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 3.181378e+01 9.870895e+00 * time: 0.002177000045776367 1 3.011008e+01 5.567599e+00 * time: 0.009013891220092773 2 2.139950e+01 5.593874e+00 * time: 0.016654014587402344 3 1.657827e+01 5.224965e+00 * time: 0.025249004364013672 4 1.182497e+01 1.876204e+00 * time: 0.033312082290649414 5 1.138761e+01 2.659294e+00 * time: 0.03926205635070801 6 1.062140e+01 1.972147e+00 * time: 0.04570603370666504 7 9.893499e+00 1.547767e+00 * time: 0.05213308334350586 8 9.400374e+00 1.540797e+00 * time: 0.05857110023498535 9 8.960376e+00 1.600791e+00 * time: 0.06446599960327148 10 8.630369e+00 1.027981e+00 * time: 0.07117009162902832 11 8.405594e+00 9.925518e-01 * time: 0.07755708694458008 12 8.215028e+00 8.771012e-01 * time: 0.08423399925231934 13 8.060957e+00 4.831716e-01 * time: 0.09109807014465332 14 7.957614e+00 3.230127e-01 * time: 0.09706807136535645 15 7.952174e+00 8.246058e-01 * time: 0.10203289985656738 16 7.905007e+00 4.635742e-01 * time: 0.10765290260314941 17 7.869904e+00 6.323642e-01 * time: 0.11279606819152832 18 7.831834e+00 3.503089e-01 * time: 0.11961007118225098 19 7.816516e+00 5.582323e-01 * time: 0.12477898597717285 20 7.792292e+00 4.536573e-01 * time: 0.13014507293701172 21 7.772558e+00 3.802472e-01 * time: 0.13544201850891113 22 7.758859e+00 3.578391e-01 * time: 0.16525697708129883 23 7.754024e+00 4.120638e-01 * time: 0.1685628890991211 24 7.747321e+00 2.915412e-01 * time: 0.1733410358428955 25 7.739785e+00 3.960328e-01 * time: 0.1766510009765625 26 7.721315e+00 2.372297e-01 * time: 0.18109989166259766 27 7.709032e+00 2.379223e-01 * time: 0.18420791625976562 28 7.699102e+00 2.837649e-01 * time: 0.18744611740112305 29 7.695808e+00 3.102473e-01 * time: 0.19069695472717285 30 7.687183e+00 1.700968e-01 * time: 0.1947329044342041 31 7.685180e+00 2.682569e-01 * time: 0.19808697700500488 32 7.682984e+00 2.788849e-01 * time: 0.20148801803588867 33 7.681360e+00 2.073162e-01 * time: 0.20453906059265137 34 7.676597e+00 1.486242e-01 * time: 0.20862197875976562 35 7.675333e+00 2.699651e-01 * time: 0.2120680809020996 36 7.670708e+00 1.214227e-01 * time: 0.21514892578125 37 7.665404e+00 1.318785e-01 * time: 0.2186110019683838 38 7.662817e+00 1.001011e-01 * time: 0.22221589088439941 39 7.662399e+00 1.446876e-01 * time: 0.22524499893188477 40 7.660194e+00 9.392956e-02 * time: 0.22931289672851562 41 7.659141e+00 1.360469e-01 * time: 0.23276400566101074 42 7.657224e+00 8.737660e-02 * time: 0.23969197273254395 43 7.656074e+00 1.343965e-01 * time: 0.2451789379119873 44 7.655046e+00 1.016003e-01 * time: 0.2507638931274414 45 7.653713e+00 8.724650e-02 * time: 0.25696897506713867 46 7.651396e+00 5.985730e-02 * time: 0.2655661106109619 47 7.650100e+00 9.278837e-02 * time: 0.27536892890930176 48 7.647707e+00 8.950711e-02 * time: 0.2822530269622803 49 7.646759e+00 1.855564e-01 * time: 0.28871893882751465 50 7.644808e+00 1.355937e-01 * time: 0.297713041305542 51 7.642266e+00 8.816050e-02 * time: 0.30451202392578125 52 7.641308e+00 1.127601e-01 * time: 0.30992889404296875 53 7.639486e+00 1.175096e-01 * time: 0.3323080539703369 54 7.638471e+00 1.849132e-01 * time: 0.3355588912963867 55 7.638311e+00 1.498368e-01 * time: 0.3388819694519043 56 7.636410e+00 1.020664e-01 * time: 0.3424839973449707 57 7.634122e+00 1.214188e-01 * time: 0.3457679748535156 58 7.632174e+00 9.744501e-02 * time: 0.348966121673584 59 7.631657e+00 1.157391e-01 * time: 0.3522000312805176 60 7.629865e+00 8.564354e-02 * time: 0.356442928314209 61 7.628720e+00 6.129150e-02 * time: 0.35999011993408203 62 7.627565e+00 7.484975e-02 * time: 0.3632240295410156 63 7.626556e+00 5.450509e-02 * time: 0.3671300411224365 64 7.626366e+00 1.247762e-01 * time: 0.37050700187683105 65 7.626152e+00 1.165183e-01 * time: 0.37384510040283203 66 7.625068e+00 6.049342e-02 * time: 0.3779571056365967 67 7.624564e+00 9.068418e-02 * time: 0.38132810592651367 68 7.623702e+00 8.241390e-02 * time: 0.38471293449401855 69 7.623181e+00 1.032951e-01 * time: 0.3881070613861084 70 7.622232e+00 4.799063e-02 * time: 0.3933279514312744 71 7.621751e+00 5.852966e-02 * time: 0.3968479633331299 72 7.621444e+00 6.514392e-02 * time: 0.4002671241760254 73 7.621335e+00 5.034334e-02 * time: 0.4071919918060303 74 7.620891e+00 5.019804e-02 * time: 0.4126110076904297 75 7.620466e+00 4.222768e-02 * time: 0.4195590019226074 76 7.620076e+00 2.615660e-02 * time: 0.4252490997314453 77 7.619831e+00 3.308669e-02 * time: 0.42986202239990234 78 7.619582e+00 2.643220e-02 * time: 0.43650293350219727 79 7.619580e+00 5.961720e-02 * time: 0.4399290084838867 80 7.619542e+00 4.142535e-02 * time: 0.4431760311126709 81 7.619308e+00 3.200206e-02 * time: 0.4461050033569336 82 7.619115e+00 6.132642e-02 * time: 0.449415922164917 83 7.618842e+00 5.767187e-02 * time: 0.4529240131378174 84 7.618754e+00 5.739387e-02 * time: 0.47153210639953613 85 7.618407e+00 3.375749e-02 * time: 0.47493505477905273 86 7.617922e+00 3.605143e-02 * time: 0.47808408737182617 87 7.617403e+00 3.743774e-02 * time: 0.48122310638427734 88 7.617212e+00 5.118934e-02 * time: 0.4925239086151123 89 7.616759e+00 4.118817e-02 * time: 0.49580812454223633 90 7.616357e+00 6.254198e-02 * time: 0.5002820491790771 91 7.616030e+00 3.408293e-02 * time: 0.5042860507965088 92 7.615747e+00 3.991483e-02 * time: 0.5073931217193604 93 7.615607e+00 6.159236e-02 * time: 0.5106620788574219 94 7.615289e+00 3.964765e-02 * time: 0.5148138999938965 95 7.615028e+00 5.277666e-02 * time: 0.5178990364074707 96 7.614550e+00 5.343814e-02 * time: 0.5211019515991211 97 7.614191e+00 6.015162e-02 * time: 0.5243370532989502 98 7.613900e+00 5.843907e-02 * time: 0.5274810791015625 99 7.613731e+00 7.718938e-02 * time: 0.5312039852142334 100 7.613350e+00 6.671927e-02 * time: 0.5343799591064453 101 7.612989e+00 5.178083e-02 * time: 0.5383169651031494 102 7.612718e+00 3.536126e-02 * time: 0.5424759387969971 103 7.612420e+00 5.074594e-02 * time: 0.546929121017456 104 7.612170e+00 5.592484e-02 * time: 0.5514869689941406 105 7.612047e+00 4.115484e-02 * time: 0.5553591251373291 106 7.611835e+00 3.714871e-02 * time: 0.5586891174316406 107 7.611523e+00 3.073197e-02 * time: 0.5629539489746094 108 7.611461e+00 4.808942e-02 * time: 0.5661389827728271 109 7.611203e+00 3.648855e-02 * time: 0.5702519416809082 110 7.610881e+00 3.144785e-02 * time: 0.5737159252166748 111 7.610516e+00 4.533274e-02 * time: 0.5774860382080078 112 7.610248e+00 4.137887e-02 * time: 0.5816209316253662 113 7.610134e+00 3.941642e-02 * time: 0.5857090950012207 114 7.609981e+00 1.991233e-02 * time: 0.6016309261322021 115 7.609887e+00 3.678904e-02 * time: 0.6048579216003418 116 7.609667e+00 3.054150e-02 * time: 0.6080119609832764 117 7.609580e+00 3.545305e-02 * time: 0.6110310554504395 118 7.609366e+00 3.374087e-02 * time: 0.6140389442443848 119 7.609111e+00 2.132177e-02 * time: 0.6176400184631348 120 7.608921e+00 3.232676e-02 * time: 0.6208279132843018 121 7.608783e+00 5.344966e-02 * time: 0.6243560314178467 122 7.608457e+00 3.736616e-02 * time: 0.6274220943450928 123 7.608287e+00 4.028187e-02 * time: 0.6305899620056152 124 7.608087e+00 4.563596e-02 * time: 0.6339280605316162 125 7.607794e+00 3.031604e-02 * time: 0.6379690170288086 126 7.607518e+00 3.119621e-02 * time: 0.6421329975128174 127 7.607324e+00 2.486876e-02 * time: 0.6456940174102783 128 7.607144e+00 3.927294e-02 * time: 0.6484870910644531 129 7.606905e+00 3.743028e-02 * time: 0.6514968872070312 130 7.606865e+00 5.001889e-02 * time: 0.6597630977630615 131 7.606684e+00 5.998625e-02 * time: 0.6630260944366455 132 7.606379e+00 4.892161e-02 * time: 0.6663720607757568 133 7.606019e+00 5.897510e-02 * time: 0.669666051864624 134 7.605895e+00 5.835775e-02 * time: 0.6737511157989502 135 7.605529e+00 6.145676e-02 * time: 0.6781129837036133 136 7.605441e+00 5.913657e-02 * time: 0.681225061416626 137 7.604960e+00 6.140341e-02 * time: 0.6850600242614746 138 7.604615e+00 7.197212e-02 * time: 0.6882419586181641 139 7.604391e+00 6.654725e-02 * time: 0.6913149356842041 140 7.604042e+00 4.288100e-02 * time: 0.6951639652252197 141 7.603491e+00 4.574636e-02 * time: 0.6983311176300049 142 7.603190e+00 3.852431e-02 * time: 0.7026081085205078 143 7.602961e+00 4.645119e-02 * time: 0.7057540416717529 144 7.602846e+00 4.211701e-02 * time: 0.7088649272918701 145 7.602700e+00 4.313572e-02 * time: 0.7223649024963379 146 7.602494e+00 4.806204e-02 * time: 0.7253549098968506 147 7.602222e+00 2.716103e-02 * time: 0.7291650772094727 148 7.602073e+00 3.925079e-02 * time: 0.732511043548584 149 7.601807e+00 4.177805e-02 * time: 0.7370409965515137 150 7.601649e+00 3.735440e-02 * time: 0.7407538890838623 151 7.601484e+00 2.376040e-02 * time: 0.74434494972229 152 7.601470e+00 4.903720e-02 * time: 0.7480709552764893 153 7.601317e+00 3.932386e-02 * time: 0.7525091171264648 154 7.601255e+00 4.931099e-02 * time: 0.7557470798492432 155 7.601215e+00 4.414912e-02 * time: 0.7586669921875 156 7.601134e+00 3.269668e-02 * time: 0.7621428966522217 157 7.601022e+00 2.164684e-02 * time: 0.7657370567321777 158 7.600899e+00 2.864477e-02 * time: 0.7692220211029053 159 7.600712e+00 2.320747e-02 * time: 0.7735199928283691 160 7.600634e+00 3.618709e-02 * time: 0.7770650386810303 161 7.600579e+00 3.187546e-02 * time: 0.7806549072265625 162 7.600535e+00 2.335649e-02 * time: 0.783958911895752 163 7.600528e+00 4.300130e-02 * time: 0.7871699333190918 164 7.600498e+00 2.436966e-02 * time: 0.790229082107544 165 7.600399e+00 4.620966e-02 * time: 0.7935009002685547 166 7.600281e+00 1.754702e-02 * time: 0.7974939346313477 167 7.600153e+00 1.268561e-02 * time: 0.8015010356903076 168 7.600101e+00 3.072953e-02 * time: 0.8048090934753418 169 7.600038e+00 1.985024e-02 * time: 0.8092870712280273 170 7.599987e+00 2.231722e-02 * time: 0.8130450248718262 171 7.599944e+00 2.888140e-02 * time: 0.8163149356842041 172 7.599870e+00 1.815027e-02 * time: 0.8202509880065918 173 7.599850e+00 2.125707e-02 * time: 0.8235421180725098 174 7.599782e+00 1.586047e-02 * time: 0.8278069496154785 175 7.599747e+00 1.386194e-02 * time: 0.842379093170166 176 7.599719e+00 1.542014e-02 * time: 0.8454849720001221 177 7.599660e+00 1.137233e-02 * time: 0.8484539985656738 178 7.599629e+00 1.981184e-02 * time: 0.8515310287475586 179 7.599583e+00 1.536164e-02 * time: 0.8555169105529785 180 7.599541e+00 1.844064e-02 * time: 0.8590860366821289 181 7.599511e+00 1.749195e-02 * time: 0.863055944442749 182 7.599475e+00 1.202908e-02 * time: 0.867048978805542 183 7.599445e+00 1.054672e-02 * time: 0.8702089786529541 184 7.599443e+00 1.801723e-02 * time: 0.8736300468444824 185 7.599416e+00 1.321944e-02 * time: 0.8765599727630615 186 7.599376e+00 1.556261e-02 * time: 0.8796429634094238 187 7.599356e+00 1.502147e-02 * time: 0.8826000690460205 188 7.599341e+00 2.127865e-02 * time: 0.8856079578399658 189 7.599299e+00 8.966958e-03 * time: 0.8898129463195801 190 7.599272e+00 9.103726e-03 * time: 0.8928489685058594 191 7.599252e+00 9.522144e-03 * time: 0.8957169055938721 192 7.599224e+00 9.413032e-03 * time: 0.8986289501190186 193 7.599214e+00 1.205275e-02 * time: 0.9016048908233643 194 7.599213e+00 1.362649e-02 * time: 0.9051210880279541 195 7.599188e+00 9.556280e-03 * time: 0.9094860553741455 196 7.599172e+00 8.555209e-03 * time: 0.9125380516052246 197 7.599156e+00 6.744782e-03 * time: 0.9158439636230469 198 7.599156e+00 1.388259e-02 * time: 0.919105052947998 199 7.599143e+00 8.155193e-03 * time: 0.9233279228210449 200 7.599136e+00 1.031590e-02 * time: 0.9265310764312744 201 7.599127e+00 7.791599e-03 * time: 0.9298279285430908 202 7.599122e+00 7.751337e-03 * time: 0.9331250190734863 203 7.599112e+00 6.378310e-03 * time: 0.9364039897918701 204 7.599108e+00 1.004145e-02 * time: 0.9395549297332764 205 7.599100e+00 6.300964e-03 * time: 0.9435939788818359 206 7.599095e+00 6.997528e-03 * time: 0.9567849636077881 207 7.599089e+00 8.928699e-03 * time: 0.9600059986114502 208 7.599082e+00 6.693594e-03 * time: 0.963141918182373 209 7.599082e+00 1.005236e-02 * time: 0.9662630558013916 210 7.599078e+00 7.104780e-03 * time: 0.9700789451599121 211 7.599073e+00 9.154709e-03 * time: 0.9733579158782959 212 7.599067e+00 5.794263e-03 * time: 0.977665901184082 213 7.599065e+00 8.799044e-03 * time: 0.980863094329834 214 7.599060e+00 6.162073e-03 * time: 0.9842259883880615 215 7.599058e+00 6.560874e-03 * time: 0.9876739978790283 216 7.599052e+00 5.394681e-03 * time: 0.9907679557800293 217 7.599045e+00 5.668608e-03 * time: 0.9946799278259277 218 7.599045e+00 8.940259e-03 * time: 0.9980630874633789 219 7.599042e+00 5.563049e-03 * time: 1.0012640953063965 220 7.599039e+00 8.681427e-03 * time: 1.0045781135559082 221 7.599038e+00 5.695018e-03 * time: 1.0076940059661865 222 7.599035e+00 6.094192e-03 * time: 1.0111329555511475 223 7.599027e+00 6.610097e-03 * time: 1.0145149230957031 224 7.599025e+00 1.275476e-02 * time: 1.0175139904022217 225 7.599022e+00 8.951125e-03 * time: 1.0208029747009277 226 7.599017e+00 8.744781e-03 * time: 1.023893117904663 227 7.599007e+00 8.258316e-03 * time: 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