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 3.159302e+01 7.954240e+00 * time: 0.002009868621826172 1 2.780263e+01 4.201989e+00 * time: 0.008098840713500977 2 1.844724e+01 3.927732e+00 * time: 0.013222932815551758 3 1.179513e+01 2.401116e+00 * time: 0.01809096336364746 4 1.176026e+01 3.197918e+00 * time: 0.021857023239135742 5 1.067411e+01 2.006407e+00 * time: 0.025758028030395508 6 9.876183e+00 1.644597e+00 * time: 0.08055496215820312 7 9.267052e+00 1.386918e+00 * time: 0.08441281318664551 8 8.693555e+00 1.211421e+00 * time: 0.08846282958984375 9 8.313659e+00 5.763612e-01 * time: 0.09302282333374023 10 8.111231e+00 3.628822e-01 * time: 0.09696197509765625 11 8.070203e+00 8.309101e-01 * time: 0.10015082359313965 12 8.013909e+00 6.662059e-01 * time: 0.10308980941772461 13 7.964668e+00 5.244057e-01 * time: 0.10687994956970215 14 7.918505e+00 6.537759e-01 * time: 0.11037087440490723 15 7.882190e+00 6.176930e-01 * time: 0.11362695693969727 16 7.861291e+00 5.792358e-01 * time: 0.11765003204345703 17 7.857775e+00 8.097500e-01 * time: 0.12124991416931152 18 7.848050e+00 5.014490e-01 * time: 0.12543487548828125 19 7.840127e+00 5.779733e-01 * time: 0.1297929286956787 20 7.824555e+00 4.517695e-01 * time: 0.13374900817871094 21 7.794490e+00 3.513582e-01 * time: 0.13894295692443848 22 7.757036e+00 3.081699e-01 * time: 0.14368200302124023 23 7.734985e+00 3.045205e-01 * time: 0.14804697036743164 24 7.718664e+00 2.388876e-01 * time: 0.15254592895507812 25 7.705537e+00 2.366791e-01 * time: 0.15892481803894043 26 7.692254e+00 1.739177e-01 * time: 0.16576004028320312 27 7.683319e+00 1.584658e-01 * time: 0.1727449893951416 28 7.677054e+00 1.507266e-01 * time: 0.17956280708312988 29 7.672841e+00 1.055739e-01 * time: 0.1869640350341797 30 7.668999e+00 9.682580e-02 * time: 0.19370388984680176 31 7.668828e+00 2.076956e-01 * time: 0.19881796836853027 32 7.666279e+00 1.110749e-01 * time: 0.20399904251098633 33 7.662265e+00 8.720288e-02 * time: 0.20971393585205078 34 7.657904e+00 8.038331e-02 * time: 0.21512484550476074 35 7.654586e+00 7.594841e-02 * time: 0.2205049991607666 36 7.651832e+00 1.043558e-01 * time: 0.2258739471435547 37 7.651782e+00 1.814430e-01 * time: 0.23126792907714844 38 7.651068e+00 1.309087e-01 * time: 0.23656702041625977 39 7.648804e+00 8.965576e-02 * time: 0.2410259246826172 40 7.647562e+00 1.303120e-01 * time: 0.27387499809265137 41 7.646463e+00 1.290807e-01 * time: 0.2770049571990967 42 7.644676e+00 6.908459e-02 * time: 0.2811448574066162 43 7.643080e+00 6.639827e-02 * time: 0.28412604331970215 44 7.642016e+00 9.022525e-02 * time: 0.2871739864349365 45 7.641164e+00 7.792273e-02 * time: 0.29041194915771484 46 7.640939e+00 7.791173e-02 * time: 0.294003963470459 47 7.640468e+00 6.759223e-02 * time: 0.29709887504577637 48 7.639373e+00 6.334034e-02 * time: 0.301088809967041 49 7.639336e+00 1.203100e-01 * time: 0.3041689395904541 50 7.638557e+00 7.425392e-02 * time: 0.3071858882904053 51 7.638444e+00 1.192827e-01 * time: 0.3139069080352783 52 7.637401e+00 6.461574e-02 * time: 0.3180990219116211 53 7.636460e+00 6.890534e-02 * time: 0.32116198539733887 54 7.636328e+00 1.082896e-01 * time: 0.32561802864074707 55 7.636240e+00 9.020901e-02 * time: 0.33007192611694336 56 7.635659e+00 8.720103e-02 * time: 0.33474183082580566 57 7.634665e+00 7.511869e-02 * time: 0.3385770320892334 58 7.634052e+00 1.004545e-01 * time: 0.3418450355529785 59 7.633051e+00 9.960058e-02 * time: 0.3453710079193115 60 7.631484e+00 7.833726e-02 * time: 0.34929490089416504 61 7.631166e+00 1.032684e-01 * time: 0.3528900146484375 62 7.630135e+00 6.785583e-02 * time: 0.35856199264526367 63 7.629366e+00 7.404326e-02 * time: 0.3621039390563965 64 7.628062e+00 5.742880e-02 * time: 0.3652470111846924 65 7.626807e+00 5.870567e-02 * time: 0.3690769672393799 66 7.625676e+00 4.600444e-02 * time: 0.3733029365539551 67 7.625311e+00 5.860455e-02 * time: 0.37666893005371094 68 7.625202e+00 8.044384e-02 * time: 0.37981486320495605 69 7.624505e+00 7.451637e-02 * time: 0.3832108974456787 70 7.623780e+00 5.906647e-02 * time: 0.38716697692871094 71 7.623195e+00 6.250966e-02 * time: 0.3905489444732666 72 7.622122e+00 5.222588e-02 * time: 0.3939328193664551 73 7.621521e+00 4.825247e-02 * time: 0.3981039524078369 74 7.621401e+00 6.166178e-02 * time: 0.4238569736480713 75 7.621026e+00 7.264602e-02 * time: 0.42705202102661133 76 7.620596e+00 4.512260e-02 * time: 0.4309418201446533 77 7.620242e+00 4.078696e-02 * time: 0.4354228973388672 78 7.619790e+00 6.124635e-02 * time: 0.4388718605041504 79 7.619088e+00 2.700327e-02 * time: 0.4429128170013428 80 7.618657e+00 3.571521e-02 * time: 0.44611597061157227 81 7.618261e+00 3.572477e-02 * time: 0.44927287101745605 82 7.617977e+00 3.869710e-02 * time: 0.4524409770965576 83 7.617864e+00 5.379989e-02 * time: 0.4558110237121582 84 7.617623e+00 3.964675e-02 * time: 0.45918703079223633 85 7.617367e+00 5.980302e-02 * time: 0.46233701705932617 86 7.617175e+00 5.003426e-02 * time: 0.4653019905090332 87 7.616841e+00 3.962685e-02 * time: 0.4692239761352539 88 7.616767e+00 6.209490e-02 * time: 0.4726858139038086 89 7.616384e+00 4.306300e-02 * time: 0.4757578372955322 90 7.616216e+00 5.521380e-02 * time: 0.4792978763580322 91 7.615980e+00 4.927200e-02 * time: 0.48257899284362793 92 7.615740e+00 3.535453e-02 * time: 0.48575401306152344 93 7.615676e+00 5.091764e-02 * time: 0.4891040325164795 94 7.615635e+00 5.922005e-02 * time: 0.4926269054412842 95 7.615382e+00 3.365309e-02 * time: 0.49715185165405273 96 7.615225e+00 4.059725e-02 * time: 0.5007200241088867 97 7.614919e+00 3.376194e-02 * time: 0.5052328109741211 98 7.614778e+00 6.411183e-02 * time: 0.5083239078521729 99 7.614484e+00 3.135854e-02 * time: 0.513124942779541 100 7.614367e+00 3.038978e-02 * time: 0.5160679817199707 101 7.614215e+00 3.256369e-02 * time: 0.5201749801635742 102 7.614127e+00 3.432930e-02 * time: 0.5234129428863525 103 7.614009e+00 3.052841e-02 * time: 0.526453971862793 104 7.613897e+00 4.097112e-02 * time: 0.5294768810272217 105 7.613811e+00 3.884824e-02 * time: 0.5325348377227783 106 7.613644e+00 3.899195e-02 * time: 0.5359549522399902 107 7.613528e+00 3.106242e-02 * time: 0.5393610000610352 108 7.613366e+00 3.288238e-02 * time: 0.5520198345184326 109 7.613356e+00 4.421898e-02 * time: 0.555513858795166 110 7.613263e+00 3.957131e-02 * time: 0.5582869052886963 111 7.613135e+00 4.413959e-02 * time: 0.5611448287963867 112 7.613004e+00 2.652923e-02 * time: 0.5642340183258057 113 7.612972e+00 4.891870e-02 * time: 0.5671529769897461 114 7.612966e+00 3.703895e-02 * time: 0.5701429843902588 115 7.612914e+00 4.892153e-02 * time: 0.5734758377075195 116 7.612712e+00 2.989786e-02 * time: 0.5783460140228271 117 7.612522e+00 4.287837e-02 * time: 0.5811920166015625 118 7.612293e+00 2.162952e-02 * time: 0.5850579738616943 119 7.612208e+00 3.290756e-02 * time: 0.5880539417266846 120 7.612017e+00 2.568855e-02 * time: 0.5917940139770508 121 7.611898e+00 2.371954e-02 * time: 0.5965328216552734 122 7.611759e+00 1.890072e-02 * time: 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7.604052e+00 1.292436e-02 * time: 0.9326198101043701 211 7.604009e+00 1.016409e-02 * time: 0.9358279705047607 212 7.603977e+00 1.082781e-02 * time: 0.9403009414672852 213 7.603928e+00 1.321790e-02 * time: 0.9434828758239746 214 7.603912e+00 2.336417e-02 * time: 0.9464948177337646 215 7.603850e+00 1.048327e-02 * time: 0.9497039318084717 216 7.603811e+00 1.478139e-02 * time: 0.9536178112030029 217 7.603777e+00 1.097894e-02 * time: 0.9573948383331299 218 7.603743e+00 1.545299e-02 * time: 0.9613440036773682 219 7.603684e+00 1.091221e-02 * time: 0.9649829864501953 220 7.603650e+00 1.213241e-02 * time: 0.9678959846496582 221 7.603600e+00 1.031612e-02 * time: 0.9721720218658447 222 7.603567e+00 2.267603e-02 * time: 0.9749879837036133 223 7.603527e+00 1.917767e-02 * time: 0.978863000869751 224 7.603461e+00 9.924378e-03 * time: 0.9830780029296875 225 7.603415e+00 1.233091e-02 * time: 0.9863159656524658 226 7.603356e+00 1.219373e-02 * time: 0.9895038604736328 227 7.603300e+00 1.277611e-02 * 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