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...)),
LocalNonlinearity(ρ -> C * ρ^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 3.158441e+01 6.529925e+00 * time: 0.005012989044189453 1 3.075073e+01 6.204884e+00 * time: 0.02259206771850586 2 2.384565e+01 6.900158e+00 * time: 0.04505801200866699 3 2.128919e+01 9.843236e+00 * time: 0.06493806838989258 4 2.018377e+01 9.144052e+00 * time: 0.08582115173339844 5 1.893492e+01 1.021940e+01 * time: 0.10528802871704102 6 1.403439e+01 5.227915e+00 * time: 0.12463808059692383 7 1.011977e+01 1.590490e+00 * time: 0.15041399002075195 8 9.348949e+00 1.183203e+00 * time: 0.1669330596923828 9 9.015211e+00 6.558213e-01 * time: 0.18684601783752441 10 8.834260e+00 5.404419e-01 * time: 0.20301198959350586 11 8.661745e+00 6.066365e-01 * time: 0.21939921379089355 12 8.507849e+00 7.603904e-01 * time: 0.23871803283691406 13 8.424249e+00 5.384395e-01 * time: 0.2515990734100342 14 8.360487e+00 6.086083e-01 * time: 0.26422810554504395 15 8.294354e+00 6.111526e-01 * time: 0.27623414993286133 16 8.231355e+00 5.778194e-01 * time: 0.2897810935974121 17 8.182206e+00 3.512043e-01 * time: 0.35185909271240234 18 8.139180e+00 3.605274e-01 * time: 0.36698102951049805 19 8.114362e+00 2.690385e-01 * time: 0.37894320487976074 20 8.090753e+00 3.617133e-01 * time: 0.39075708389282227 21 8.063628e+00 3.784565e-01 * time: 0.40259408950805664 22 8.042975e+00 4.132328e-01 * time: 0.4145979881286621 23 8.035295e+00 6.769483e-01 * time: 0.4265320301055908 24 8.007101e+00 2.451503e-01 * time: 0.4390590190887451 25 7.983448e+00 3.981588e-01 * time: 0.45063114166259766 26 7.948248e+00 3.615840e-01 * time: 0.4624199867248535 27 7.912236e+00 2.997334e-01 * time: 0.4745349884033203 28 7.874021e+00 1.940306e-01 * time: 0.48646020889282227 29 7.849063e+00 2.490011e-01 * time: 0.4983952045440674 30 7.830412e+00 1.608276e-01 * time: 0.5144321918487549 31 7.819493e+00 3.117066e-01 * time: 0.5260941982269287 32 7.812808e+00 4.556886e-01 * time: 0.5385231971740723 33 7.809314e+00 4.468907e-01 * time: 0.5503671169281006 34 7.798518e+00 3.331207e-01 * time: 0.5623400211334229 35 7.774004e+00 3.206037e-01 * time: 0.5744590759277344 36 7.749812e+00 3.385668e-01 * time: 0.5860781669616699 37 7.738444e+00 3.758820e-01 * time: 0.5978569984436035 38 7.732835e+00 3.319419e-01 * time: 0.6095490455627441 39 7.715152e+00 2.167460e-01 * time: 0.6221082210540771 40 7.701509e+00 1.572975e-01 * time: 0.6351540088653564 41 7.695084e+00 2.799058e-01 * time: 0.6480941772460938 42 7.684083e+00 2.292058e-01 * time: 0.7040841579437256 43 7.673713e+00 1.905271e-01 * time: 0.7198870182037354 44 7.665272e+00 1.519861e-01 * time: 0.7359151840209961 45 7.659671e+00 1.091244e-01 * time: 0.7511610984802246 46 7.655865e+00 1.740893e-01 * time: 0.7630951404571533 47 7.655305e+00 2.404561e-01 * time: 0.7750611305236816 48 7.651406e+00 1.465628e-01 * time: 0.7904829978942871 49 7.651069e+00 1.717657e-01 * time: 0.8024301528930664 50 7.647354e+00 1.367918e-01 * time: 0.8144631385803223 51 7.643276e+00 1.150762e-01 * time: 0.8261480331420898 52 7.642837e+00 1.533728e-01 * time: 0.8413150310516357 53 7.640802e+00 1.504505e-01 * time: 0.8530540466308594 54 7.638910e+00 1.012863e-01 * time: 0.8687491416931152 55 7.638536e+00 1.618244e-01 * time: 0.880620002746582 56 7.636860e+00 9.836260e-02 * time: 0.8962421417236328 57 7.636789e+00 1.669978e-01 * time: 0.9077601432800293 58 7.636011e+00 1.747419e-01 * time: 0.9195291996002197 59 7.634030e+00 1.117289e-01 * time: 0.9313950538635254 60 7.630807e+00 6.890239e-02 * time: 0.9467902183532715 61 7.629135e+00 9.575292e-02 * time: 0.9588930606842041 62 7.627555e+00 8.994076e-02 * time: 0.9720280170440674 63 7.625488e+00 5.338541e-02 * time: 0.9894599914550781 64 7.625127e+00 7.262604e-02 * time: 1.0024471282958984 65 7.624339e+00 5.492033e-02 * time: 1.045790195465088 66 7.623521e+00 5.575755e-02 * time: 1.061622142791748 67 7.623088e+00 6.203352e-02 * time: 1.0735511779785156 68 7.622127e+00 5.924829e-02 * time: 1.0891871452331543 69 7.621472e+00 2.904818e-02 * time: 1.1050782203674316 70 7.621003e+00 4.336102e-02 * time: 1.117004156112671 71 7.620444e+00 4.070069e-02 * time: 1.1290581226348877 72 7.620383e+00 8.000880e-02 * time: 1.144928216934204 73 7.620274e+00 6.530196e-02 * time: 1.1573781967163086 74 7.619809e+00 6.272210e-02 * time: 1.1708860397338867 75 7.619484e+00 5.758136e-02 * time: 1.1828231811523438 76 7.619142e+00 5.606010e-02 * time: 1.1949050426483154 77 7.618864e+00 4.618268e-02 * time: 1.2071380615234375 78 7.618336e+00 4.077179e-02 * time: 1.2189970016479492 79 7.617988e+00 4.958259e-02 * time: 1.2307190895080566 80 7.617845e+00 4.203126e-02 * time: 1.2429661750793457 81 7.617545e+00 4.210287e-02 * time: 1.254892110824585 82 7.617344e+00 2.124653e-02 * time: 1.2704792022705078 83 7.617246e+00 4.418489e-02 * time: 1.2823851108551025 84 7.617027e+00 3.055669e-02 * time: 1.2980060577392578 85 7.616882e+00 3.383558e-02 * time: 1.3104500770568848 86 7.616689e+00 2.833114e-02 * time: 1.327991008758545 87 7.616596e+00 3.499899e-02 * time: 1.3413481712341309 88 7.616593e+00 3.264173e-02 * time: 1.3771851062774658 89 7.616522e+00 2.592447e-02 * time: 1.3891031742095947 90 7.616405e+00 4.241944e-02 * time: 1.4010791778564453 91 7.616319e+00 3.687591e-02 * time: 1.4128060340881348 92 7.616162e+00 3.262028e-02 * time: 1.4243831634521484 93 7.615949e+00 2.230139e-02 * time: 1.4361050128936768 94 7.615927e+00 4.021824e-02 * time: 1.4479191303253174 95 7.615858e+00 5.021080e-02 * time: 1.4597342014312744 96 7.615690e+00 3.264359e-02 * time: 1.4713711738586426 97 7.615613e+00 3.229184e-02 * time: 1.4830760955810547 98 7.615436e+00 2.663123e-02 * time: 1.4988842010498047 99 7.615249e+00 1.706134e-02 * time: 1.5144309997558594 100 7.615131e+00 1.558290e-02 * time: 1.5300052165985107 101 7.615088e+00 3.960597e-02 * time: 1.541923999786377 102 7.614951e+00 2.573776e-02 * time: 1.5571980476379395 103 7.614847e+00 2.211328e-02 * time: 1.569031000137329 104 7.614744e+00 2.184947e-02 * time: 1.580848217010498 105 7.614706e+00 3.118701e-02 * time: 1.5924930572509766 106 7.614583e+00 2.225687e-02 * time: 1.6084351539611816 107 7.614444e+00 3.721184e-02 * time: 1.6203851699829102 108 7.614270e+00 2.238069e-02 * time: 1.6362311840057373 109 7.614136e+00 3.332699e-02 * time: 1.6495351791381836 110 7.613996e+00 3.047618e-02 * time: 1.6624751091003418 111 7.613852e+00 2.431327e-02 * time: 1.6754281520843506 112 7.613679e+00 3.551999e-02 * time: 1.7102580070495605 113 7.613513e+00 2.328950e-02 * time: 1.7221720218658447 114 7.613490e+00 4.512833e-02 * time: 1.7338461875915527 115 7.613388e+00 4.183490e-02 * time: 1.7461740970611572 116 7.613230e+00 2.065543e-02 * time: 1.761625051498413 117 7.613047e+00 2.239544e-02 * time: 1.7735559940338135 118 7.612909e+00 3.036614e-02 * time: 1.785431146621704 119 7.612740e+00 1.804457e-02 * time: 1.7971811294555664 120 7.612726e+00 5.013613e-02 * time: 1.8088171482086182 121 7.612510e+00 4.108673e-02 * time: 1.8241770267486572 122 7.612388e+00 3.100265e-02 * time: 1.8399240970611572 123 7.612273e+00 3.618959e-02 * time: 1.8522601127624512 124 7.612139e+00 4.680698e-02 * time: 1.863948106765747 125 7.611903e+00 2.270899e-02 * time: 1.879164218902588 126 7.611732e+00 2.316536e-02 * time: 1.8907160758972168 127 7.611678e+00 4.431792e-02 * time: 1.90214204788208 128 7.611507e+00 3.821697e-02 * time: 1.9172132015228271 129 7.611307e+00 3.305492e-02 * time: 1.9324150085449219 130 7.611202e+00 5.913222e-02 * time: 1.9448540210723877 131 7.610937e+00 3.714063e-02 * time: 1.9600050449371338 132 7.610834e+00 4.066158e-02 * time: 1.973717212677002 133 7.610721e+00 3.380214e-02 * time: 1.9908959865570068 134 7.610424e+00 3.130255e-02 * time: 2.003547191619873 135 7.610233e+00 3.079234e-02 * time: 2.040894031524658 136 7.610069e+00 2.612226e-02 * time: 2.0601181983947754 137 7.609930e+00 2.650606e-02 * time: 2.079094171524048 138 7.609803e+00 1.910044e-02 * time: 2.094766139984131 139 7.609730e+00 3.099989e-02 * time: 2.1066830158233643 140 7.609633e+00 3.299785e-02 * time: 2.1182310581207275 141 7.609497e+00 2.455384e-02 * time: 2.12986421585083 142 7.609379e+00 3.025385e-02 * time: 2.145726203918457 143 7.609237e+00 1.065880e-02 * time: 2.161553144454956 144 7.609107e+00 1.687673e-02 * time: 2.173441171646118 145 7.609083e+00 2.562332e-02 * time: 2.1851260662078857 146 7.608934e+00 2.855580e-02 * time: 2.196895122528076 147 7.608820e+00 1.760804e-02 * time: 2.2124030590057373 148 7.608657e+00 1.998861e-02 * time: 2.223755121231079 149 7.608605e+00 2.835028e-02 * time: 2.235621213912964 150 7.608461e+00 2.620076e-02 * time: 2.2516682147979736 151 7.608396e+00 4.002600e-02 * time: 2.2631471157073975 152 7.608276e+00 3.675353e-02 * time: 2.2749290466308594 153 7.608185e+00 3.570448e-02 * time: 2.286708116531372 154 7.608020e+00 2.730947e-02 * time: 2.299337148666382 155 7.607962e+00 4.760897e-02 * time: 2.3125171661376953 156 7.607782e+00 2.783927e-02 * time: 2.3298439979553223 157 7.607659e+00 2.816178e-02 * time: 2.3674161434173584 158 7.607546e+00 3.266044e-02 * time: 2.384275197982788 159 7.607407e+00 2.782225e-02 * time: 2.39614200592041 160 7.607286e+00 1.697699e-02 * time: 2.4114601612091064 161 7.607168e+00 2.324873e-02 * time: 2.4230501651763916 162 7.607131e+00 4.170998e-02 * time: 2.4354031085968018 163 7.607051e+00 3.509715e-02 * time: 2.446927070617676 164 7.606945e+00 3.507676e-02 * time: 2.458728075027466 165 7.606803e+00 3.277189e-02 * time: 2.474316120147705 166 7.606683e+00 2.995915e-02 * time: 2.4860000610351562 167 7.606528e+00 3.020213e-02 * time: 2.5013771057128906 168 7.606507e+00 4.120695e-02 * time: 2.512902021408081 169 7.606481e+00 2.714590e-02 * time: 2.5243802070617676 170 7.606357e+00 2.121519e-02 * time: 2.536015033721924 171 7.606268e+00 2.347918e-02 * time: 2.54746413230896 172 7.606117e+00 2.037599e-02 * time: 2.5633931159973145 173 7.606072e+00 3.525702e-02 * time: 2.574989080429077 174 7.605993e+00 2.386536e-02 * time: 2.5900931358337402 175 7.605908e+00 1.637711e-02 * time: 2.6052310466766357 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2.8704540729522705 194 7.605066e+00 2.152810e-02 * time: 2.882331132888794 195 7.604987e+00 1.236865e-02 * time: 2.89829421043396 196 7.604977e+00 2.291799e-02 * time: 2.9102840423583984 197 7.604922e+00 2.114200e-02 * time: 2.9222681522369385 198 7.604885e+00 2.833928e-02 * time: 2.934333086013794 199 7.604853e+00 3.081825e-02 * time: 2.9463260173797607 200 7.604738e+00 2.444369e-02 * time: 2.9593491554260254 201 7.604632e+00 2.768807e-02 * time: 2.9765121936798096 202 7.604604e+00 4.545622e-02 * time: 2.9892361164093018 203 7.604518e+00 3.115240e-02 * time: 3.023972988128662 204 7.604403e+00 3.129996e-02 * time: 3.0395500659942627 205 7.604384e+00 2.621662e-02 * time: 3.051220178604126 206 7.604297e+00 2.283514e-02 * time: 3.063847064971924 207 7.604269e+00 3.426511e-02 * time: 3.0755650997161865 208 7.604166e+00 2.709227e-02 * time: 3.087613105773926 209 7.603995e+00 2.238371e-02 * time: 3.1033310890197754 210 7.603860e+00 2.106487e-02 * time: 3.1190621852874756 211 7.603845e+00 4.031376e-02 * time: 3.130902051925659 212 7.603767e+00 3.556432e-02 * time: 3.1429131031036377 213 7.603742e+00 5.084857e-02 * time: 3.154780149459839 214 7.603736e+00 4.147890e-02 * time: 3.1666250228881836 215 7.603588e+00 3.549621e-02 * time: 3.178352117538452 216 7.603360e+00 3.328011e-02 * time: 3.189937114715576 217 7.603026e+00 2.357955e-02 * time: 3.2058651447296143 218 7.602898e+00 2.350465e-02 * time: 3.2216861248016357 219 7.602762e+00 1.790864e-02 * time: 3.237380027770996 220 7.602671e+00 2.674959e-02 * time: 3.249364137649536 221 7.602552e+00 2.060617e-02 * time: 3.2613790035247803 222 7.602433e+00 1.838683e-02 * time: 3.283520221710205 223 7.602294e+00 1.365126e-02 * time: 3.299960136413574 224 7.602156e+00 2.724136e-02 * time: 3.3127481937408447 225 7.602036e+00 1.850297e-02 * time: 3.3292860984802246 226 7.601976e+00 3.024826e-02 * time: 3.3643641471862793 227 7.601865e+00 3.175196e-02 * time: 3.3764140605926514 228 7.601675e+00 2.446966e-02 * time: 3.388507127761841 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