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.
using DFTK
using StaticArrays
using Plots
Unit cell. Having one of the lattice vectors as zero means a 2D system
a = 10
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
Apot(x, y, z) = .2 * @SVector [y - a/2, -(x - a/2), 0]
Apot(X) = Apot(X...);
Parameters
Ecut = 20 # Increase this for production
C = 500.0
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 3.573390e+01 1.270673e+01 * time: 0.001837015151977539 1 2.901446e+01 4.798925e+00 * time: 0.007480144500732422 2 2.450068e+01 3.153774e+00 * time: 0.013027191162109375 3 2.104124e+01 2.086765e+00 * time: 0.01845717430114746 4 1.971808e+01 1.601496e+00 * time: 0.02364516258239746 5 1.961157e+01 2.231076e+00 * time: 0.028060197830200195 6 1.923222e+01 1.582225e+00 * time: 0.03232002258300781 7 1.887210e+01 1.194757e+00 * time: 0.036570072174072266 8 1.854645e+01 9.641637e-01 * time: 0.040798187255859375 9 1.832456e+01 8.119555e-01 * time: 0.045101165771484375 10 1.817173e+01 6.088977e-01 * time: 0.04932117462158203 11 1.806599e+01 5.899637e-01 * time: 0.05356311798095703 12 1.797911e+01 6.802276e-01 * time: 0.05781817436218262 13 1.790603e+01 6.100871e-01 * time: 0.062090158462524414 14 1.784231e+01 8.564619e-01 * time: 0.06553006172180176 15 1.777079e+01 6.576499e-01 * time: 0.0688772201538086 16 1.777053e+01 1.274989e+00 * time: 0.07218813896179199 17 1.773873e+01 8.869446e-01 * time: 0.0755300521850586 18 1.766156e+01 6.417243e-01 * time: 0.07887816429138184 19 1.760219e+01 5.753854e-01 * time: 0.08272218704223633 20 1.754185e+01 6.411906e-01 * time: 0.08606505393981934 21 1.751483e+01 6.893562e-01 * time: 0.08940315246582031 22 1.751279e+01 1.053726e+00 * time: 0.0926971435546875 23 1.747748e+01 8.654686e-01 * time: 0.09598612785339355 24 1.744169e+01 5.448395e-01 * time: 0.09927511215209961 25 1.740650e+01 6.822720e-01 * time: 0.10259222984313965 26 1.736136e+01 5.234986e-01 * time: 0.1058812141418457 27 1.733550e+01 5.390121e-01 * time: 0.10922002792358398 28 1.731185e+01 4.685549e-01 * time: 0.11258721351623535 29 1.728311e+01 3.716967e-01 * time: 0.11589908599853516 30 1.726047e+01 3.496719e-01 * time: 0.11917400360107422 31 1.722939e+01 3.366926e-01 * time: 0.12243103981018066 32 1.720351e+01 3.596577e-01 * time: 0.1257190704345703 33 1.718220e+01 4.880529e-01 * time: 0.1548910140991211 34 1.715565e+01 4.972392e-01 * time: 0.15912914276123047 35 1.714955e+01 7.172081e-01 * time: 0.16212701797485352 36 1.712251e+01 6.253669e-01 * time: 0.1650981903076172 37 1.709023e+01 4.175085e-01 * time: 0.167985200881958 38 1.706482e+01 4.698043e-01 * time: 0.17085504531860352 39 1.704905e+01 4.169481e-01 * time: 0.17381811141967773 40 1.704240e+01 3.240295e-01 * time: 0.17688202857971191 41 1.703334e+01 3.225695e-01 * time: 0.17975521087646484 42 1.702521e+01 2.289613e-01 * time: 0.18270301818847656 43 1.701931e+01 2.190146e-01 * time: 0.18561816215515137 44 1.701574e+01 2.904510e-01 * time: 0.18850421905517578 45 1.701168e+01 1.591044e-01 * time: 0.19142818450927734 46 1.700721e+01 1.481492e-01 * time: 0.19463014602661133 47 1.700423e+01 1.868759e-01 * time: 0.19762611389160156 48 1.700214e+01 1.998513e-01 * time: 0.20072412490844727 49 1.699931e+01 1.653740e-01 * time: 0.2037220001220703 50 1.699643e+01 2.414779e-01 * time: 0.20667719841003418 51 1.699600e+01 2.798112e-01 * time: 0.2097182273864746 52 1.699350e+01 2.299809e-01 * time: 0.21262407302856445 53 1.699052e+01 1.844732e-01 * time: 0.21547913551330566 54 1.698797e+01 1.645113e-01 * time: 0.21838712692260742 55 1.698409e+01 1.666185e-01 * time: 0.2213151454925537 56 1.698193e+01 1.436701e-01 * time: 0.2242591381072998 57 1.697929e+01 1.307648e-01 * time: 0.22721099853515625 58 1.697730e+01 1.755405e-01 * time: 0.2308790683746338 59 1.697544e+01 2.296418e-01 * time: 0.23380303382873535 60 1.697296e+01 2.196035e-01 * time: 0.23676013946533203 61 1.696894e+01 2.198135e-01 * time: 0.23974323272705078 62 1.696547e+01 2.316806e-01 * time: 0.2428731918334961 63 1.696220e+01 2.772998e-01 * time: 0.24659299850463867 64 1.695616e+01 2.498578e-01 * time: 0.2494950294494629 65 1.695132e+01 2.212388e-01 * time: 0.2524731159210205 66 1.694798e+01 2.717563e-01 * time: 0.2553980350494385 67 1.694469e+01 2.411086e-01 * time: 0.2583141326904297 68 1.694070e+01 1.665585e-01 * time: 0.2612190246582031 69 1.693845e+01 1.135444e-01 * time: 0.26415109634399414 70 1.693713e+01 1.694117e-01 * time: 0.2670400142669678 71 1.693621e+01 1.128945e-01 * time: 0.27016615867614746 72 1.693526e+01 8.944661e-02 * time: 0.27352213859558105 73 1.693458e+01 8.048647e-02 * time: 0.27692508697509766 74 1.693425e+01 9.138856e-02 * time: 0.2802541255950928 75 1.693383e+01 6.577505e-02 * time: 0.2836000919342041 76 1.693363e+01 4.420535e-02 * time: 0.2869260311126709 77 1.693338e+01 4.532063e-02 * time: 0.29027223587036133 78 1.693328e+01 5.948380e-02 * time: 0.2936060428619385 79 1.693310e+01 3.220068e-02 * time: 0.29696011543273926 80 1.693295e+01 3.027421e-02 * time: 0.3005251884460449 81 1.693284e+01 4.270845e-02 * time: 0.30445003509521484 82 1.693280e+01 4.339660e-02 * time: 0.3256101608276367 83 1.693271e+01 3.652206e-02 * time: 0.3288991451263428 84 1.693259e+01 3.493256e-02 * time: 0.3318760395050049 85 1.693249e+01 3.794068e-02 * time: 0.334827184677124 86 1.693241e+01 3.759091e-02 * time: 0.33777523040771484 87 1.693232e+01 3.701097e-02 * time: 0.3406500816345215 88 1.693219e+01 3.761892e-02 * time: 0.3436000347137451 89 1.693208e+01 4.113818e-02 * time: 0.346555233001709 90 1.693195e+01 3.776853e-02 * time: 0.3494901657104492 91 1.693175e+01 3.728770e-02 * time: 0.35241007804870605 92 1.693146e+01 3.350343e-02 * time: 0.35527801513671875 93 1.693139e+01 6.842466e-02 * time: 0.35816407203674316 94 1.693114e+01 5.740468e-02 * time: 0.36105823516845703 95 1.693084e+01 5.069526e-02 * time: 0.3639712333679199 96 1.693076e+01 5.639160e-02 * time: 0.36688899993896484 97 1.693046e+01 5.951617e-02 * time: 0.36978912353515625 98 1.693020e+01 4.161410e-02 * time: 0.372650146484375 99 1.693001e+01 4.745062e-02 * time: 0.37562108039855957 100 1.692975e+01 3.940615e-02 * time: 0.37859416007995605 101 1.692954e+01 3.620303e-02 * time: 0.3814990520477295 102 1.692937e+01 3.342614e-02 * time: 0.38440704345703125 103 1.692915e+01 3.526521e-02 * time: 0.38726115226745605 104 1.692908e+01 6.454634e-02 * time: 0.39017510414123535 105 1.692888e+01 5.649424e-02 * time: 0.393125057220459 106 1.692867e+01 4.331902e-02 * time: 0.39606714248657227 107 1.692852e+01 3.738067e-02 * time: 0.3988950252532959 108 1.692821e+01 3.447704e-02 * time: 0.4018540382385254 109 1.692807e+01 3.511740e-02 * time: 0.40476107597351074 110 1.692791e+01 3.743523e-02 * time: 0.40758609771728516 111 1.692781e+01 4.463446e-02 * time: 0.41046905517578125 112 1.692767e+01 5.632572e-02 * time: 0.413316011428833 113 1.692751e+01 4.691502e-02 * time: 0.4161670207977295 114 1.692739e+01 3.933166e-02 * time: 0.4190521240234375 115 1.692725e+01 3.586532e-02 * time: 0.4226870536804199 116 1.692709e+01 2.875680e-02 * time: 0.42563414573669434 117 1.692700e+01 3.401281e-02 * time: 0.4284830093383789 118 1.692698e+01 3.876681e-02 * time: 0.4313240051269531 119 1.692691e+01 3.035561e-02 * time: 0.43418407440185547 120 1.692684e+01 2.962075e-02 * time: 0.4370710849761963 121 1.692672e+01 2.553967e-02 * time: 0.4400041103363037 122 1.692665e+01 3.246369e-02 * 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140 1.692541e+01 3.008570e-02 * time: 0.5116491317749023 141 1.692539e+01 3.045868e-02 * time: 0.5146431922912598 142 1.692535e+01 2.356405e-02 * time: 0.5175840854644775 143 1.692530e+01 2.770573e-02 * time: 0.5205421447753906 144 1.692526e+01 2.328018e-02 * time: 0.5234570503234863 145 1.692522e+01 1.714682e-02 * time: 0.5263731479644775 146 1.692521e+01 2.370733e-02 * time: 0.5292720794677734 147 1.692518e+01 2.044555e-02 * time: 0.5321781635284424 148 1.692517e+01 2.917003e-02 * time: 0.5351200103759766 149 1.692512e+01 2.145829e-02 * time: 0.5380451679229736 150 1.692508e+01 2.228256e-02 * time: 0.540971040725708 151 1.692505e+01 1.760307e-02 * time: 0.5439231395721436 152 1.692503e+01 3.134357e-02 * time: 0.5468010902404785 153 1.692501e+01 2.444356e-02 * time: 0.5496740341186523 154 1.692499e+01 2.549673e-02 * time: 0.5525221824645996 155 1.692495e+01 1.662706e-02 * time: 0.5553710460662842 156 1.692494e+01 1.256782e-02 * time: 0.5582640171051025 157 1.692492e+01 1.810513e-02 * 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1.692475e+01 9.825440e-03 * time: 0.6142990589141846 176 1.692474e+01 1.087044e-02 * time: 0.6171841621398926 177 1.692474e+01 9.297098e-03 * time: 0.6201410293579102 178 1.692473e+01 6.103006e-03 * time: 0.6230380535125732 179 1.692473e+01 1.027376e-02 * time: 0.6259441375732422 180 1.692473e+01 1.371966e-02 * time: 0.6288602352142334 181 1.692472e+01 1.407364e-02 * time: 0.6443090438842773 182 1.692472e+01 6.141369e-03 * time: 0.6480991840362549 183 1.692471e+01 7.387532e-03 * time: 0.6509921550750732 184 1.692471e+01 3.829857e-03 * time: 0.653968095779419 185 1.692471e+01 5.074726e-03 * time: 0.6568841934204102 186 1.692470e+01 4.787742e-03 * time: 0.6597230434417725 187 1.692470e+01 5.518929e-03 * time: 0.6626491546630859 188 1.692470e+01 4.305865e-03 * time: 0.6655561923980713 189 1.692470e+01 4.682996e-03 * time: 0.6684811115264893 190 1.692469e+01 7.500676e-03 * time: 0.6713101863861084 191 1.692469e+01 1.035813e-02 * time: 0.6741800308227539 192 1.692469e+01 7.430379e-03 * 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1.692462e+01 6.008671e-03 * time: 0.7300601005554199 211 1.692462e+01 4.255594e-03 * time: 0.7329092025756836 212 1.692462e+01 4.357171e-03 * time: 0.7357931137084961 213 1.692461e+01 4.610036e-03 * time: 0.7386910915374756 214 1.692461e+01 4.192991e-03 * time: 0.741657018661499 215 1.692461e+01 7.146883e-03 * time: 0.7445251941680908 216 1.692461e+01 7.641390e-03 * time: 0.7473471164703369 217 1.692461e+01 4.846327e-03 * time: 0.7502012252807617 218 1.692460e+01 4.209031e-03 * time: 0.7530801296234131 219 1.692460e+01 2.824560e-03 * time: 0.7560281753540039 220 1.692460e+01 3.174169e-03 * time: 0.7588942050933838 221 1.692460e+01 2.681620e-03 * time: 0.7617740631103516 222 1.692460e+01 3.192796e-03 * time: 0.7646291255950928 223 1.692459e+01 2.728014e-03 * time: 0.7674651145935059 224 1.692459e+01 2.927397e-03 * time: 0.7703170776367188 225 1.692459e+01 3.432120e-03 * time: 0.7732861042022705 226 1.692459e+01 6.250026e-03 * time: 0.7762560844421387 227 1.692459e+01 7.522768e-03 * 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