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 3.122979e+01 8.041349e+00 * time: 0.0014331340789794922 1 2.962691e+01 5.194341e+00 * time: 0.00603795051574707 2 2.182118e+01 6.600144e+00 * time: 0.012025117874145508 3 1.563171e+01 4.917492e+00 * time: 0.01752305030822754 4 1.186117e+01 2.149640e+00 * time: 0.022284984588623047 5 1.137053e+01 2.929259e+00 * time: 0.026194095611572266 6 1.053403e+01 1.750955e+00 * time: 0.031018972396850586 7 9.747891e+00 1.552575e+00 * time: 0.034928083419799805 8 9.090514e+00 1.679429e+00 * time: 0.038886070251464844 9 8.779532e+00 1.127845e+00 * time: 0.04311013221740723 10 8.582690e+00 9.808984e-01 * time: 0.0481419563293457 11 8.381509e+00 8.715960e-01 * time: 0.052304983139038086 12 8.223375e+00 6.743384e-01 * time: 0.05624508857727051 13 8.197315e+00 8.439144e-01 * time: 0.05989503860473633 14 8.139930e+00 7.023806e-01 * time: 0.06377601623535156 15 8.074470e+00 6.752596e-01 * time: 0.09373712539672852 16 8.036873e+00 5.538479e-01 * time: 0.09615302085876465 17 7.995625e+00 5.610557e-01 * time: 0.0984959602355957 18 7.942597e+00 4.213812e-01 * time: 0.10098791122436523 19 7.923442e+00 8.019301e-01 * time: 0.10300397872924805 20 7.907011e+00 8.261580e-01 * time: 0.10527896881103516 21 7.870397e+00 4.219163e-01 * time: 0.10834407806396484 22 7.855176e+00 5.505108e-01 * time: 0.11079597473144531 23 7.815045e+00 4.594445e-01 * time: 0.11294412612915039 24 7.799625e+00 4.662650e-01 * time: 0.1157069206237793 25 7.777751e+00 3.029118e-01 * time: 0.11850595474243164 26 7.756934e+00 3.360724e-01 * time: 0.12151503562927246 27 7.749661e+00 3.981501e-01 * time: 0.12383699417114258 28 7.748521e+00 3.674228e-01 * time: 0.12640595436096191 29 7.743930e+00 4.164525e-01 * time: 0.12885308265686035 30 7.724561e+00 2.842537e-01 * time: 0.1319410800933838 31 7.721135e+00 5.095137e-01 * time: 0.13440299034118652 32 7.708546e+00 4.804166e-01 * time: 0.13692307472229004 33 7.691247e+00 2.350169e-01 * time: 0.13994407653808594 34 7.682490e+00 3.041951e-01 * time: 0.14243292808532715 35 7.679673e+00 2.418369e-01 * time: 0.14464807510375977 36 7.674242e+00 1.786314e-01 * time: 0.14735102653503418 37 7.664293e+00 1.583925e-01 * time: 0.1502389907836914 38 7.661325e+00 2.861231e-01 * time: 0.15380597114562988 39 7.652962e+00 2.294544e-01 * time: 0.1591949462890625 40 7.647011e+00 1.843652e-01 * time: 0.16402602195739746 41 7.640637e+00 1.971624e-01 * time: 0.16797494888305664 42 7.636014e+00 1.372626e-01 * time: 0.17357897758483887 43 7.633162e+00 1.062178e-01 * time: 0.1783759593963623 44 7.631505e+00 1.006395e-01 * time: 0.1817479133605957 45 7.631221e+00 1.342639e-01 * time: 0.1857929229736328 46 7.629368e+00 8.469784e-02 * time: 0.20447993278503418 47 7.628685e+00 9.651779e-02 * time: 0.20651507377624512 48 7.627694e+00 8.109871e-02 * time: 0.20859003067016602 49 7.626311e+00 4.839437e-02 * time: 0.21058011054992676 50 7.625166e+00 5.113374e-02 * time: 0.21263599395751953 51 7.624780e+00 7.262042e-02 * time: 0.21499013900756836 52 7.624503e+00 6.358642e-02 * time: 0.2175159454345703 53 7.624236e+00 8.465746e-02 * time: 0.21996712684631348 54 7.624051e+00 7.048943e-02 * time: 0.22196507453918457 55 7.623163e+00 7.060252e-02 * time: 0.22400403022766113 56 7.622416e+00 3.661848e-02 * time: 0.22653603553771973 57 7.621963e+00 7.270198e-02 * time: 0.22852802276611328 58 7.621737e+00 8.689250e-02 * time: 0.23093008995056152 59 7.621224e+00 5.613937e-02 * time: 0.23328304290771484 60 7.620919e+00 6.026545e-02 * time: 0.2357640266418457 61 7.620240e+00 5.781363e-02 * time: 0.23807501792907715 62 7.619508e+00 4.033903e-02 * time: 0.24078893661499023 63 7.619377e+00 6.979910e-02 * time: 0.24297404289245605 64 7.618908e+00 6.716091e-02 * time: 0.24570703506469727 65 7.618355e+00 3.383349e-02 * time: 0.24873113632202148 66 7.617652e+00 3.791594e-02 * time: 0.251068115234375 67 7.617143e+00 4.593312e-02 * time: 0.253277063369751 68 7.616893e+00 8.083180e-02 * time: 0.255648136138916 69 7.616190e+00 4.484075e-02 * time: 0.2586350440979004 70 7.616032e+00 7.532133e-02 * time: 0.2624959945678711 71 7.615566e+00 4.377557e-02 * time: 0.2677431106567383 72 7.615047e+00 5.773214e-02 * time: 0.27170705795288086 73 7.614853e+00 5.599879e-02 * time: 0.275130033493042 74 7.614841e+00 7.548428e-02 * time: 0.27919602394104004 75 7.614441e+00 7.990188e-02 * time: 0.28328704833984375 76 7.614056e+00 7.432898e-02 * time: 0.2869751453399658 77 7.613328e+00 6.679307e-02 * time: 0.29041504859924316 78 7.612781e+00 5.976445e-02 * time: 0.30577611923217773 79 7.612486e+00 5.926440e-02 * time: 0.3080129623413086 80 7.612055e+00 6.092114e-02 * time: 0.31112003326416016 81 7.611601e+00 4.362699e-02 * time: 0.3140439987182617 82 7.611309e+00 3.585826e-02 * time: 0.31603407859802246 83 7.610903e+00 5.130359e-02 * time: 0.31803107261657715 84 7.610518e+00 3.544783e-02 * time: 0.3205270767211914 85 7.610339e+00 3.879526e-02 * time: 0.3227670192718506 86 7.609995e+00 3.580910e-02 * time: 0.325747013092041 87 7.609913e+00 7.756345e-02 * time: 0.3281710147857666 88 7.609824e+00 5.500128e-02 * time: 0.3305840492248535 89 7.609462e+00 5.539547e-02 * time: 0.33307504653930664 90 7.609008e+00 5.098483e-02 * time: 0.3355889320373535 91 7.608538e+00 3.716341e-02 * time: 0.33866405487060547 92 7.608480e+00 5.906393e-02 * time: 0.3411281108856201 93 7.608180e+00 4.713079e-02 * time: 0.3442881107330322 94 7.607972e+00 5.505148e-02 * time: 0.3466460704803467 95 7.607676e+00 5.182515e-02 * time: 0.3488340377807617 96 7.607194e+00 3.502500e-02 * time: 0.35094308853149414 97 7.607184e+00 5.021192e-02 * time: 0.35303807258605957 98 7.606871e+00 5.456627e-02 * time: 0.3560910224914551 99 7.606568e+00 3.041682e-02 * time: 0.3591940402984619 100 7.606377e+00 5.002653e-02 * time: 0.3623509407043457 101 7.606121e+00 3.774674e-02 * time: 0.3657550811767578 102 7.606108e+00 5.366885e-02 * time: 0.3691270351409912 103 7.605826e+00 3.891657e-02 * time: 0.3742949962615967 104 7.605723e+00 4.869222e-02 * time: 0.37764906883239746 105 7.605583e+00 7.508928e-02 * time: 0.38048505783081055 106 7.605452e+00 5.287392e-02 * time: 0.3834221363067627 107 7.605135e+00 3.823169e-02 * time: 0.38662004470825195 108 7.604837e+00 2.049159e-02 * time: 0.39923095703125 109 7.604801e+00 5.590117e-02 * time: 0.4014449119567871 110 7.604780e+00 4.867948e-02 * time: 0.40383100509643555 111 7.604596e+00 4.126217e-02 * time: 0.40619611740112305 112 7.604281e+00 4.214552e-02 * time: 0.4082491397857666 113 7.604201e+00 5.422412e-02 * time: 0.4102311134338379 114 7.604004e+00 6.177909e-02 * time: 0.4121971130371094 115 7.603796e+00 6.492113e-02 * time: 0.4141850471496582 116 7.603532e+00 4.775545e-02 * time: 0.416748046875 117 7.603377e+00 5.655964e-02 * time: 0.41919398307800293 118 7.603273e+00 5.289166e-02 * time: 0.4216439723968506 119 7.603077e+00 5.213989e-02 * time: 0.4237360954284668 120 7.603010e+00 6.035143e-02 * time: 0.42572808265686035 121 7.602684e+00 4.559793e-02 * time: 0.42824792861938477 122 7.602468e+00 4.612345e-02 * time: 0.4308459758758545 123 7.602223e+00 4.130215e-02 * time: 0.4330711364746094 124 7.601883e+00 4.218979e-02 * time: 0.43625593185424805 125 7.601691e+00 2.352147e-02 * time: 0.4395010471343994 126 7.601578e+00 2.887224e-02 * time: 0.4419851303100586 127 7.601397e+00 2.530797e-02 * time: 0.44446301460266113 128 7.601246e+00 2.756150e-02 * time: 0.4475901126861572 129 7.601098e+00 3.283176e-02 * time: 0.45073699951171875 130 7.600982e+00 3.638977e-02 * time: 0.4540529251098633 131 7.600751e+00 2.043097e-02 * time: 0.45861005783081055 132 7.600673e+00 3.205637e-02 * time: 0.4622640609741211 133 7.600524e+00 3.257767e-02 * time: 0.4670069217681885 134 7.600368e+00 4.152013e-02 * time: 0.4708230495452881 135 7.600159e+00 3.068077e-02 * time: 0.4757080078125 136 7.599995e+00 2.019210e-02 * time: 0.47872400283813477 137 7.599949e+00 3.422529e-02 * time: 0.48124098777770996 138 7.599757e+00 2.716672e-02 * time: 0.48394107818603516 139 7.599620e+00 3.177704e-02 * 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7.593871e+00 1.607887e-02 * time: 0.6465799808502197 193 7.593842e+00 1.681870e-02 * time: 0.649183988571167 194 7.593808e+00 1.131391e-02 * time: 0.6524429321289062 195 7.593778e+00 9.706294e-03 * time: 0.6557459831237793 196 7.593753e+00 9.213728e-03 * time: 0.6583030223846436 197 7.593729e+00 1.640549e-02 * time: 0.6610541343688965 198 7.593727e+00 1.756481e-02 * time: 0.6640870571136475 199 7.593699e+00 1.183837e-02 * time: 0.6672670841217041 200 7.593675e+00 1.020887e-02 * time: 0.6791789531707764 201 7.593641e+00 1.286421e-02 * time: 0.6817100048065186 202 7.593602e+00 1.505342e-02 * time: 0.6842019557952881 203 7.593580e+00 1.100117e-02 * time: 0.6875169277191162 204 7.593543e+00 9.048546e-03 * time: 0.6906700134277344 205 7.593537e+00 2.092083e-02 * time: 0.6931331157684326 206 7.593531e+00 1.725994e-02 * time: 0.6956090927124023 207 7.593504e+00 1.538315e-02 * time: 0.6981029510498047 208 7.593502e+00 1.934268e-02 * time: 0.7005951404571533 209 7.593463e+00 1.756513e-02 * time: 0.703963041305542 210 7.593426e+00 1.343708e-02 * time: 0.7064769268035889 211 7.593404e+00 1.335173e-02 * time: 0.7089779376983643 212 7.593382e+00 1.235428e-02 * time: 0.7114889621734619 213 7.593377e+00 1.183915e-02 * time: 0.7139899730682373 214 7.593353e+00 1.006407e-02 * time: 0.7165710926055908 215 7.593324e+00 1.223770e-02 * time: 0.7192609310150146 216 7.593300e+00 1.016437e-02 * time: 0.7224149703979492 217 7.593283e+00 1.331974e-02 * time: 0.7255489826202393 218 7.593274e+00 1.318231e-02 * time: 0.7280299663543701 219 7.593251e+00 9.042707e-03 * time: 0.7312679290771484 220 7.593242e+00 6.715157e-03 * time: 0.7346489429473877 221 7.593226e+00 4.483438e-03 * time: 0.7372019290924072 222 7.593214e+00 9.274588e-03 * time: 0.7397110462188721 223 7.593200e+00 8.244464e-03 * time: 0.7422099113464355 224 7.593197e+00 1.429226e-02 * time: 0.7447259426116943 225 7.593190e+00 9.599830e-03 * time: 0.7472310066223145 226 7.593164e+00 7.083517e-03 * time: 0.7499289512634277 227 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