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, 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)
n Energy log10(ΔE) log10(Δρ) Δtime --- --------------- --------- --------- ------ 1 +30.40743520513 -1.42 5.13s 2 +23.95051791622 0.81 -0.77 10.6ms 3 +21.65829285710 0.36 -0.37 10.4ms 4 +19.79805467132 0.27 -0.28 10.5ms 5 +18.34725768879 0.16 -0.49 10.4ms 6 +13.44183896546 0.69 -0.45 10.4ms 7 +10.03875646258 0.53 -0.45 10.4ms 8 +9.224587654307 -0.09 -0.71 8.44ms 9 +8.915536203565 -0.51 -0.95 8.51ms 10 +8.748496019147 -0.78 -1.01 8.41ms 11 +8.624414735914 -0.91 -0.98 8.36ms 12 +8.496505081782 -0.89 -0.98 8.36ms 13 +8.403619841622 -1.03 -1.06 6.37ms 14 +8.318072596839 -1.07 -1.07 6.37ms 15 +8.220172203342 -1.01 -1.01 6.49ms 16 +8.174244157943 -1.34 -1.06 6.54ms 17 +8.122124893775 -1.28 -1.05 6.39ms 18 +8.057408145205 -1.19 -0.78 49.5ms 19 +8.016103387287 -1.38 -1.00 6.91ms 20 +7.976589050989 -1.40 -1.14 6.66ms 21 +7.942160965503 -1.46 -1.22 6.51ms 22 +7.912519223739 -1.53 -1.09 8.39ms 23 +7.885169958563 -1.56 -1.19 6.38ms 24 +7.857929374049 -1.56 -1.18 6.41ms 25 +7.838054375681 -1.70 -1.18 6.45ms 26 +7.813489391633 -1.61 -1.27 6.36ms 27 +7.807638907214 -2.23 -1.26 6.36ms 28 +7.794523541314 -1.88 -1.35 6.37ms 29 +7.786136511033 -2.08 -1.36 6.35ms 30 +7.770365002571 -1.80 -1.12 8.48ms 31 +7.763995631407 -2.20 -1.24 6.40ms 32 +7.749705060282 -1.84 -1.31 6.39ms 33 +7.742339367976 -2.13 -1.34 6.43ms 34 +7.731506958657 -1.97 -1.14 8.37ms 35 +7.722796413393 -2.06 -1.40 6.37ms 36 +7.720441219131 -2.63 -1.28 33.1ms 37 +7.710635638641 -2.01 -1.42 6.79ms 38 +7.707967460885 -2.57 -1.27 6.41ms 39 +7.699905369066 -2.09 -1.48 6.39ms 40 +7.699751183325 -3.81 -1.34 6.40ms 41 +7.692279641004 -2.13 -1.47 6.36ms 42 +7.686639013682 -2.25 -1.34 8.35ms 43 +7.683328563494 -2.48 -1.43 6.37ms 44 +7.681589036952 -2.76 -1.46 6.40ms 45 +7.679111709509 -2.61 -1.46 6.35ms 46 +7.675881512906 -2.49 -1.56 6.34ms 47 +7.671921382044 -2.40 -1.32 8.30ms 48 +7.669112597370 -2.55 -1.68 6.32ms 49 +7.668966744259 -3.84 -1.45 6.38ms 50 +7.666071357831 -2.54 -1.77 6.37ms 51 +7.662929783659 -2.50 -1.36 8.33ms 52 +7.660899070272 -2.69 -1.59 6.42ms 53 +7.658443827569 -2.61 -1.73 6.38ms 54 +7.656117400000 -2.63 -1.60 6.37ms 55 +7.654093412043 -2.69 -1.36 8.37ms 56 +7.652194642305 -2.72 -1.50 110ms 57 +7.652008351222 -3.73 -1.49 8.56ms 58 +7.650008522980 -2.70 -1.62 8.11ms 59 +7.648742068357 -2.90 -1.57 8.22ms 60 +7.646056039246 -2.57 -1.17 10.6ms 61 +7.644011513007 -2.69 -1.67 8.26ms 62 +7.643283208520 -3.14 -1.64 8.36ms 63 +7.640607721221 -2.57 -1.42 8.26ms 64 +7.638793894247 -2.74 -1.52 10.8ms 65 +7.637288344953 -2.82 -1.68 8.28ms 66 +7.635702379870 -2.80 -1.56 10.6ms 67 +7.635635896940 -4.18 -1.71 8.31ms 68 +7.635506725063 -3.89 -1.54 8.11ms 69 +7.634244756424 -2.90 -1.58 10.9ms 70 +7.632347478761 -2.72 -1.80 8.37ms 71 +7.630822064931 -2.82 -1.64 551ms 72 +7.629498567770 -2.88 -1.89 6.59ms 73 +7.628177859889 -2.88 -1.46 8.70ms 74 +7.627243957282 -3.03 -1.77 6.52ms 75 +7.626306735581 -3.03 -1.88 6.41ms 76 +7.625576061958 -3.14 -2.04 6.42ms 77 +7.625056631662 -3.28 -1.92 6.51ms 78 +7.624463816544 -3.23 -1.47 8.58ms 79 +7.623792793674 -3.17 -1.49 8.51ms 80 +7.623104801284 -3.16 -2.03 7.46ms 81 +7.622682516599 -3.37 -1.78 8.59ms 82 +7.622354999403 -3.48 -1.78 8.00ms 83 +7.622139013520 -3.67 -2.04 7.96ms 84 +7.621749301055 -3.41 -2.09 7.95ms 85 +7.621549590557 -3.70 -2.02 7.91ms 86 +7.621384635449 -3.78 -2.06 8.03ms 87 +7.621052695397 -3.48 -2.09 8.10ms 88 +7.620835760113 -3.66 -1.91 8.07ms 89 +7.620682807220 -3.82 -1.81 8.14ms 90 +7.620619929810 -4.20 -2.19 8.09ms 91 +7.620223192538 -3.40 -2.11 8.24ms 92 +7.620096760435 -3.90 -2.18 8.03ms 93 +7.619835935341 -3.58 -1.90 10.6ms 94 +7.619748661897 -4.06 -1.93 6.99ms 95 +7.619590529046 -3.80 -2.29 6.94ms 96 +7.619471623799 -3.92 -2.09 7.00ms 97 +7.619262466071 -3.68 -1.92 9.29ms 98 +7.619157128193 -3.98 -2.08 7.04ms 99 +7.619016316774 -3.85 -2.06 7.01ms 100 +7.618741575751 -3.56 -2.02 7.11ms 101 +7.618538029263 -3.69 -1.90 9.34ms 102 +7.618387688675 -3.82 -2.16 6.90ms 103 +7.618312103968 -4.12 -2.03 6.85ms 104 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-1.81 8.45ms 130 +7.610284001260 -3.71 -1.93 8.40ms 131 +7.610263135250 -4.68 -1.78 6.47ms 132 +7.609959375595 -3.52 -2.14 6.43ms 133 +7.609915086966 -4.35 -1.48 6.45ms 134 +7.609599466041 -3.50 -1.78 6.37ms 135 +7.609174677972 -3.37 -1.84 8.36ms 136 +7.608677873783 -3.30 -1.93 6.52ms 137 +7.608437035874 -3.62 -1.98 6.58ms 138 +7.608298119723 -3.86 -2.01 6.48ms 139 +7.607990735010 -3.51 -1.76 8.44ms 140 +7.607660477391 -3.48 -2.03 6.47ms 141 +7.607445773850 -3.67 -2.09 8.54ms 142 +7.607270337122 -3.76 -1.87 6.47ms 143 +7.607186400597 -4.08 -2.12 6.43ms 144 +7.606946753872 -3.62 -1.80 9.01ms 145 +7.606722581589 -3.65 -1.88 8.59ms 146 +7.606624294590 -4.01 -2.04 6.48ms 147 +7.606394872724 -3.64 -1.90 8.42ms 148 +7.606334516054 -4.22 -2.08 6.76ms 149 +7.606243060616 -4.04 -1.95 6.47ms 150 +7.606209479334 -4.47 -2.20 6.43ms 151 +7.606017073368 -3.72 -2.09 6.41ms 152 +7.605956753803 -4.22 -2.24 6.43ms 153 +7.605849161817 -3.97 -2.07 6.51ms 154 +7.605732581268 -3.93 -2.04 6.38ms 155 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-1.61 8.36ms 181 +7.604065857909 -4.70 -2.15 6.35ms 182 +7.603960120423 -3.98 -1.57 8.40ms 183 +7.603908442865 -4.29 -2.20 6.35ms 184 +7.603826087998 -4.08 -2.23 6.47ms 185 +7.603800245040 -4.59 -2.32 6.36ms 186 +7.603712066841 -4.05 -2.33 6.37ms 187 +7.603651682821 -4.22 -2.07 8.41ms 188 +7.603604458943 -4.33 -2.34 6.40ms 189 +7.603553427850 -4.29 -2.32 20.2ms 190 +7.603493610866 -4.22 -2.19 6.53ms 191 +7.603454734658 -4.41 -2.26 6.45ms 192 +7.603401976902 -4.28 -2.12 6.43ms 193 +7.603347386849 -4.26 -2.33 8.40ms 194 +7.603309883062 -4.43 -2.05 6.40ms 195 +7.603262659122 -4.33 -2.17 6.39ms 196 +7.603258057164 -5.34 -2.11 6.41ms 197 +7.603186603127 -4.15 -2.50 6.43ms 198 +7.603176662550 -5.00 -2.44 6.41ms 199 +7.603108292370 -4.17 -2.14 6.42ms 200 +7.603080127346 -4.55 -2.33 6.35ms 201 +7.603023836162 -4.25 -2.42 6.37ms 202 +7.602952786680 -4.15 -2.16 8.35ms 203 +7.602892316555 -4.22 -2.09 8.39ms 204 +7.602869888511 -4.65 -2.48 6.40ms 205 +7.602798119397 -4.14 -1.91 8.41ms 206 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-2.37 6.43ms 232 +7.602325884138 -5.03 -2.73 15.4ms 233 +7.602312260251 -4.87 -2.54 6.47ms 234 +7.602284879580 -4.56 -2.32 8.46ms 235 +7.602273325539 -4.94 -2.49 6.40ms 236 +7.602256291123 -4.77 -2.42 8.37ms 237 +7.602244110998 -4.91 -2.57 6.50ms 238 +7.602241928492 -5.66 -2.40 6.41ms 239 +7.602231087326 -4.96 -2.59 6.44ms 240 +7.602216406811 -4.83 -2.40 8.41ms 241 +7.602211965176 -5.35 -2.32 6.37ms 242 +7.602183452543 -4.54 -2.06 8.41ms 243 +7.602163667876 -4.70 -2.44 6.38ms 244 +7.602159106577 -5.34 -2.27 6.41ms 245 +7.602149565301 -5.02 -2.65 6.40ms 246 +7.602136424664 -4.88 -2.09 6.36ms 247 +7.602103794698 -4.49 -2.56 14.1ms 248 +7.602056252331 -4.32 -2.52 6.43ms 249 +7.602007354953 -4.31 -2.00 8.43ms 250 +7.601999037657 -5.08 -2.47 6.41ms 251 +7.601983945660 -4.82 -2.75 6.42ms 252 +7.601957317728 -4.57 -2.06 8.44ms 253 +7.601955496522 -5.74 -2.58 6.38ms 254 +7.601933641044 -4.66 -2.46 6.41ms 255 +7.601905930048 -4.56 -2.03 8.39ms 256 +7.601904477951 -5.84 -2.21 6.37ms 257 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-2.66 6.45ms 283 +7.601601220497 -5.64 -2.69 6.42ms 284 +7.601588704595 -4.90 -3.06 6.38ms 285 +7.601575252569 -4.87 -2.40 8.41ms 286 +7.601574907551 -6.46 -2.69 6.40ms 287 +7.601565714989 -5.04 -2.39 8.45ms 288 +7.601561034040 -5.33 -2.83 6.35ms 289 +7.601551447881 -5.02 -2.29 14.6ms 290 +7.601551061891 -6.41 -2.50 6.41ms 291 +7.601543884715 -5.14 -3.03 6.44ms 292 +7.601532169456 -4.93 -2.33 8.35ms 293 +7.601522391628 -5.01 -2.27 8.33ms 294 +7.601521963778 -6.37 -2.47 6.38ms 295 +7.601517430435 -5.34 -2.99 6.38ms 296 +7.601514201520 -5.49 -2.98 6.41ms 297 +7.601511259941 -5.53 -2.36 6.41ms 298 +7.601506162738 -5.29 -2.70 27.8ms 299 +7.601501560098 -5.34 -2.64 6.94ms 300 +7.601497332656 -5.37 -2.81 8.62ms 301 +7.601493710643 -5.44 -2.79 6.61ms 302 +7.601490787684 -5.53 -2.93 6.46ms 303 +7.601487034489 -5.43 -2.95 8.45ms 304 +7.601486891384 -6.84 -2.88 6.43ms 305 +7.601480418330 -5.19 -2.51 8.45ms 306 +7.601476763172 -5.44 -2.72 6.48ms 307 +7.601470725411 -5.22 -2.99 6.47ms 308 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