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 3.076795e+01 7.503771e+00 * time: 0.0020389556884765625 1 2.850236e+01 6.173472e+00 * time: 0.008480072021484375 2 2.104104e+01 6.919747e+00 * time: 0.015841007232666016 3 1.491597e+01 4.598344e+00 * time: 0.023647069931030273 4 1.126953e+01 1.938035e+00 * time: 0.030544042587280273 5 1.053462e+01 2.504776e+00 * time: 0.03644394874572754 6 9.941404e+00 1.738964e+00 * time: 0.0425260066986084 7 9.324278e+00 1.413210e+00 * time: 0.048799991607666016 8 8.865733e+00 1.267031e+00 * time: 0.05532193183898926 9 8.667788e+00 1.081462e+00 * time: 0.061949968338012695 10 8.541607e+00 9.737555e-01 * time: 0.06846785545349121 11 8.421264e+00 7.820865e-01 * time: 0.07505106925964355 12 8.416931e+00 1.149503e+00 * time: 0.11214494705200195 13 8.342651e+00 8.691841e-01 * time: 0.11568093299865723 14 8.299733e+00 7.508414e-01 * time: 0.11880302429199219 15 8.246963e+00 6.333104e-01 * time: 0.12193489074707031 16 8.216101e+00 9.787482e-01 * time: 0.12495803833007812 17 8.165582e+00 4.814081e-01 * time: 0.12902498245239258 18 8.132928e+00 7.601908e-01 * time: 0.13222789764404297 19 8.097300e+00 6.906450e-01 * time: 0.1353929042816162 20 8.072718e+00 5.018843e-01 * time: 0.13840985298156738 21 8.068286e+00 6.941872e-01 * time: 0.14182806015014648 22 8.051294e+00 4.447927e-01 * time: 0.14503693580627441 23 8.013217e+00 3.658953e-01 * time: 0.1486189365386963 24 7.986649e+00 4.642554e-01 * time: 0.1518239974975586 25 7.973411e+00 4.339732e-01 * time: 0.15495800971984863 26 7.955554e+00 4.414018e-01 * time: 0.15821003913879395 27 7.945662e+00 4.620689e-01 * time: 0.16254901885986328 28 7.935328e+00 3.823380e-01 * time: 0.1665658950805664 29 7.915427e+00 2.302696e-01 * time: 0.17302489280700684 30 7.899154e+00 4.128918e-01 * time: 0.17808103561401367 31 7.895051e+00 5.744434e-01 * time: 0.1831209659576416 32 7.874350e+00 3.463292e-01 * time: 0.18949103355407715 33 7.863409e+00 5.354438e-01 * time: 0.1951148509979248 34 7.836191e+00 3.107415e-01 * time: 0.20070600509643555 35 7.816445e+00 5.281816e-01 * time: 0.20599102973937988 36 7.794423e+00 2.746795e-01 * time: 0.21248602867126465 37 7.779161e+00 3.549642e-01 * time: 0.21760201454162598 38 7.768131e+00 5.159801e-01 * time: 0.22307085990905762 39 7.762015e+00 3.996403e-01 * time: 0.22880196571350098 40 7.750701e+00 3.953044e-01 * time: 0.2501869201660156 41 7.733739e+00 3.273122e-01 * time: 0.2543659210205078 42 7.716369e+00 1.958495e-01 * time: 0.25845885276794434 43 7.697180e+00 2.908962e-01 * time: 0.2618858814239502 44 7.684226e+00 2.217537e-01 * time: 0.2657458782196045 45 7.677914e+00 2.393541e-01 * time: 0.26906585693359375 46 7.674764e+00 2.432048e-01 * time: 0.2722499370574951 47 7.664538e+00 1.752451e-01 * time: 0.27538394927978516 48 7.657095e+00 2.166913e-01 * time: 0.27858400344848633 49 7.650741e+00 1.622593e-01 * time: 0.2824840545654297 50 7.643812e+00 1.112002e-01 * time: 0.2868359088897705 51 7.641499e+00 2.613629e-01 * time: 0.28988194465637207 52 7.638484e+00 1.510957e-01 * time: 0.2929980754852295 53 7.636713e+00 1.230388e-01 * time: 0.29607295989990234 54 7.635442e+00 1.422057e-01 * time: 0.2991650104522705 55 7.633194e+00 1.276798e-01 * time: 0.30245494842529297 56 7.632034e+00 1.391272e-01 * time: 0.3057548999786377 57 7.630154e+00 9.138629e-02 * time: 0.30994486808776855 58 7.628769e+00 9.773973e-02 * time: 0.31313395500183105 59 7.627658e+00 8.352539e-02 * time: 0.3167390823364258 60 7.627235e+00 8.787924e-02 * time: 0.32022786140441895 61 7.626353e+00 6.979234e-02 * time: 0.3234279155731201 62 7.626059e+00 1.382001e-01 * time: 0.3266599178314209 63 7.625978e+00 1.189826e-01 * time: 0.3299880027770996 64 7.625710e+00 8.271356e-02 * time: 0.33326292037963867 65 7.625074e+00 9.444492e-02 * time: 0.3370189666748047 66 7.624624e+00 1.342753e-01 * time: 0.3420999050140381 67 7.623915e+00 6.508073e-02 * time: 0.3468198776245117 68 7.623039e+00 6.571763e-02 * time: 0.3662528991699219 69 7.622989e+00 1.299484e-01 * time: 0.36942386627197266 70 7.622075e+00 8.719087e-02 * time: 0.3733830451965332 71 7.621134e+00 7.483142e-02 * time: 0.37730908393859863 72 7.620764e+00 6.928794e-02 * time: 0.38079404830932617 73 7.620018e+00 7.769109e-02 * time: 0.38504600524902344 74 7.619123e+00 3.989617e-02 * time: 0.38902997970581055 75 7.618322e+00 4.010574e-02 * time: 0.3929250240325928 76 7.617969e+00 6.326625e-02 * time: 0.39818286895751953 77 7.617751e+00 1.106100e-01 * time: 0.4016439914703369 78 7.616654e+00 5.653593e-02 * time: 0.40582990646362305 79 7.615974e+00 9.941706e-02 * time: 0.4090230464935303 80 7.614859e+00 6.690946e-02 * time: 0.4134409427642822 81 7.613689e+00 6.148705e-02 * time: 0.4168229103088379 82 7.612411e+00 5.938995e-02 * time: 0.42017197608947754 83 7.611339e+00 4.599660e-02 * time: 0.42407989501953125 84 7.610906e+00 9.858877e-02 * time: 0.42763686180114746 85 7.610138e+00 7.441001e-02 * time: 0.4307389259338379 86 7.608952e+00 1.010586e-01 * time: 0.43398404121398926 87 7.607961e+00 8.436170e-02 * time: 0.4378838539123535 88 7.607823e+00 1.221717e-01 * time: 0.44114208221435547 89 7.607665e+00 1.406282e-01 * time: 0.44571399688720703 90 7.606951e+00 9.656510e-02 * time: 0.4487769603729248 91 7.605692e+00 9.165149e-02 * time: 0.4527699947357178 92 7.605677e+00 1.172422e-01 * time: 0.45705604553222656 93 7.604491e+00 7.218918e-02 * time: 0.46202993392944336 94 7.604445e+00 8.763104e-02 * time: 0.46704888343811035 95 7.603270e+00 7.334709e-02 * time: 0.4817018508911133 96 7.602100e+00 6.818422e-02 * time: 0.48502206802368164 97 7.601671e+00 8.317083e-02 * time: 0.4884529113769531 98 7.600964e+00 5.071106e-02 * time: 0.4915900230407715 99 7.600558e+00 3.375613e-02 * time: 0.49589085578918457 100 7.600410e+00 7.902292e-02 * time: 0.49932289123535156 101 7.600070e+00 4.917086e-02 * time: 0.503870964050293 102 7.599725e+00 4.286126e-02 * time: 0.5071380138397217 103 7.599528e+00 4.586485e-02 * time: 0.5102810859680176 104 7.599294e+00 4.198435e-02 * time: 0.5134360790252686 105 7.599258e+00 5.215479e-02 * time: 0.5165350437164307 106 7.598808e+00 5.459585e-02 * time: 0.5198838710784912 107 7.598658e+00 7.249420e-02 * time: 0.5230779647827148 108 7.598333e+00 6.560099e-02 * time: 0.5263268947601318 109 7.597955e+00 5.330009e-02 * time: 0.5297050476074219 110 7.597452e+00 4.287359e-02 * time: 0.5329439640045166 111 7.596891e+00 3.159582e-02 * time: 0.5377428531646729 112 7.596546e+00 3.778488e-02 * time: 0.5410349369049072 113 7.596485e+00 6.844052e-02 * time: 0.5444939136505127 114 7.596141e+00 4.456374e-02 * time: 0.5489699840545654 115 7.596026e+00 6.481510e-02 * time: 0.5523998737335205 116 7.595730e+00 3.991431e-02 * time: 0.5566129684448242 117 7.595453e+00 3.637096e-02 * time: 0.5601239204406738 118 7.595414e+00 5.759842e-02 * time: 0.5635690689086914 119 7.595247e+00 3.401749e-02 * time: 0.5679140090942383 120 7.595036e+00 3.452864e-02 * time: 0.572303056716919 121 7.594859e+00 2.305893e-02 * time: 0.5772349834442139 122 7.594775e+00 2.379900e-02 * time: 0.5824658870697021 123 7.594679e+00 1.941102e-02 * time: 0.6004219055175781 124 7.594601e+00 1.504851e-02 * time: 0.6042790412902832 125 7.594560e+00 1.865639e-02 * time: 0.6072709560394287 126 7.594496e+00 1.732401e-02 * time: 0.6111619472503662 127 7.594461e+00 2.208592e-02 * time: 0.6144008636474609 128 7.594420e+00 2.196065e-02 * time: 0.6175000667572021 129 7.594373e+00 1.782270e-02 * time: 0.6206729412078857 130 7.594330e+00 1.241412e-02 * time: 0.6246340274810791 131 7.594272e+00 1.449663e-02 * time: 0.6294310092926025 132 7.594225e+00 1.875029e-02 * time: 0.6329820156097412 133 7.594214e+00 2.233585e-02 * time: 0.6362209320068359 134 7.594168e+00 1.792658e-02 * time: 0.6396150588989258 135 7.594139e+00 1.867884e-02 * time: 0.6431219577789307 136 7.594121e+00 2.588331e-02 * time: 0.6467568874359131 137 7.594092e+00 2.447014e-02 * time: 0.649914026260376 138 7.594019e+00 2.051747e-02 * time: 0.654015064239502 139 7.593962e+00 1.930053e-02 * time: 0.6583089828491211 140 7.593892e+00 8.507175e-03 * time: 0.6631200313568115 141 7.593846e+00 1.436769e-02 * time: 0.6676769256591797 142 7.593802e+00 1.676631e-02 * time: 0.6722679138183594 143 7.593790e+00 2.207647e-02 * time: 0.6762259006500244 144 7.593727e+00 1.069258e-02 * time: 0.6803669929504395 145 7.593704e+00 1.980815e-02 * time: 0.6836190223693848 146 7.593672e+00 1.292378e-02 * time: 0.6875550746917725 147 7.593626e+00 1.496855e-02 * time: 0.692633867263794 148 7.593590e+00 8.754166e-03 * time: 0.6987550258636475 149 7.593579e+00 1.513008e-02 * time: 0.703685998916626 150 7.593549e+00 1.202735e-02 * time: 0.719235897064209 151 7.593533e+00 1.737725e-02 * time: 0.7223470211029053 152 7.593503e+00 1.296021e-02 * time: 0.7256600856781006 153 7.593497e+00 1.585223e-02 * time: 0.7288649082183838 154 7.593485e+00 1.093878e-02 * time: 0.731956958770752 155 7.593454e+00 8.177583e-03 * time: 0.7353079319000244 156 7.593441e+00 1.480453e-02 * time: 0.7389318943023682 157 7.593411e+00 1.169251e-02 * time: 0.742480993270874 158 7.593383e+00 1.182235e-02 * time: 0.7458679676055908 159 7.593379e+00 1.606276e-02 * time: 0.7489550113677979 160 7.593360e+00 1.050815e-02 * time: 0.7531130313873291 161 7.593347e+00 8.194181e-03 * time: 0.7573778629302979 162 7.593332e+00 8.292948e-03 * time: 0.7613949775695801 163 7.593327e+00 1.285735e-02 * time: 0.7646160125732422 164 7.593312e+00 1.158504e-02 * time: 0.7687680721282959 165 7.593309e+00 1.077731e-02 * time: 0.7721860408782959 166 7.593302e+00 1.038465e-02 * time: 0.7756569385528564 167 7.593286e+00 1.138088e-02 * time: 0.7791550159454346 168 7.593268e+00 6.093210e-03 * time: 0.782322883605957 169 7.593243e+00 1.093810e-02 * time: 0.7856988906860352 170 7.593219e+00 6.540312e-03 * time: 0.7897288799285889 171 7.593208e+00 9.746676e-03 * time: 0.7930150032043457 172 7.593206e+00 1.131440e-02 * time: 0.7962000370025635 173 7.593199e+00 1.448494e-02 * time: 0.7993860244750977 174 7.593184e+00 9.689039e-03 * time: 0.8038129806518555 175 7.593170e+00 9.467887e-03 * time: 0.8102579116821289 176 7.593150e+00 4.707899e-03 * time: 0.8178329467773438 177 7.593139e+00 7.741943e-03 * time: 0.8326079845428467 178 7.593124e+00 8.677130e-03 * time: 0.8358449935913086 179 7.593117e+00 7.275979e-03 * time: 0.8391749858856201 180 7.593102e+00 6.141387e-03 * time: 0.8423478603363037 181 7.593096e+00 1.267706e-02 * time: 0.84552001953125 182 7.593095e+00 1.527469e-02 * time: 0.8491699695587158 183 7.593077e+00 1.022908e-02 * time: 0.8534619808197021 184 7.593065e+00 7.343864e-03 * time: 0.8582069873809814 185 7.593048e+00 7.149303e-03 * time: 0.8616359233856201 186 7.593030e+00 1.100032e-02 * time: 0.8659038543701172 187 7.593024e+00 1.325340e-02 * time: 0.8692200183868408 188 7.593009e+00 9.471548e-03 * time: 0.8731689453125 189 7.593007e+00 1.716832e-02 * time: 0.8767108917236328 190 7.593006e+00 1.463446e-02 * time: 0.8799948692321777 191 7.592987e+00 1.084598e-02 * time: 0.8831548690795898 192 7.592981e+00 2.109459e-02 * time: 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1.457746e-02 * time: 0.9670748710632324 211 7.592745e+00 1.442534e-02 * time: 0.9708189964294434 212 7.592717e+00 1.008898e-02 * time: 0.9741039276123047 213 7.592702e+00 1.763376e-02 * time: 0.9773359298706055 214 7.592679e+00 1.052433e-02 * time: 0.9804098606109619 215 7.592663e+00 1.365699e-02 * time: 0.9836368560791016 216 7.592639e+00 1.005924e-02 * time: 0.9875578880310059 217 7.592625e+00 1.063797e-02 * time: 0.9909000396728516 218 7.592621e+00 1.332081e-02 * time: 0.9941320419311523 219 7.592600e+00 9.790048e-03 * time: 0.997169017791748 220 7.592582e+00 1.025096e-02 * time: 1.0009779930114746 221 7.592579e+00 1.898079e-02 * time: 1.004188060760498 222 7.592566e+00 1.162037e-02 * time: 1.0082180500030518 223 7.592555e+00 1.043071e-02 * time: 1.0114409923553467 224 7.592545e+00 8.585214e-03 * time: 1.0145390033721924 225 7.592536e+00 1.036358e-02 * time: 1.0177838802337646 226 7.592529e+00 8.093503e-03 * time: 1.0213499069213867 227 7.592517e+00 1.072320e-02 * time: 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