873826e-01 * time: 0.17045807838439941 40 7.630676e+00 1.907696e-01 * time: 0.1729731559753418 41 7.627276e+00 1.216255e-01 * time: 0.17716002464294434 42 7.625329e+00 1.120364e-01 * time: 0.17985010147094727 43 7.625260e+00 1.344570e-01 * time: 0.18230819702148438 44 7.625175e+00 1.650475e-01 * time: 0.18477702140808105 45 7.623656e+00 8.448788e-02 * time: 0.18832111358642578 46 7.621858e+00 9.490464e-02 * time: 0.19078612327575684 47 7.620198e+00 6.964938e-02 * time: 0.1932840347290039 48 7.619273e+00 7.693805e-02 * time: 0.19625520706176758 49 7.619174e+00 1.024584e-01 * time: 0.1987450122833252 50 7.618532e+00 7.846299e-02 * time: 0.20122408866882324 51 7.617676e+00 5.774121e-02 * time: 0.20453810691833496 52 7.617220e+00 5.871266e-02 * time: 0.20714402198791504 53 7.616509e+00 7.402475e-02 * time: 0.2098691463470459 54 7.615856e+00 5.584623e-02 * time: 0.21361207962036133 55 7.615180e+00 1.991162e-02 * time: 0.21724319458007812 56 7.614999e+00 5.794120e-02 * time: 0.24823617935180664 57 7.614607e+00 3.400781e-02 * time: 0.25087714195251465 58 7.614333e+00 3.286529e-02 * time: 0.2541041374206543 59 7.614191e+00 6.456124e-02 * time: 0.2568662166595459 60 7.613795e+00 4.241773e-02 * time: 0.26003098487854004 61 7.613759e+00 5.302181e-02 * time: 0.2625131607055664 62 7.613444e+00 3.546896e-02 * time: 0.26500606536865234 63 7.613181e+00 3.568412e-02 * time: 0.26836204528808594 64 7.613113e+00 4.186827e-02 * time: 0.270855188369751 65 7.612850e+00 2.751318e-02 * time: 0.2733910083770752 66 7.612661e+00 3.530788e-02 * time: 0.2758781909942627 67 7.612502e+00 5.803999e-02 * time: 0.2795531749725342 68 7.612432e+00 3.986693e-02 * time: 0.28203606605529785 69 7.612256e+00 3.069685e-02 * time: 0.2845320701599121 70 7.612052e+00 2.478624e-02 * time: 0.2877640724182129 71 7.611846e+00 3.526024e-02 * time: 0.2902381420135498 72 7.611808e+00 5.641826e-02 * time: 0.292741060256958 73 7.611669e+00 3.760128e-02 * time: 0.29518699645996094 74 7.611489e+00 3.921610e-02 * time: 0.29811716079711914 75 7.611255e+00 2.570178e-02 * time: 0.3013191223144531 76 7.611249e+00 4.516344e-02 * time: 0.3038320541381836 77 7.611247e+00 4.865136e-02 * time: 0.30628514289855957 78 7.611073e+00 4.837257e-02 * time: 0.30893898010253906 79 7.610860e+00 3.213950e-02 * time: 0.3120880126953125 80 7.610694e+00 4.415006e-02 * time: 0.3145902156829834 81 7.610480e+00 3.869551e-02 * time: 0.317249059677124 82 7.610466e+00 4.853458e-02 * time: 0.31983304023742676 83 7.610169e+00 4.608561e-02 * time: 0.3229491710662842 84 7.610115e+00 4.571226e-02 * time: 0.3254671096801758 85 7.610068e+00 3.800216e-02 * time: 0.32854700088500977 86 7.609864e+00 3.433130e-02 * time: 0.33103203773498535 87 7.609557e+00 3.257779e-02 * time: 0.334240198135376 88 7.609286e+00 1.800476e-02 * time: 0.33776211738586426 89 7.609003e+00 3.259305e-02 * time: 0.340238094329834 90 7.608850e+00 5.413609e-02 * time: 0.34267616271972656 91 7.608618e+00 3.242661e-02 * time: 0.3464012145996094 92 7.608509e+00 4.729983e-02 * time: 0.3493931293487549 93 7.608368e+00 4.219664e-02 * time: 0.3525660037994385 94 7.608137e+00 3.481083e-02 * time: 0.35571718215942383 95 7.607720e+00 4.960533e-02 * time: 0.3584151268005371 96 7.607265e+00 3.568045e-02 * time: 0.3616490364074707 97 7.606896e+00 3.645053e-02 * time: 0.3878641128540039 98 7.606616e+00 6.424043e-02 * time: 0.3910250663757324 99 7.606402e+00 3.628539e-02 * time: 0.3941609859466553 100 7.606284e+00 5.187189e-02 * time: 0.39662599563598633 101 7.606106e+00 3.630876e-02 * time: 0.3995201587677002 102 7.605952e+00 6.238896e-02 * time: 0.4019811153411865 103 7.605201e+00 4.605068e-02 * time: 0.4044811725616455 104 7.604930e+00 2.125700e-02 * time: 0.40774106979370117 105 7.604794e+00 4.266766e-02 * time: 0.41025710105895996 106 7.604500e+00 3.235220e-02 * time: 0.4131321907043457 107 7.604314e+00 3.402249e-02 * time: 0.41560816764831543 108 7.604218e+00 3.828807e-02 * time: 0.4184300899505615 109 7.603939e+00 2.880304e-02 * time: 0.4211289882659912 110 7.603680e+00 3.570032e-02 * time: 0.423626184463501 111 7.603661e+00 5.223243e-02 * time: 0.42611002922058105 112 7.603573e+00 4.781460e-02 * time: 0.4288899898529053 113 7.603511e+00 5.308732e-02 * time: 0.4313681125640869 114 7.603218e+00 2.934813e-02 * time: 0.43389320373535156 115 7.602998e+00 4.714035e-02 * time: 0.43636417388916016 116 7.602779e+00 4.436420e-02 * time: 0.4392821788787842 117 7.602540e+00 2.649035e-02 * time: 0.44249510765075684 118 7.602269e+00 3.206203e-02 * time: 0.4457371234893799 119 7.602057e+00 1.525701e-02 * time: 0.4492931365966797 120 7.601916e+00 1.808723e-02 * time: 0.45178914070129395 121 7.601725e+00 2.235921e-02 * time: 0.4542710781097412 122 7.601540e+00 2.605546e-02 * time: 0.45746707916259766 123 7.601372e+00 2.081683e-02 * time: 0.46089911460876465 124 7.601361e+00 3.519971e-02 * time: 0.46361303329467773 125 7.601237e+00 2.963502e-02 * time: 0.46613121032714844 126 7.601113e+00 4.504419e-02 * time: 0.4687201976776123 127 7.600856e+00 3.194459e-02 * time: 0.47191715240478516 128 7.600688e+00 4.435267e-02 * time: 0.4743931293487549 129 7.600584e+00 3.807013e-02 * time: 0.47687602043151855 130 7.600405e+00 1.986216e-02 * time: 0.48204708099365234 131 7.600350e+00 3.975766e-02 * time: 0.4845280647277832 132 7.600168e+00 2.307156e-02 * time: 0.48699307441711426 133 7.599986e+00 2.683787e-02 * time: 0.4895491600036621 134 7.599839e+00 3.495979e-02 * time: 0.49206018447875977 135 7.599664e+00 2.017059e-02 * time: 0.49516916275024414 136 7.599588e+00 2.447524e-02 * time: 0.4976620674133301 137 7.599492e+00 2.536040e-02 * time: 0.5007021427154541 138 7.599341e+00 2.911457e-02 * time: 0.5032320022583008 139 7.599167e+00 1.657637e-02 * time: 0.5178442001342773 140 7.599140e+00 4.894781e-02 * time: 0.5207371711730957 141 7.599083e+00 5.652759e-02 * time: 0.5232131481170654 142 7.599049e+00 3.108235e-02 * time: 0.5256941318511963 143 7.598828e+00 3.971102e-02 * time: 0.5281581878662109 144 7.598669e+00 4.291273e-02 * time: 0.530764102935791 145 7.598515e+00 3.655640e-02 * time: 0.5332331657409668 146 7.598321e+00 3.458225e-02 * time: 0.535736083984375 147 7.598163e+00 3.773841e-02 * time: 0.5383071899414062 148 7.597995e+00 2.708749e-02 * time: 0.5416920185089111 149 7.597961e+00 4.778295e-02 * time: 0.5441761016845703 150 7.597921e+00 5.273311e-02 * time: 0.5466911792755127 151 7.597653e+00 2.963431e-02 * time: 0.550321102142334 152 7.597628e+00 3.952414e-02 * time: 0.5528440475463867 153 7.597499e+00 3.415718e-02 * time: 0.5553441047668457 154 7.597309e+00 3.217545e-02 * time: 0.557833194732666 155 7.597068e+00 2.679157e-02 * time: 0.5611810684204102 156 7.596879e+00 2.136610e-02 * time: 0.5643270015716553 157 7.596676e+00 1.526044e-02 * time: 0.5675060749053955 158 7.596658e+00 3.343681e-02 * time: 0.5733461380004883 159 7.596648e+00 3.243151e-02 * time: 0.5769190788269043 160 7.596594e+00 4.235168e-02 * time: 0.5793859958648682 161 7.596431e+00 3.235905e-02 * time: 0.5834751129150391 162 7.596411e+00 3.665062e-02 * time: 0.5861070156097412 163 7.596396e+00 6.312340e-02 * time: 0.588935136795044 164 7.596349e+00 4.656405e-02 * time: 0.5918810367584229 165 7.596157e+00 3.916277e-02 * time: 0.5945100784301758 166 7.595950e+00 3.717835e-02 * time: 0.5971870422363281 167 7.595838e+00 4.562296e-02 * time: 0.6003291606903076 168 7.595681e+00 2.792892e-02 * time: 0.6030020713806152 169 7.595471e+00 3.492325e-02 * time: 0.6063439846038818 170 7.595432e+00 4.008381e-02 * time: 0.6088321208953857 171 7.595385e+00 3.147767e-02 * time: 0.6114530563354492 172 7.595315e+00 3.703344e-02 * time: 0.6139659881591797 173 7.595098e+00 3.107451e-02 * time: 0.6172080039978027 174 7.594905e+00 2.908897e-02 * time: 0.6205980777740479 175 7.594661e+00 1.938092e-02 * time: 0.6237621307373047 176 7.594588e+00 2.449097e-02 * time: 0.626270055770874 177 7.594533e+00 4.594181e-02 * time: 0.6287860870361328 178 7.594502e+00 3.678008e-02 * time: 0.6314060688018799 179 7.594475e+00 3.162107e-02 * time: 0.6339080333709717 180 7.594475e+00 2.947044e-02 * time: 0.6364831924438477 181 7.594238e+00 3.069006e-02 * time: 0.6508641242980957 182 7.594183e+00 3.919755e-02 * time: 0.6533160209655762 183 7.594036e+00 3.039133e-02 * time: 0.6557660102844238 184 7.593930e+00 2.458984e-02 * time: 0.6582322120666504 185 7.593836e+00 3.060697e-02 * time: 0.6610691547393799 186 7.593726e+00 3.207243e-02 * time: 0.6636030673980713 187 7.593601e+00 1.303089e-02 * time: 0.6668531894683838 188 7.593514e+00 1.554783e-02 * time: 0.6701090335845947 189 7.593466e+00 1.778889e-02 * time: 0.6727910041809082 190 7.593460e+00 2.528457e-02 * time: 0.6752719879150391 191 7.593383e+00 1.752822e-02 * time: 0.6785240173339844 192 7.593311e+00 1.495942e-02 * time: 0.6816670894622803 193 7.593263e+00 2.459590e-02 * time: 0.6843180656433105 194 7.593235e+00 2.024434e-02 * time: 0.6868062019348145 195 7.593175e+00 1.062793e-02 * time: 0.6899580955505371 196 7.593119e+00 1.566207e-02 * time: 0.6928291320800781 197 7.593080e+00 9.183803e-03 * time: 0.69606614112854 198 7.593029e+00 1.006380e-02 * time: 0.6985161304473877 199 7.592992e+00 1.523107e-02 * time: 0.7011620998382568 200 7.592943e+00 1.223783e-02 * time: 0.7043380737304688 201 7.592896e+00 9.064214e-03 * time: 0.7075140476226807 202 7.592889e+00 1.228219e-02 * time: 0.7100281715393066 203 7.592868e+00 1.200123e-02 * time: 0.7128970623016357 204 7.592833e+00 1.453247e-02 * time: 0.7161011695861816 205 7.592797e+00 7.114914e-03 * time: 0.7192440032958984 206 7.592757e+00 7.954386e-03 * time: 0.7218871116638184 207 7.592734e+00 8.141967e-03 * time: 0.7243821620941162 208 7.592719e+00 8.903407e-03 * time: 0.7268610000610352 209 7.592687e+00 1.142010e-02 * time: 0.7293779850006104 210 7.592658e+00 7.328099e-03 * time: 0.7326741218566895 211 7.592632e+00 1.124453e-02 * time: 0.7351620197296143 212 7.592616e+00 1.487056e-02 * time: 0.7376441955566406 213 7.592590e+00 6.394042e-03 * time: 0.7408850193023682 214 7.592587e+00 1.080734e-02 * time: 0.7438621520996094 215 7.592572e+00 6.954836e-03 * time: 0.7463181018829346 216 7.592556e+00 8.242226e-03 * time: 0.7488231658935547 217 7.592549e+00 1.286166e-02 * time: 0.7513670921325684 218 7.592543e+00 8.024781e-03 * time: 0.7542610168457031 219 7.592533e+00 7.596814e-03 * time: 0.7568001747131348 220 7.592527e+00 9.322897e-03 * time: 0.7593450546264648 221 7.592518e+00 8.904697e-03 * time: 0.7621901035308838 222 7.592510e+00 1.029235e-02 * time: 0.7763280868530273 223 7.592506e+00 9.447667e-03 * time: 0.783703088760376 224 7.592501e+00 8.012172e-03 * time: 0.7862629890441895 225 7.592488e+00 5.456165e-03 * time: 0.789445161819458 226 7.592481e+00 6.026109e-03 * time: 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