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[:, :, 1, 1], c=:blues)
Iter Function value Gradient norm 0 3.105161e+01 8.135250e+00 * time: 0.0018990039825439453 1 2.314313e+01 4.185435e+00 * time: 0.008825063705444336 2 1.648258e+01 3.595554e+00 * time: 0.015731096267700195 3 1.273536e+01 2.382911e+00 * time: 0.021572113037109375 4 9.961725e+00 1.291876e+00 * time: 0.026746034622192383 5 9.223828e+00 1.860277e+00 * time: 0.031507015228271484 6 8.741099e+00 1.175847e+00 * time: 0.03589200973510742 7 8.472479e+00 7.961030e-01 * time: 0.04087519645690918 8 8.311689e+00 9.525920e-01 * time: 0.09021902084350586 9 8.275526e+00 8.296795e-01 * time: 0.0932002067565918 10 8.246208e+00 7.421042e-01 * time: 0.09605216979980469 11 8.240311e+00 1.001146e+00 * time: 0.0989840030670166 12 8.150341e+00 4.959490e-01 * time: 0.10269021987915039 13 8.126966e+00 7.912038e-01 * time: 0.10555601119995117 14 8.084134e+00 6.355446e-01 * time: 0.10855603218078613 15 8.039537e+00 5.498641e-01 * time: 0.11141109466552734 16 7.980361e+00 5.325098e-01 * time: 0.1150510311126709 17 7.963144e+00 4.207574e-01 * time: 0.11781501770019531 18 7.924472e+00 4.457566e-01 * time: 0.12082314491271973 19 7.893223e+00 3.922636e-01 * time: 0.12359809875488281 20 7.869237e+00 4.798811e-01 * time: 0.12682509422302246 21 7.841623e+00 3.568805e-01 * time: 0.1300210952758789 22 7.818467e+00 3.179261e-01 * time: 0.13436317443847656 23 7.802398e+00 2.418383e-01 * time: 0.13863611221313477 24 7.783704e+00 3.603037e-01 * time: 0.14192700386047363 25 7.777750e+00 3.137056e-01 * time: 0.145064115524292 26 7.770279e+00 2.679355e-01 * time: 0.14835119247436523 27 7.759338e+00 2.958856e-01 * time: 0.1517200469970703 28 7.742685e+00 1.641332e-01 * time: 0.15586400032043457 29 7.741704e+00 4.026390e-01 * time: 0.15929722785949707 30 7.735567e+00 2.870783e-01 * time: 0.16289806365966797 31 7.727467e+00 2.017193e-01 * time: 0.1666111946105957 32 7.715692e+00 1.545526e-01 * time: 0.170166015625 33 7.707414e+00 1.633176e-01 * time: 0.17330622673034668 34 7.701193e+00 1.298772e-01 * time: 0.17709612846374512 35 7.692984e+00 1.095916e-01 * time: 0.18113422393798828 36 7.686663e+00 9.252611e-02 * time: 0.18497705459594727 37 7.682248e+00 7.543221e-02 * time: 0.18881607055664062 38 7.681415e+00 1.741614e-01 * time: 0.19238615036010742 39 7.678852e+00 1.164613e-01 * time: 0.1958920955657959 40 7.674706e+00 9.776009e-02 * time: 0.22667908668518066 41 7.673233e+00 1.114376e-01 * time: 0.22996902465820312 42 7.672548e+00 1.340283e-01 * time: 0.2328171730041504 43 7.669825e+00 1.073752e-01 * time: 0.2353651523590088 44 7.667984e+00 9.663599e-02 * time: 0.23780608177185059 45 7.667408e+00 1.086421e-01 * time: 0.24035120010375977 46 7.666101e+00 1.145175e-01 * time: 0.24303102493286133 47 7.664064e+00 8.339079e-02 * time: 0.24574804306030273 48 7.662725e+00 1.231989e-01 * time: 0.2483510971069336 49 7.659795e+00 9.562690e-02 * time: 0.2526891231536865 50 7.659078e+00 1.404238e-01 * time: 0.2553441524505615 51 7.656099e+00 1.103888e-01 * time: 0.25800609588623047 52 7.653176e+00 8.668524e-02 * time: 0.2617311477661133 53 7.650511e+00 7.947619e-02 * time: 0.2653062343597412 54 7.648505e+00 8.803798e-02 * time: 0.26812314987182617 55 7.647721e+00 1.558565e-01 * time: 0.2708871364593506 56 7.645242e+00 1.237427e-01 * time: 0.27442407608032227 57 7.642671e+00 8.562518e-02 * time: 0.27784013748168945 58 7.639461e+00 1.496980e-01 * time: 0.2806510925292969 59 7.637211e+00 1.154253e-01 * time: 0.28344106674194336 60 7.634552e+00 8.822295e-02 * time: 0.28612303733825684 61 7.632310e+00 1.140565e-01 * time: 0.28902101516723633 62 7.630435e+00 7.065791e-02 * time: 0.29329919815063477 63 7.628798e+00 9.121952e-02 * time: 0.29630112648010254 64 7.627685e+00 1.170035e-01 * time: 0.29877209663391113 65 7.626295e+00 7.804397e-02 * time: 0.3016390800476074 66 7.624865e+00 4.720245e-02 * time: 0.30547118186950684 67 7.622896e+00 6.227335e-02 * time: 0.3084390163421631 68 7.622133e+00 1.004908e-01 * time: 0.31171417236328125 69 7.620874e+00 8.568546e-02 * time: 0.31481218338012695 70 7.619603e+00 5.406312e-02 * time: 0.31878113746643066 71 7.618457e+00 7.451042e-02 * time: 0.32225823402404785 72 7.617883e+00 8.275176e-02 * time: 0.32557201385498047 73 7.617249e+00 4.917774e-02 * time: 0.3506031036376953 74 7.616998e+00 7.541408e-02 * time: 0.35337114334106445 75 7.616222e+00 5.886760e-02 * time: 0.3566911220550537 76 7.616076e+00 5.785629e-02 * time: 0.3590400218963623 77 7.615873e+00 6.288340e-02 * time: 0.3616490364074707 78 7.615467e+00 3.940573e-02 * time: 0.3644142150878906 79 7.615114e+00 4.007669e-02 * time: 0.36685705184936523 80 7.614864e+00 3.507605e-02 * time: 0.3695840835571289 81 7.614590e+00 1.946838e-02 * time: 0.37322402000427246 82 7.614205e+00 3.036793e-02 * time: 0.3761141300201416 83 7.614005e+00 3.361125e-02 * time: 0.3790090084075928 84 7.613957e+00 3.774864e-02 * time: 0.3817331790924072 85 7.613711e+00 3.637523e-02 * time: 0.38542604446411133 86 7.613461e+00 3.933052e-02 * time: 0.388214111328125 87 7.613056e+00 3.161323e-02 * time: 0.3915441036224365 88 7.612684e+00 2.196268e-02 * time: 0.3952171802520752 89 7.612494e+00 3.286099e-02 * time: 0.39803123474121094 90 7.612237e+00 3.605917e-02 * time: 0.4013381004333496 91 7.611882e+00 1.669469e-02 * time: 0.40493321418762207 92 7.611645e+00 2.610068e-02 * time: 0.4072701930999756 93 7.611348e+00 2.468908e-02 * time: 0.4098222255706787 94 7.611160e+00 5.301449e-02 * time: 0.4126100540161133 95 7.610883e+00 3.554461e-02 * time: 0.4160802364349365 96 7.610636e+00 3.287128e-02 * time: 0.4184250831604004 97 7.610599e+00 5.401654e-02 * time: 0.4208970069885254 98 7.610475e+00 4.326270e-02 * time: 0.4235551357269287 99 7.610326e+00 3.837223e-02 * time: 0.426530122756958 100 7.610199e+00 4.102934e-02 * time: 0.4295802116394043 101 7.610146e+00 4.947994e-02 * time: 0.432833194732666 102 7.609949e+00 2.699965e-02 * time: 0.4366450309753418 103 7.609878e+00 5.601682e-02 * time: 0.4394512176513672 104 7.609659e+00 3.411716e-02 * time: 0.44228410720825195 105 7.609640e+00 3.025929e-02 * time: 0.45594000816345215 106 7.609444e+00 2.560685e-02 * time: 0.4584650993347168 107 7.609402e+00 3.994197e-02 * time: 0.46112608909606934 108 7.609225e+00 2.016656e-02 * time: 0.4647231101989746 109 7.609103e+00 1.844439e-02 * time: 0.4679450988769531 110 7.609102e+00 3.893100e-02 * time: 0.4706861972808838 111 7.609098e+00 3.559330e-02 * time: 0.4739551544189453 112 7.609040e+00 1.972980e-02 * time: 0.4774460792541504 113 7.608979e+00 2.505345e-02 * time: 0.48082518577575684 114 7.608854e+00 2.358798e-02 * time: 0.4842720031738281 115 7.608724e+00 1.913385e-02 * time: 0.4877440929412842 116 7.608677e+00 3.886704e-02 * time: 0.4902992248535156 117 7.608587e+00 3.548366e-02 * time: 0.49326515197753906 118 7.608576e+00 2.523140e-02 * time: 0.4960181713104248 119 7.608479e+00 2.157015e-02 * time: 0.4984281063079834 120 7.608390e+00 2.717405e-02 * time: 0.5008001327514648 121 7.608247e+00 2.251121e-02 * time: 0.5044100284576416 122 7.608109e+00 1.950110e-02 * time: 0.5079312324523926 123 7.607974e+00 1.798669e-02 * time: 0.5114550590515137 124 7.607944e+00 3.301787e-02 * time: 0.5146310329437256 125 7.607922e+00 2.950096e-02 * time: 0.5173652172088623 126 7.607817e+00 2.240230e-02 * time: 0.5198860168457031 127 7.607629e+00 2.142822e-02 * time: 0.5232431888580322 128 7.607541e+00 2.595930e-02 * time: 0.5258312225341797 129 7.607372e+00 1.950580e-02 * time: 0.5293631553649902 130 7.607354e+00 3.889747e-02 * time: 0.5318241119384766 131 7.607168e+00 2.416859e-02 * time: 0.5356321334838867 132 7.607047e+00 2.548437e-02 * time: 0.5386090278625488 133 7.606951e+00 3.757998e-02 * time: 0.541600227355957 134 7.606857e+00 2.454002e-02 * time: 0.5448851585388184 135 7.606706e+00 3.017237e-02 * time: 0.5483200550079346 136 7.606549e+00 2.093485e-02 * time: 0.5514931678771973 137 7.606460e+00 3.634124e-02 * time: 0.5653150081634521 138 7.606391e+00 4.135379e-02 * time: 0.5678691864013672 139 7.606287e+00 4.141551e-02 * time: 0.5704801082611084 140 7.606270e+00 3.488872e-02 * time: 0.5729410648345947 141 7.606256e+00 3.433710e-02 * time: 0.576120138168335 142 7.606151e+00 3.133224e-02 * time: 0.5785551071166992 143 7.606061e+00 3.056618e-02 * time: 0.5812702178955078 144 7.606030e+00 5.119188e-02 * time: 0.5838971138000488 145 7.605886e+00 3.245798e-02 * time: 0.5872189998626709 146 7.605858e+00 3.144503e-02 * time: 0.5896902084350586 147 7.605730e+00 2.435330e-02 * time: 0.5920751094818115 148 7.605636e+00 4.338568e-02 * time: 0.5948851108551025 149 7.605567e+00 2.436046e-02 * time: 0.5974130630493164 150 7.605482e+00 1.918499e-02 * time: 0.5997841358184814 151 7.605437e+00 3.934304e-02 * time: 0.6024432182312012 152 7.605373e+00 2.995829e-02 * time: 0.6050770282745361 153 7.605259e+00 2.396337e-02 * time: 0.6082181930541992 154 7.605181e+00 3.531049e-02 * time: 0.6108341217041016 155 7.605061e+00 2.253787e-02 * time: 0.614328145980835 156 7.604976e+00 1.682726e-02 * time: 0.6170690059661865 157 7.604935e+00 2.973387e-02 * time: 0.6197750568389893 158 7.604873e+00 2.313383e-02 * time: 0.622528076171875 159 7.604791e+00 2.734255e-02 * time: 0.6253671646118164 160 7.604682e+00 1.430313e-02 * time: 0.628911018371582 161 7.604633e+00 2.979951e-02 * time: 0.6312961578369141 162 7.604546e+00 1.461828e-02 * time: 0.6352560520172119 163 7.604489e+00 2.104824e-02 * time: 0.6380300521850586 164 7.604419e+00 1.614867e-02 * time: 0.6414940357208252 165 7.604369e+00 1.317928e-02 * time: 0.6448941230773926 166 7.604325e+00 1.880639e-02 * time: 0.647834062576294 167 7.604262e+00 1.106014e-02 * time: 0.6514310836791992 168 7.604218e+00 2.193885e-02 * time: 0.65431809425354 169 7.604185e+00 1.288283e-02 * time: 0.6571221351623535 170 7.604131e+00 1.445975e-02 * time: 0.6710870265960693 171 7.604057e+00 1.564109e-02 * time: 0.6736660003662109 172 7.604052e+00 2.595458e-02 * time: 0.6766831874847412 173 7.604017e+00 1.512751e-02 * time: 0.6791121959686279 174 7.603964e+00 1.830220e-02 * time: 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7.603159e+00 1.611538e-02 * time: 0.7344250679016113 193 7.603158e+00 1.982741e-02 * time: 0.738037109375 194 7.603130e+00 1.389574e-02 * time: 0.7407591342926025 195 7.603101e+00 1.557255e-02 * time: 0.7437372207641602 196 7.603055e+00 1.140129e-02 * time: 0.7466931343078613 197 7.603037e+00 1.213112e-02 * time: 0.7495810985565186 198 7.603014e+00 2.186182e-02 * time: 0.752251148223877 199 7.602979e+00 9.909920e-03 * time: 0.7558391094207764 200 7.602963e+00 1.447839e-02 * time: 0.7592170238494873 201 7.602934e+00 9.358954e-03 * time: 0.7628991603851318 202 7.602915e+00 7.787919e-03 * time: 0.7667181491851807 203 7.602894e+00 7.530685e-03 * time: 0.7806620597839355 204 7.602861e+00 7.078135e-03 * time: 0.7837722301483154 205 7.602858e+00 1.217531e-02 * time: 0.7863330841064453 206 7.602856e+00 1.211488e-02 * time: 0.788870096206665 207 7.602840e+00 1.472989e-02 * time: 0.7913720607757568 208 7.602821e+00 1.116947e-02 * time: 0.7937140464782715 209 7.602798e+00 1.428451e-02 * time: 0.7967932224273682 210 7.602777e+00 1.644955e-02 * time: 0.799314022064209 211 7.602760e+00 1.817088e-02 * time: 0.8018860816955566 212 7.602739e+00 1.727356e-02 * time: 0.8048431873321533 213 7.602687e+00 1.441276e-02 * time: 0.8077700138092041 214 7.602616e+00 1.111332e-02 * time: 0.8105521202087402 215 7.602587e+00 1.944929e-02 * time: 0.8133060932159424 216 7.602519e+00 2.105641e-02 * time: 0.8170931339263916 217 7.602479e+00 9.963882e-03 * time: 0.8204522132873535 218 7.602476e+00 1.880738e-02 * time: 0.8231351375579834 219 7.602438e+00 1.553928e-02 * time: 0.8266701698303223 220 7.602410e+00 1.637626e-02 * time: 0.8293631076812744 221 7.602406e+00 2.926086e-02 * time: 0.8318970203399658 222 7.602404e+00 2.392553e-02 * time: 0.8348062038421631 223 7.602334e+00 1.606513e-02 * time: 0.838820219039917 224 7.602282e+00 2.045806e-02 * time: 0.8414180278778076 225 7.602223e+00 1.742602e-02 * time: 0.8441140651702881 226 7.602170e+00 9.252147e-03 * time: 0.8476741313934326 227 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