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 3.186678e+01 9.383155e+00 * time: 0.0016548633575439453 1 2.763206e+01 4.860778e+00 * time: 0.006888866424560547 2 1.885185e+01 4.280055e+00 * time: 0.012089967727661133 3 1.288344e+01 2.940758e+00 * time: 0.01645803451538086 4 1.194721e+01 2.801569e+00 * time: 0.019794940948486328 5 1.089841e+01 1.770251e+00 * time: 0.023251056671142578 6 1.011533e+01 1.271781e+00 * time: 0.026489973068237305 7 9.340925e+00 1.162382e+00 * time: 0.030128002166748047 8 8.725210e+00 6.931441e-01 * time: 0.033540964126586914 9 8.467250e+00 4.123209e-01 * time: 0.03746485710144043 10 8.358130e+00 4.780912e-01 * time: 0.04129290580749512 11 8.286957e+00 4.688065e-01 * time: 0.04557991027832031 12 8.272010e+00 6.452613e-01 * time: 0.04852890968322754 13 8.188374e+00 4.655708e-01 * time: 0.05157303810119629 14 8.162237e+00 5.967006e-01 * time: 0.05424189567565918 15 8.085695e+00 7.480522e-01 * time: 0.056715965270996094 16 8.013657e+00 5.301728e-01 * time: 0.060133934020996094 17 7.987975e+00 7.962903e-01 * time: 0.1053159236907959 18 7.929826e+00 6.741991e-01 * time: 0.10892200469970703 19 7.848895e+00 4.961541e-01 * time: 0.11214494705200195 20 7.831137e+00 6.316091e-01 * time: 0.11490297317504883 21 7.818284e+00 5.626819e-01 * time: 0.11750292778015137 22 7.790311e+00 4.648493e-01 * time: 0.12085604667663574 23 7.754822e+00 2.852908e-01 * time: 0.12427401542663574 24 7.730264e+00 3.372240e-01 * time: 0.12671494483947754 25 7.716068e+00 2.843639e-01 * time: 0.12906289100646973 26 7.701747e+00 2.010284e-01 * time: 0.13161396980285645 27 7.700124e+00 2.883239e-01 * time: 0.1341390609741211 28 7.691607e+00 2.326198e-01 * time: 0.13651299476623535 29 7.684879e+00 1.332036e-01 * time: 0.13988590240478516 30 7.682395e+00 2.440068e-01 * time: 0.1424999237060547 31 7.673349e+00 2.106831e-01 * time: 0.14510488510131836 32 7.666693e+00 1.711247e-01 * time: 0.14756393432617188 33 7.665821e+00 2.187834e-01 * time: 0.15007495880126953 34 7.662436e+00 1.341418e-01 * time: 0.15333986282348633 35 7.660110e+00 1.645868e-01 * time: 0.15594887733459473 36 7.659362e+00 2.084078e-01 * time: 0.1585710048675537 37 7.656457e+00 1.626125e-01 * time: 0.16135597229003906 38 7.656188e+00 2.102620e-01 * time: 0.16412687301635742 39 7.652600e+00 1.180453e-01 * time: 0.16776490211486816 40 7.650010e+00 1.336260e-01 * time: 0.17072606086730957 41 7.645992e+00 1.265253e-01 * time: 0.17450690269470215 42 7.644085e+00 1.230184e-01 * time: 0.1774458885192871 43 7.642765e+00 1.558829e-01 * time: 0.18036198616027832 44 7.640004e+00 6.930845e-02 * time: 0.184251070022583 45 7.638946e+00 9.780124e-02 * time: 0.18727707862854004 46 7.637115e+00 7.337029e-02 * time: 0.19099688529968262 47 7.635318e+00 5.779216e-02 * time: 0.19481396675109863 48 7.634469e+00 9.015674e-02 * time: 0.19773197174072266 49 7.632977e+00 4.968419e-02 * time: 0.2015080451965332 50 7.632184e+00 5.849560e-02 * time: 0.20448589324951172 51 7.631207e+00 3.947136e-02 * time: 0.20828986167907715 52 7.630482e+00 6.061611e-02 * time: 0.21109485626220703 53 7.629480e+00 4.417135e-02 * time: 0.23879003524780273 54 7.629195e+00 6.342209e-02 * time: 0.24136900901794434 55 7.628554e+00 5.408703e-02 * time: 0.24381494522094727 56 7.627807e+00 5.328629e-02 * time: 0.24652791023254395 57 7.627642e+00 5.465785e-02 * time: 0.24916386604309082 58 7.627028e+00 4.686141e-02 * time: 0.2518930435180664 59 7.626499e+00 5.711766e-02 * time: 0.2545499801635742 60 7.626113e+00 3.760274e-02 * time: 0.25714802742004395 61 7.625687e+00 4.041762e-02 * time: 0.2595798969268799 62 7.625313e+00 4.640540e-02 * time: 0.26212286949157715 63 7.625045e+00 5.613019e-02 * time: 0.26473402976989746 64 7.624572e+00 3.153462e-02 * time: 0.2673459053039551 65 7.624414e+00 8.770346e-02 * time: 0.2699470520019531 66 7.624001e+00 6.677856e-02 * time: 0.2732388973236084 67 7.623648e+00 6.925827e-02 * time: 0.2758610248565674 68 7.622986e+00 4.283296e-02 * time: 0.2793090343475342 69 7.622691e+00 5.392292e-02 * time: 0.2817220687866211 70 7.622416e+00 5.525521e-02 * time: 0.2841958999633789 71 7.621938e+00 4.712146e-02 * time: 0.2868359088897705 72 7.621588e+00 5.292050e-02 * time: 0.2893838882446289 73 7.621377e+00 9.317087e-02 * time: 0.2919809818267822 74 7.621063e+00 6.277133e-02 * time: 0.29491496086120605 75 7.620276e+00 5.789929e-02 * time: 0.2977108955383301 76 7.619513e+00 3.276276e-02 * time: 0.3014090061187744 77 7.618927e+00 4.876947e-02 * time: 0.30429601669311523 78 7.618346e+00 4.942239e-02 * time: 0.3080019950866699 79 7.617934e+00 4.279016e-02 * time: 0.310960054397583 80 7.617496e+00 3.773494e-02 * time: 0.31472206115722656 81 7.617094e+00 2.405566e-02 * time: 0.3184850215911865 82 7.617053e+00 5.394157e-02 * time: 0.3213980197906494 83 7.616967e+00 5.312728e-02 * time: 0.3242359161376953 84 7.616922e+00 6.436742e-02 * time: 0.32726502418518066 85 7.616531e+00 6.642598e-02 * time: 0.33019304275512695 86 7.616523e+00 7.204034e-02 * time: 0.3329908847808838 87 7.616272e+00 9.439531e-02 * time: 0.33581995964050293 88 7.615383e+00 4.137449e-02 * time: 0.33868408203125 89 7.615060e+00 5.195357e-02 * time: 0.3416099548339844 90 7.614599e+00 5.764841e-02 * time: 0.3444099426269531 91 7.614263e+00 3.488419e-02 * time: 0.3680250644683838 92 7.614200e+00 7.111445e-02 * time: 0.37058591842651367 93 7.613861e+00 4.745466e-02 * time: 0.3739299774169922 94 7.613758e+00 7.308721e-02 * time: 0.3766360282897949 95 7.613495e+00 4.235777e-02 * time: 0.37992191314697266 96 7.613284e+00 5.425698e-02 * time: 0.3826110363006592 97 7.613238e+00 6.422631e-02 * time: 0.3851280212402344 98 7.613127e+00 4.979302e-02 * time: 0.3876760005950928 99 7.612736e+00 4.531421e-02 * time: 0.39010095596313477 100 7.612289e+00 5.095473e-02 * time: 0.39246296882629395 101 7.612222e+00 4.808735e-02 * time: 0.39496707916259766 102 7.611996e+00 6.912635e-02 * time: 0.3975050449371338 103 7.611706e+00 2.985130e-02 * time: 0.4007599353790283 104 7.611560e+00 3.449005e-02 * time: 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* time: 0.515143871307373 140 7.603884e+00 3.926253e-02 * time: 0.5178439617156982 141 7.603753e+00 4.465385e-02 * time: 0.5204100608825684 142 7.603644e+00 4.676564e-02 * time: 0.522974967956543 143 7.603606e+00 3.177344e-02 * time: 0.5255379676818848 144 7.603503e+00 3.557479e-02 * time: 0.5282430648803711 145 7.603356e+00 3.606652e-02 * time: 0.5309679508209229 146 7.603291e+00 4.589027e-02 * time: 0.5334959030151367 147 7.603101e+00 3.482555e-02 * time: 0.5369060039520264 148 7.603052e+00 4.365922e-02 * time: 0.5397450923919678 149 7.603047e+00 3.430691e-02 * time: 0.5425539016723633 150 7.602956e+00 4.213008e-02 * time: 0.5454549789428711 151 7.602784e+00 3.915344e-02 * time: 0.5484879016876221 152 7.602743e+00 4.001797e-02 * time: 0.5513160228729248 153 7.602743e+00 4.687891e-02 * time: 0.5541350841522217 154 7.602557e+00 4.547882e-02 * time: 0.5567560195922852 155 7.602279e+00 3.396413e-02 * time: 0.5602278709411621 156 7.602109e+00 3.248041e-02 * time: 0.5634810924530029 157 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0.734483003616333 210 7.598959e+00 8.618926e-03 * time: 0.736907958984375 211 7.598951e+00 1.343123e-02 * time: 0.7394678592681885 212 7.598950e+00 7.262747e-03 * time: 0.7420458793640137 213 7.598938e+00 8.706761e-03 * time: 0.7445158958435059 214 7.598930e+00 1.213919e-02 * time: 0.7471120357513428 215 7.598917e+00 7.614596e-03 * time: 0.7504429817199707 216 7.598907e+00 8.597830e-03 * time: 0.7528760433197021 217 7.598906e+00 8.504951e-03 * time: 0.7554900646209717 218 7.598893e+00 5.709894e-03 * time: 0.7580668926239014 219 7.598891e+00 1.042595e-02 * time: 0.7605738639831543 220 7.598885e+00 1.002559e-02 * time: 0.7631959915161133 221 7.598877e+00 6.454602e-03 * time: 0.7659459114074707 222 7.598873e+00 1.066817e-02 * time: 0.768779993057251 223 7.598862e+00 5.488670e-03 * time: 0.7722969055175781 224 7.598858e+00 8.595326e-03 * time: 0.7749669551849365 225 7.598857e+00 8.508869e-03 * time: 0.7774660587310791 226 7.598853e+00 6.631499e-03 * time: 0.7801470756530762 227 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