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 3.077016e+01 7.099970e+00 * time: 0.0017120838165283203 1 2.776218e+01 4.512457e+00 * time: 0.0064890384674072266 2 1.935778e+01 4.836689e+00 * time: 0.01180410385131836 3 1.343812e+01 3.448865e+00 * time: 0.017058134078979492 4 1.157216e+01 3.099052e+00 * time: 0.02149510383605957 5 1.066942e+01 2.204533e+00 * time: 0.05390596389770508 6 9.501471e+00 1.473333e+00 * time: 0.05660605430603027 7 8.777708e+00 1.091840e+00 * time: 0.059127092361450195 8 8.440348e+00 7.715083e-01 * time: 0.06161093711853027 9 8.403502e+00 1.330032e+00 * time: 0.06386399269104004 10 8.303916e+00 1.135657e+00 * time: 0.06653308868408203 11 8.271293e+00 1.160964e+00 * time: 0.06931710243225098 12 8.137761e+00 6.803516e-01 * time: 0.07297110557556152 13 8.048212e+00 6.851147e-01 * time: 0.07725000381469727 14 8.004141e+00 7.643399e-01 * time: 0.08101511001586914 15 7.962402e+00 4.475359e-01 * time: 0.0842750072479248 16 7.929738e+00 5.183875e-01 * time: 0.08750295639038086 17 7.902781e+00 3.430982e-01 * time: 0.09162712097167969 18 7.871907e+00 2.653334e-01 * time: 0.09636998176574707 19 7.846709e+00 4.816162e-01 * time: 0.0997779369354248 20 7.821301e+00 3.078363e-01 * time: 0.10334396362304688 21 7.797091e+00 4.357181e-01 * time: 0.10680794715881348 22 7.788851e+00 4.507833e-01 * time: 0.11038804054260254 23 7.767902e+00 2.896440e-01 * time: 0.11412310600280762 24 7.748439e+00 4.247666e-01 * time: 0.11757612228393555 25 7.744732e+00 4.381962e-01 * time: 0.1211240291595459 26 7.734628e+00 3.889163e-01 * time: 0.12470698356628418 27 7.711182e+00 2.092718e-01 * time: 0.12864112854003906 28 7.699710e+00 3.116059e-01 * time: 0.13201093673706055 29 7.686328e+00 2.396813e-01 * time: 0.13657903671264648 30 7.675046e+00 2.855162e-01 * time: 0.14107799530029297 31 7.663566e+00 1.737080e-01 * time: 0.1461191177368164 32 7.655602e+00 2.191812e-01 * time: 0.14963793754577637 33 7.649349e+00 1.262314e-01 * time: 0.1541740894317627 34 7.647846e+00 1.985115e-01 * time: 0.15772700309753418 35 7.645812e+00 1.488947e-01 * time: 0.176285982131958 36 7.643513e+00 1.658047e-01 * time: 0.17835712432861328 37 7.641865e+00 1.402109e-01 * time: 0.1803750991821289 38 7.638447e+00 1.417783e-01 * time: 0.1824660301208496 39 7.637949e+00 1.295858e-01 * time: 0.18458199501037598 40 7.635803e+00 7.828126e-02 * time: 0.18709206581115723 41 7.634192e+00 1.751718e-01 * time: 0.18947505950927734 42 7.633979e+00 1.848441e-01 * time: 0.19156098365783691 43 7.633042e+00 1.365978e-01 * time: 0.193648099899292 44 7.631218e+00 1.103198e-01 * time: 0.19582104682922363 45 7.628926e+00 6.951779e-02 * time: 0.19847702980041504 46 7.628550e+00 1.140155e-01 * time: 0.2005629539489746 47 7.628082e+00 1.216560e-01 * time: 0.20261907577514648 48 7.626822e+00 8.612412e-02 * time: 0.20560693740844727 49 7.625805e+00 9.981885e-02 * time: 0.20760512351989746 50 7.624409e+00 4.687745e-02 * time: 0.21011710166931152 51 7.623660e+00 7.903797e-02 * time: 0.21327805519104004 52 7.622912e+00 5.746376e-02 * time: 0.2176680564880371 53 7.621963e+00 4.712308e-02 * time: 0.22241592407226562 54 7.621056e+00 2.792622e-02 * time: 0.22679710388183594 55 7.620473e+00 4.247146e-02 * time: 0.23029112815856934 56 7.620007e+00 4.527969e-02 * time: 0.23382210731506348 57 7.619345e+00 4.876799e-02 * time: 0.23890399932861328 58 7.618601e+00 3.379615e-02 * time: 0.24353694915771484 59 7.617984e+00 3.279501e-02 * time: 0.24718308448791504 60 7.617432e+00 3.803242e-02 * time: 0.2507619857788086 61 7.616830e+00 3.485162e-02 * time: 0.2542901039123535 62 7.616441e+00 3.949164e-02 * time: 0.25768303871154785 63 7.615949e+00 5.935234e-02 * time: 0.2600820064544678 64 7.615870e+00 6.170036e-02 * time: 0.2635040283203125 65 7.615670e+00 5.540708e-02 * time: 0.2667360305786133 66 7.615193e+00 4.567557e-02 * time: 0.27041196823120117 67 7.614811e+00 4.141398e-02 * time: 0.2846550941467285 68 7.614615e+00 3.737054e-02 * time: 0.2867860794067383 69 7.614409e+00 4.051927e-02 * time: 0.2888491153717041 70 7.614267e+00 4.847423e-02 * time: 0.29106807708740234 71 7.614074e+00 3.389768e-02 * time: 0.29305100440979004 72 7.613861e+00 4.345371e-02 * time: 0.2950451374053955 73 7.613659e+00 3.658832e-02 * time: 0.29713892936706543 74 7.613401e+00 3.278005e-02 * time: 0.29962992668151855 75 7.613152e+00 2.286754e-02 * time: 0.30224013328552246 76 7.612922e+00 2.451509e-02 * time: 0.3047349452972412 77 7.612815e+00 3.533614e-02 * time: 0.3068680763244629 78 7.612645e+00 4.091742e-02 * time: 0.30910801887512207 79 7.612290e+00 2.756706e-02 * time: 0.3118460178375244 80 7.612222e+00 4.471561e-02 * time: 0.31412696838378906 81 7.611944e+00 3.815673e-02 * time: 0.31670212745666504 82 7.611844e+00 5.912756e-02 * time: 0.3186929225921631 83 7.611519e+00 5.506328e-02 * time: 0.32071995735168457 84 7.611479e+00 5.425979e-02 * time: 0.32273411750793457 85 7.611125e+00 4.706506e-02 * time: 0.32487010955810547 86 7.610800e+00 3.669273e-02 * time: 0.3270409107208252 87 7.610594e+00 5.225045e-02 * time: 0.3293769359588623 88 7.610233e+00 2.602166e-02 * time: 0.3322739601135254 89 7.610010e+00 3.228904e-02 * time: 0.33500194549560547 90 7.609850e+00 3.296205e-02 * time: 0.33707404136657715 91 7.609641e+00 2.189554e-02 * time: 0.3397400379180908 92 7.609456e+00 2.068626e-02 * time: 0.34183192253112793 93 7.609412e+00 3.659626e-02 * time: 0.34400510787963867 94 7.609405e+00 4.413358e-02 * time: 0.3465549945831299 95 7.609303e+00 4.463400e-02 * time: 0.3487880229949951 96 7.609146e+00 2.616280e-02 * time: 0.3507699966430664 97 7.608947e+00 3.418183e-02 * time: 0.3527860641479492 98 7.608737e+00 1.961115e-02 * time: 0.3562641143798828 99 7.608591e+00 2.283681e-02 * time: 0.36591100692749023 100 7.608438e+00 1.791892e-02 * time: 0.3681190013885498 101 7.608436e+00 3.356030e-02 * time: 0.3700859546661377 102 7.608348e+00 2.633680e-02 * time: 0.37213706970214844 103 7.608270e+00 1.926483e-02 * time: 0.3746640682220459 104 7.608202e+00 3.104673e-02 * time: 0.37706995010375977 105 7.608122e+00 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