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 3.092949e+01 7.144394e+00 * time: 0.0018329620361328125 1 3.039245e+01 4.509089e+00 * time: 0.007828950881958008 2 2.078743e+01 4.382530e+00 * time: 0.014898061752319336 3 1.507186e+01 3.588408e+00 * time: 0.02097296714782715 4 1.200423e+01 1.828797e+00 * time: 0.027698993682861328 5 1.147697e+01 2.192761e+00 * time: 0.034364938735961914 6 1.056654e+01 1.398145e+00 * time: 0.039016008377075195 7 9.477255e+00 1.622747e+00 * time: 0.0434720516204834 8 8.929882e+00 1.568717e+00 * time: 0.047866106033325195 9 8.678980e+00 1.114768e+00 * time: 0.05199003219604492 10 8.465320e+00 1.062911e+00 * time: 0.055989980697631836 11 8.287403e+00 1.022523e+00 * time: 0.06016397476196289 12 8.123438e+00 6.219178e-01 * time: 0.06507611274719238 13 8.016256e+00 4.476339e-01 * time: 0.06985902786254883 14 7.937866e+00 3.560243e-01 * time: 0.07485198974609375 15 7.901210e+00 6.028070e-01 * time: 0.0787210464477539 16 7.885194e+00 4.175437e-01 * time: 0.08263301849365234 17 7.877753e+00 7.210553e-01 * time: 0.0866250991821289 18 7.843726e+00 4.307479e-01 * time: 0.09159207344055176 19 7.810886e+00 4.612220e-01 * time: 0.09560203552246094 20 7.771616e+00 3.468957e-01 * time: 0.09949588775634766 21 7.753548e+00 3.208861e-01 * time: 0.10350394248962402 22 7.734905e+00 3.313122e-01 * time: 0.10852289199829102 23 7.718152e+00 2.835561e-01 * time: 0.11344695091247559 24 7.706856e+00 3.901759e-01 * time: 0.11706089973449707 25 7.701661e+00 3.109875e-01 * time: 0.1208341121673584 26 7.700990e+00 2.890605e-01 * time: 0.15642595291137695 27 7.695612e+00 2.386133e-01 * time: 0.1594541072845459 28 7.691636e+00 2.331825e-01 * time: 0.16239690780639648 29 7.683439e+00 1.674815e-01 * time: 0.1654679775238037 30 7.677170e+00 2.318238e-01 * time: 0.16843295097351074 31 7.672712e+00 1.914697e-01 * time: 0.17133307456970215 32 7.670791e+00 1.980980e-01 * time: 0.17427396774291992 33 7.668600e+00 1.394941e-01 * time: 0.17716693878173828 34 7.665276e+00 9.027116e-02 * time: 0.18077397346496582 35 7.664891e+00 2.126395e-01 * time: 0.18364405632019043 36 7.662996e+00 1.709069e-01 * time: 0.18675804138183594 37 7.661378e+00 1.907077e-01 * time: 0.18974900245666504 38 7.660759e+00 1.745404e-01 * time: 0.19270896911621094 39 7.660378e+00 1.150397e-01 * time: 0.1957099437713623 40 7.658113e+00 1.092295e-01 * time: 0.19860291481018066 41 7.656312e+00 1.714183e-01 * time: 0.20154404640197754 42 7.654793e+00 1.345714e-01 * time: 0.20441889762878418 43 7.651451e+00 1.158949e-01 * time: 0.20821905136108398 44 7.651082e+00 1.159841e-01 * time: 0.21104693412780762 45 7.649434e+00 1.281182e-01 * time: 0.21421289443969727 46 7.646652e+00 7.067507e-02 * time: 0.2182319164276123 47 7.644570e+00 7.518722e-02 * time: 0.22190594673156738 48 7.643033e+00 6.021664e-02 * time: 0.22555088996887207 49 7.641780e+00 8.117246e-02 * time: 0.22931408882141113 50 7.640602e+00 2.624572e-02 * time: 0.23301196098327637 51 7.640491e+00 6.791138e-02 * time: 0.23584699630737305 52 7.640415e+00 7.633086e-02 * time: 0.2388620376586914 53 7.639948e+00 7.220428e-02 * time: 0.24179291725158691 54 7.639241e+00 6.705693e-02 * time: 0.24467706680297852 55 7.638300e+00 9.608251e-02 * time: 0.24758696556091309 56 7.638262e+00 5.528776e-02 * time: 0.2506740093231201 57 7.637424e+00 3.926671e-02 * time: 0.2789750099182129 58 7.636904e+00 6.511189e-02 * time: 0.28186798095703125 59 7.636484e+00 6.006467e-02 * time: 0.28484606742858887 60 7.636103e+00 8.170569e-02 * time: 0.28774189949035645 61 7.635239e+00 6.511558e-02 * time: 0.2906229496002197 62 7.634345e+00 3.559750e-02 * time: 0.29429197311401367 63 7.633650e+00 3.705961e-02 * time: 0.29814791679382324 64 7.633483e+00 6.586718e-02 * time: 0.3009200096130371 65 7.632622e+00 4.458498e-02 * time: 0.3036930561065674 66 7.631810e+00 5.054105e-02 * time: 0.3073410987854004 67 7.631024e+00 3.047601e-02 * time: 0.3111259937286377 68 7.630530e+00 4.498980e-02 * time: 0.313978910446167 69 7.629999e+00 3.212314e-02 * time: 0.31752490997314453 70 7.629596e+00 3.507266e-02 * time: 0.32129693031311035 71 7.629121e+00 4.462318e-02 * time: 0.32411789894104004 72 7.628863e+00 4.731378e-02 * time: 0.3269350528717041 73 7.628691e+00 6.594313e-02 * time: 0.32981109619140625 74 7.628356e+00 2.989370e-02 * time: 0.3334200382232666 75 7.627832e+00 5.030390e-02 * time: 0.3362081050872803 76 7.627673e+00 3.972048e-02 * time: 0.3391430377960205 77 7.627116e+00 3.337398e-02 * time: 0.3419530391693115 78 7.626590e+00 3.622094e-02 * time: 0.34461402893066406 79 7.626237e+00 3.772358e-02 * time: 0.34746694564819336 80 7.626068e+00 5.355742e-02 * time: 0.3503739833831787 81 7.625921e+00 6.319780e-02 * time: 0.35317492485046387 82 7.625501e+00 5.445749e-02 * time: 0.3560030460357666 83 7.625219e+00 6.489779e-02 * time: 0.35884690284729004 84 7.624850e+00 6.726056e-02 * time: 0.36181092262268066 85 7.624435e+00 5.616170e-02 * time: 0.36464595794677734 86 7.624060e+00 5.613096e-02 * time: 0.36748290061950684 87 7.623566e+00 3.248903e-02 * time: 0.388261079788208 88 7.622762e+00 5.289825e-02 * time: 0.3912529945373535 89 7.622200e+00 3.355599e-02 * time: 0.39495301246643066 90 7.621552e+00 4.324419e-02 * time: 0.3977320194244385 91 7.621128e+00 4.930253e-02 * time: 0.4014160633087158 92 7.620677e+00 1.796776e-02 * time: 0.4050929546356201 93 7.620090e+00 2.794969e-02 * time: 0.40794897079467773 94 7.619696e+00 4.094311e-02 * time: 0.41083788871765137 95 7.619402e+00 4.664244e-02 * time: 0.41356492042541504 96 7.619316e+00 9.689157e-02 * time: 0.4163360595703125 97 7.618821e+00 4.791369e-02 * time: 0.42000603675842285 98 7.618453e+00 5.079010e-02 * time: 0.42371392250061035 99 7.617900e+00 4.724327e-02 * time: 0.4265270233154297 100 7.617323e+00 4.062095e-02 * time: 0.42929601669311523 101 7.616996e+00 3.213397e-02 * time: 0.4322481155395508 102 7.616759e+00 4.587416e-02 * time: 0.435197114944458 103 7.616481e+00 2.313925e-02 * time: 0.4387509822845459 104 7.616187e+00 3.874921e-02 * time: 0.44160008430480957 105 7.615872e+00 3.122982e-02 * time: 0.44446492195129395 106 7.615684e+00 3.908353e-02 * time: 0.44733691215515137 107 7.615346e+00 2.199401e-02 * time: 0.45098304748535156 108 7.615007e+00 2.563637e-02 * time: 0.45381999015808105 109 7.614907e+00 3.994865e-02 * time: 0.45656609535217285 110 7.614840e+00 5.328746e-02 * time: 0.4593510627746582 111 7.614720e+00 3.870166e-02 * time: 0.46227002143859863 112 7.614596e+00 5.527046e-02 * time: 0.4651370048522949 113 7.614375e+00 3.061071e-02 * time: 0.467972993850708 114 7.614209e+00 5.321926e-02 * time: 0.4708380699157715 115 7.613845e+00 2.474032e-02 * time: 0.47449588775634766 116 7.613610e+00 3.077614e-02 * time: 0.4772670269012451 117 7.613259e+00 3.313034e-02 * time: 0.48003506660461426 118 7.612879e+00 2.420330e-02 * time: 0.49411606788635254 119 7.612837e+00 4.552855e-02 * time: 0.49697208404541016 120 7.612749e+00 6.665934e-02 * time: 0.4997549057006836 121 7.612511e+00 4.065434e-02 * time: 0.5034348964691162 122 7.612304e+00 4.023747e-02 * time: 0.5070860385894775 123 7.612093e+00 4.559837e-02 * time: 0.5099430084228516 124 7.611757e+00 4.962074e-02 * time: 0.5137150287628174 125 7.611453e+00 1.860271e-02 * time: 0.5172760486602783 126 7.611201e+00 3.199354e-02 * time: 0.5201330184936523 127 7.611017e+00 3.525804e-02 * time: 0.5230669975280762 128 7.610882e+00 3.147620e-02 * time: 0.5258979797363281 129 7.610602e+00 3.271631e-02 * time: 0.528764009475708 130 7.610499e+00 5.527226e-02 * time: 0.5317621231079102 131 7.610371e+00 4.641534e-02 * time: 0.534682035446167 132 7.610278e+00 6.806592e-02 * time: 0.5375819206237793 133 7.610005e+00 3.757165e-02 * time: 0.5405130386352539 134 7.609653e+00 2.228037e-02 * time: 0.5442090034484863 135 7.609384e+00 2.447838e-02 * time: 0.547144889831543 136 7.609224e+00 2.839178e-02 * time: 0.5500121116638184 137 7.609101e+00 4.026911e-02 * time: 0.5529220104217529 138 7.608961e+00 1.848177e-02 * time: 0.5566360950469971 139 7.608831e+00 1.695676e-02 * 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7.607503e+00 1.560984e-02 * time: 0.6286380290985107 158 7.607485e+00 1.506689e-02 * time: 0.6315860748291016 159 7.607429e+00 1.359827e-02 * time: 0.6344900131225586 160 7.607340e+00 1.604387e-02 * time: 0.637347936630249 161 7.607330e+00 2.319956e-02 * time: 0.6402308940887451 162 7.607325e+00 2.307167e-02 * time: 0.6431660652160645 163 7.607304e+00 2.358642e-02 * time: 0.6462080478668213 164 7.607173e+00 1.581495e-02 * time: 0.6490380764007568 165 7.607044e+00 1.904193e-02 * time: 0.6518549919128418 166 7.607010e+00 3.189679e-02 * time: 0.6548349857330322 167 7.606896e+00 2.190862e-02 * time: 0.6585090160369873 168 7.606753e+00 2.583350e-02 * time: 0.6613221168518066 169 7.606626e+00 1.627174e-02 * time: 0.6650149822235107 170 7.606575e+00 2.266247e-02 * time: 0.6678969860076904 171 7.606545e+00 2.153687e-02 * time: 0.6706759929656982 172 7.606456e+00 1.768502e-02 * time: 0.6742970943450928 173 7.606414e+00 2.668723e-02 * time: 0.6770970821380615 174 7.606333e+00 2.265223e-02 * 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7.604565e+00 2.636902e-02 * time: 0.7485499382019043 193 7.604426e+00 2.640297e-02 * time: 0.751439094543457 194 7.604288e+00 2.448490e-02 * time: 0.7553260326385498 195 7.604209e+00 2.910371e-02 * time: 0.7583310604095459 196 7.604078e+00 2.841721e-02 * time: 0.7611560821533203 197 7.603978e+00 2.278817e-02 * time: 0.764847993850708 198 7.603862e+00 2.605204e-02 * time: 0.7677369117736816 199 7.603851e+00 4.544651e-02 * time: 0.7705249786376953 200 7.603698e+00 2.478883e-02 * time: 0.7734129428863525 201 7.603662e+00 4.545850e-02 * time: 0.7763159275054932 202 7.603443e+00 3.030984e-02 * time: 0.7799999713897705 203 7.603293e+00 3.436837e-02 * time: 0.7836019992828369 204 7.603137e+00 2.810137e-02 * time: 0.7865109443664551 205 7.602982e+00 3.637739e-02 * time: 0.7893979549407959 206 7.602979e+00 3.396018e-02 * time: 0.7922489643096924 207 7.602875e+00 3.194552e-02 * time: 0.7950499057769775 208 7.602831e+00 3.224236e-02 * time: 0.7980639934539795 209 7.602682e+00 2.430060e-02 * time: 0.8121390342712402 210 7.602634e+00 3.961676e-02 * time: 0.8150210380554199 211 7.602402e+00 3.559408e-02 * time: 0.8180301189422607 212 7.602197e+00 2.968846e-02 * time: 0.8216249942779541 213 7.602050e+00 3.246387e-02 * time: 0.8245069980621338 214 7.601963e+00 3.666965e-02 * time: 0.8275051116943359 215 7.601887e+00 2.387274e-02 * time: 0.8305950164794922 216 7.601670e+00 2.413248e-02 * time: 0.8343410491943359 217 7.601430e+00 2.227738e-02 * time: 0.8380100727081299 218 7.601192e+00 2.585857e-02 * time: 0.8417870998382568 219 7.600923e+00 1.747981e-02 * time: 0.8455410003662109 220 7.600768e+00 2.519239e-02 * time: 0.8486959934234619 221 7.600622e+00 2.783515e-02 * time: 0.852341890335083 222 7.600593e+00 4.221140e-02 * time: 0.8553268909454346 223 7.600548e+00 3.351037e-02 * time: 0.8583250045776367 224 7.600431e+00 2.502463e-02 * time: 0.8611910343170166 225 7.600249e+00 4.038561e-02 * time: 0.8640611171722412 226 7.600016e+00 2.906412e-02 * time: 0.8679280281066895 227 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