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 3.102290e+01 7.041416e+00 * time: 0.0018489360809326172 1 2.840777e+01 4.949200e+00 * time: 0.006920814514160156 2 2.097527e+01 5.716790e+00 * time: 0.012439966201782227 3 1.370089e+01 2.886502e+00 * time: 0.016038894653320312 4 1.116671e+01 1.133748e+00 * time: 0.01960587501525879 5 1.040005e+01 2.048981e+00 * time: 0.022694826126098633 6 9.888087e+00 1.255660e+00 * time: 0.025656938552856445 7 9.461884e+00 1.011384e+00 * time: 0.028729915618896484 8 9.000984e+00 8.456923e-01 * time: 0.03168487548828125 9 8.655796e+00 9.045388e-01 * time: 0.03452897071838379 10 8.390322e+00 5.943918e-01 * time: 0.03753781318664551 11 8.350848e+00 1.030523e+00 * time: 0.040007829666137695 12 8.294046e+00 8.644571e-01 * time: 0.04257392883300781 13 8.198117e+00 6.974578e-01 * time: 0.04592585563659668 14 8.048967e+00 4.116745e-01 * time: 0.04960989952087402 15 7.957076e+00 3.845398e-01 * time: 0.05331587791442871 16 7.943642e+00 6.021342e-01 * time: 0.092681884765625 17 7.911156e+00 5.087298e-01 * time: 0.09547591209411621 18 7.879390e+00 5.412756e-01 * time: 0.09806990623474121 19 7.842516e+00 3.481325e-01 * time: 0.10102200508117676 20 7.812414e+00 2.676037e-01 * time: 0.10408997535705566 21 7.782916e+00 3.394597e-01 * time: 0.10641288757324219 22 7.765237e+00 3.985629e-01 * time: 0.10876202583312988 23 7.746721e+00 1.605332e-01 * time: 0.11177182197570801 24 7.732907e+00 2.088353e-01 * time: 0.1141350269317627 25 7.728162e+00 2.831760e-01 * time: 0.11638784408569336 26 7.725938e+00 2.903319e-01 * time: 0.11867284774780273 27 7.715058e+00 2.002263e-01 * time: 0.12154793739318848 28 7.705152e+00 2.600964e-01 * time: 0.12373900413513184 29 7.699972e+00 2.184188e-01 * time: 0.12601494789123535 30 7.692585e+00 2.279659e-01 * time: 0.12842392921447754 31 7.689894e+00 2.321909e-01 * time: 0.1306748390197754 32 7.683332e+00 2.787574e-01 * time: 0.13326787948608398 33 7.680395e+00 1.890848e-01 * time: 0.13609600067138672 34 7.672437e+00 1.547245e-01 * time: 0.13898992538452148 35 7.669930e+00 2.164050e-01 * time: 0.14186501502990723 36 7.665492e+00 1.855944e-01 * time: 0.14486002922058105 37 7.661820e+00 1.272048e-01 * time: 0.14773893356323242 38 7.657584e+00 1.362699e-01 * time: 0.15016889572143555 39 7.654934e+00 8.766086e-02 * time: 0.15271401405334473 40 7.654480e+00 1.598830e-01 * time: 0.15536999702453613 41 7.652239e+00 9.934691e-02 * time: 0.1589670181274414 42 7.650453e+00 7.527233e-02 * time: 0.16277003288269043 43 7.649083e+00 1.150946e-01 * time: 0.165618896484375 44 7.648177e+00 8.135321e-02 * time: 0.16861701011657715 45 7.646946e+00 7.720579e-02 * time: 0.17166686058044434 46 7.645058e+00 5.406924e-02 * time: 0.17465686798095703 47 7.643652e+00 5.052017e-02 * time: 0.17757081985473633 48 7.642202e+00 6.189246e-02 * time: 0.18045401573181152 49 7.641855e+00 9.368634e-02 * time: 0.18334603309631348 50 7.641054e+00 6.412553e-02 * time: 0.18594789505004883 51 7.639683e+00 3.917063e-02 * time: 0.20866894721984863 52 7.638715e+00 5.194167e-02 * time: 0.21099591255187988 53 7.638035e+00 6.588134e-02 * time: 0.21354293823242188 54 7.636989e+00 6.142308e-02 * time: 0.2164309024810791 55 7.636072e+00 4.057438e-02 * time: 0.21939587593078613 56 7.635475e+00 4.872530e-02 * time: 0.222398042678833 57 7.634980e+00 3.900013e-02 * time: 0.22528791427612305 58 7.634404e+00 4.278678e-02 * time: 0.2277059555053711 59 7.634009e+00 5.571754e-02 * time: 0.2299790382385254 60 7.633680e+00 5.079089e-02 * time: 0.23236584663391113 61 7.633375e+00 5.332203e-02 * time: 0.23460984230041504 62 7.632928e+00 7.453823e-02 * time: 0.2369098663330078 63 7.632317e+00 5.953921e-02 * time: 0.2391979694366455 64 7.631522e+00 4.226178e-02 * time: 0.24220800399780273 65 7.631328e+00 7.342402e-02 * time: 0.24451398849487305 66 7.630578e+00 4.370234e-02 * time: 0.2468578815460205 67 7.629892e+00 6.013812e-02 * time: 0.24912190437316895 68 7.629020e+00 3.830866e-02 * time: 0.2514188289642334 69 7.628330e+00 4.472795e-02 * time: 0.25385403633117676 70 7.627872e+00 5.994983e-02 * time: 0.25613903999328613 71 7.627804e+00 7.089002e-02 * time: 0.25854992866516113 72 7.627268e+00 6.465028e-02 * time: 0.261538028717041 73 7.627131e+00 9.168751e-02 * time: 0.2640049457550049 74 7.627032e+00 8.133178e-02 * time: 0.2662949562072754 75 7.626524e+00 6.939501e-02 * time: 0.26857900619506836 76 7.625646e+00 8.837095e-02 * time: 0.2708468437194824 77 7.624706e+00 5.111281e-02 * time: 0.27394986152648926 78 7.624605e+00 1.144389e-01 * time: 0.27618885040283203 79 7.623676e+00 6.356848e-02 * time: 0.27858996391296387 80 7.622880e+00 6.605025e-02 * time: 0.2809908390045166 81 7.622372e+00 6.413931e-02 * time: 0.2839798927307129 82 7.621806e+00 4.222750e-02 * time: 0.286984920501709 83 7.621357e+00 5.332367e-02 * time: 0.2892758846282959 84 7.620831e+00 4.491713e-02 * time: 0.30832791328430176 85 7.620395e+00 3.744157e-02 * time: 0.311262845993042 86 7.619932e+00 2.750860e-02 * time: 0.314162015914917 87 7.619625e+00 4.952384e-02 * time: 0.31640100479125977 88 7.619325e+00 4.109088e-02 * time: 0.3193199634552002 89 7.618805e+00 4.077561e-02 * time: 0.3223879337310791 90 7.618440e+00 6.256811e-02 * time: 0.32460999488830566 91 7.618277e+00 5.624196e-02 * time: 0.3268589973449707 92 7.617923e+00 4.926166e-02 * time: 0.3297419548034668 93 7.617529e+00 3.595190e-02 * time: 0.33199286460876465 94 7.617030e+00 4.915827e-02 * time: 0.33424997329711914 95 7.616617e+00 4.009074e-02 * time: 0.3372659683227539 96 7.616325e+00 3.765083e-02 * time: 0.3400909900665283 97 7.615827e+00 4.831876e-02 * time: 0.3425748348236084 98 7.615529e+00 4.149599e-02 * time: 0.3454148769378662 99 7.615277e+00 5.030796e-02 * time: 0.3477199077606201 100 7.614993e+00 2.840701e-02 * time: 0.35056400299072266 101 7.614957e+00 6.053424e-02 * time: 0.3530080318450928 102 7.614873e+00 4.829080e-02 * time: 0.3552408218383789 103 7.614364e+00 4.870874e-02 * time: 0.3575890064239502 104 7.614015e+00 6.855373e-02 * time: 0.3598289489746094 105 7.613472e+00 5.783060e-02 * time: 0.36226892471313477 106 7.613176e+00 5.116245e-02 * time: 0.36455202102661133 107 7.612838e+00 3.983134e-02 * time: 0.36757683753967285 108 7.612509e+00 3.729807e-02 * time: 0.36981201171875 109 7.612240e+00 3.433138e-02 * time: 0.37215685844421387 110 7.611957e+00 4.622372e-02 * time: 0.37439489364624023 111 7.611711e+00 3.217555e-02 * time: 0.3774528503417969 112 7.611399e+00 3.286435e-02 * time: 0.380511999130249 113 7.611322e+00 4.109409e-02 * time: 0.3829958438873291 114 7.611141e+00 3.494357e-02 * time: 0.3852348327636719 115 7.610745e+00 5.965660e-02 * time: 0.38759398460388184 116 7.610352e+00 4.204375e-02 * time: 0.3898189067840576 117 7.610031e+00 3.620723e-02 * time: 0.3923499584197998 118 7.609874e+00 4.097104e-02 * time: 0.4032559394836426 119 7.609354e+00 4.943510e-02 * time: 0.4055008888244629 120 7.609015e+00 2.944610e-02 * time: 0.40834784507751465 121 7.608768e+00 4.623617e-02 * time: 0.41059398651123047 122 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3.363934e-02 * time: 0.4571249485015869 140 7.605885e+00 3.210235e-02 * time: 0.4593830108642578 141 7.605735e+00 3.743706e-02 * time: 0.46168994903564453 142 7.605672e+00 2.553263e-02 * time: 0.4640069007873535 143 7.605569e+00 3.283540e-02 * time: 0.46627283096313477 144 7.605387e+00 2.316794e-02 * time: 0.468552827835083 145 7.605260e+00 3.224783e-02 * time: 0.4708118438720703 146 7.605209e+00 3.186753e-02 * time: 0.47321295738220215 147 7.605165e+00 3.512431e-02 * time: 0.4754478931427002 148 7.605161e+00 5.130535e-02 * time: 0.4778409004211426 149 7.605003e+00 2.895404e-02 * time: 0.4800739288330078 150 7.604861e+00 4.363684e-02 * time: 0.482496976852417 151 7.604709e+00 1.558589e-02 * time: 0.4855179786682129 152 7.604545e+00 2.500268e-02 * time: 0.49652791023254395 153 7.604422e+00 2.204940e-02 * time: 0.4995279312133789 154 7.604363e+00 3.075254e-02 * time: 0.5018858909606934 155 7.604280e+00 3.219137e-02 * time: 0.5042359828948975 156 7.604230e+00 3.849645e-02 * time: 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time: 0.6052608489990234 192 7.598010e+00 6.289213e-02 * time: 0.6083559989929199 193 7.597793e+00 5.683004e-02 * time: 0.6106388568878174 194 7.597546e+00 6.211485e-02 * time: 0.613090991973877 195 7.597473e+00 5.042880e-02 * time: 0.6154630184173584 196 7.597134e+00 4.642671e-02 * time: 0.6178610324859619 197 7.597072e+00 6.263889e-02 * time: 0.620189905166626 198 7.596587e+00 5.182877e-02 * time: 0.6230578422546387 199 7.596243e+00 4.544697e-02 * time: 0.6260459423065186 200 7.596068e+00 3.643112e-02 * time: 0.6284899711608887 201 7.596041e+00 6.219180e-02 * time: 0.6307530403137207 202 7.595757e+00 5.405962e-02 * time: 0.633836030960083 203 7.595583e+00 3.919552e-02 * time: 0.6368780136108398 204 7.595555e+00 5.148544e-02 * time: 0.6393508911132812 205 7.595329e+00 4.462371e-02 * time: 0.6418328285217285 206 7.595030e+00 4.471488e-02 * time: 0.6442320346832275 207 7.594872e+00 3.658050e-02 * time: 0.6473250389099121 208 7.594684e+00 3.072993e-02 * time: 0.6495640277862549 209 7.594620e+00 3.742502e-02 * time: 0.6520109176635742 210 7.594491e+00 3.668803e-02 * time: 0.6549868583679199 211 7.594338e+00 2.719559e-02 * time: 0.6582369804382324 212 7.594192e+00 1.184969e-02 * time: 0.6614518165588379 213 7.594130e+00 1.735984e-02 * time: 0.6637728214263916 214 7.594117e+00 3.105997e-02 * time: 0.6662440299987793 215 7.594115e+00 2.907798e-02 * time: 0.6686220169067383 216 7.594099e+00 2.492571e-02 * time: 0.6709568500518799 217 7.594037e+00 1.858997e-02 * time: 0.6733658313751221 218 7.593967e+00 1.892479e-02 * time: 0.675743818283081 219 7.593841e+00 1.893340e-02 * time: 0.6869869232177734 220 7.593720e+00 7.826913e-03 * time: 0.6900689601898193 221 7.593704e+00 1.684473e-02 * time: 0.6925289630889893 222 7.593659e+00 1.214727e-02 * time: 0.694943904876709 223 7.593654e+00 1.549655e-02 * time: 0.6974649429321289 224 7.593621e+00 1.408484e-02 * time: 0.6998920440673828 225 7.593587e+00 8.559957e-03 * time: 0.7030520439147949 226 7.593542e+00 1.016134e-02 * time: 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