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 +27.99933216796 -1.42 6.26s 2 +18.89976936421 0.96 -0.92 11.8ms 3 +12.95759703710 0.77 -0.59 11.8ms 4 +11.44808011500 0.18 -0.47 9.39ms 5 +10.82303007302 -0.20 -0.62 9.36ms 6 +9.765991457194 0.02 -0.78 9.40ms 7 +8.840205055131 -0.03 -0.85 9.35ms 8 +8.376191649832 -0.33 -0.87 9.34ms 9 +8.209496936857 -0.78 -0.91 9.38ms 10 +8.193246727659 -1.79 -1.03 7.10ms 11 +8.121551312672 -1.14 -1.25 7.07ms 12 +8.090307338613 -1.51 -1.12 7.26ms 13 +8.023500248605 -1.18 -1.08 9.32ms 14 +7.996669591178 -1.57 -1.03 7.10ms 15 +7.950744630625 -1.34 -1.12 7.09ms 16 +7.922970046125 -1.56 -1.26 7.07ms 17 +7.892137163434 -1.51 -1.04 9.33ms 18 +7.874292532596 -1.75 -1.16 7.09ms 19 +7.850233572315 -1.62 -1.16 7.08ms 20 +7.823787396215 -1.58 -1.20 7.10ms 21 +7.798502765792 -1.60 -1.18 9.29ms 22 +7.782654468256 -1.80 -1.30 7.07ms 23 +7.773073879048 -2.02 -1.20 7.06ms 24 +7.763671356375 -2.03 -1.36 7.05ms 25 +7.746300426175 -1.76 -1.11 9.28ms 26 +7.736932239114 -2.03 -1.30 7.08ms 27 +7.725855217956 -1.96 -1.41 7.06ms 28 +7.720989195794 -2.31 -1.32 7.11ms 29 +7.711509529617 -2.02 -1.15 9.31ms 30 +7.704487508453 -2.15 -1.36 7.07ms 31 +7.699836872493 -2.33 -1.38 7.04ms 32 +7.693825179297 -2.22 -1.49 7.08ms 33 +7.691326276123 -2.60 -1.39 7.07ms 34 +7.685411667170 -2.23 -1.25 9.31ms 35 +7.680689145586 -2.33 -1.46 7.07ms 36 +7.677597529688 -2.51 -1.37 7.66ms 37 +7.675627602301 -2.71 -1.46 7.92ms 38 +7.673855179826 -2.75 -1.48 7.99ms 39 +7.670062759167 -2.42 -1.39 11.7ms 40 +7.665379305700 -2.33 -1.60 11.2ms 41 +7.661030754958 -2.36 -1.58 11.3ms 42 +7.657549871631 -2.46 -1.45 11.6ms 43 +7.656526894667 -2.99 -1.51 8.00ms 44 +7.655197041377 -2.88 -1.51 8.01ms 45 +7.651712488239 -2.46 -1.58 7.95ms 46 +7.648910749746 -2.55 -1.53 34.7ms 47 +7.645442636922 -2.46 -1.28 9.52ms 48 +7.645432586174 -5.00 -1.37 7.21ms 49 +7.645341599713 -4.04 -1.38 7.18ms 50 +7.643338920394 -2.70 -1.47 7.16ms 51 +7.641834873134 -2.82 -1.39 7.15ms 52 +7.639009008608 -2.55 -1.51 7.07ms 53 +7.635712566518 -2.48 -1.61 7.11ms 54 +7.632905761867 -2.55 -1.56 7.10ms 55 +7.631038072689 -2.73 -1.68 9.37ms 56 +7.628666262689 -2.62 -1.67 7.12ms 57 +7.627512953455 -2.94 -1.78 7.19ms 58 +7.627043509253 -3.33 -1.82 7.15ms 59 +7.626059730163 -3.01 -1.74 9.55ms 60 +7.625149102403 -3.04 -1.70 9.48ms 61 +7.624451111979 -3.16 -1.69 9.48ms 62 +7.623699982152 -3.12 -1.56 9.37ms 63 +7.623189134118 -3.29 -1.65 7.15ms 64 +7.622848630916 -3.47 -1.89 7.14ms 65 +7.622288958352 -3.25 -1.83 7.15ms 66 +7.621782058003 -3.30 -2.01 7.12ms 67 +7.621110730899 -3.17 -1.97 7.14ms 68 +7.620737272804 -3.43 -1.94 7.16ms 69 +7.620535439162 -3.70 -1.89 7.15ms 70 +7.620235701090 -3.52 -2.06 7.08ms 71 +7.619814488779 -3.38 -2.01 7.11ms 72 +7.619776610225 -4.42 -1.94 7.12ms 73 +7.619629065208 -3.83 -1.90 7.18ms 74 +7.619189830432 -3.36 -1.79 9.40ms 75 +7.619017104359 -3.76 -1.89 7.14ms 76 +7.618918299299 -4.01 -1.86 7.11ms 77 +7.618487299523 -3.37 -2.02 7.12ms 78 +7.618129037471 -3.45 -2.02 7.11ms 79 +7.617867393041 -3.58 -1.93 7.16ms 80 +7.617467538670 -3.40 -1.85 9.36ms 81 +7.617215585699 -3.60 -1.97 9.35ms 82 +7.617206214795 -5.03 -2.01 7.12ms 83 +7.617006872970 -3.70 -1.92 9.41ms 84 +7.616917852181 -4.05 -2.04 7.16ms 85 +7.616704683003 -3.67 -2.29 7.08ms 86 +7.616497129801 -3.68 -2.00 7.62ms 87 +7.616220060462 -3.56 -1.90 10.6ms 88 +7.616070764063 -3.83 -2.13 7.99ms 89 +7.616049614195 -4.67 -2.13 7.95ms 90 +7.615965775227 -4.08 -1.94 7.93ms 91 +7.615782385666 -3.74 -2.28 7.95ms 92 +7.615629442463 -3.82 -2.03 7.93ms 93 +7.615312127681 -3.50 -1.71 10.4ms 94 +7.615236517612 -4.12 -2.14 7.94ms 95 +7.615039507507 -3.71 -1.90 24.9ms 96 +7.614904126465 -3.87 -2.21 9.56ms 97 +7.614760820316 -3.84 -2.01 7.23ms 98 +7.614675942511 -4.07 -2.20 7.19ms 99 +7.614486900808 -3.72 -2.28 7.18ms 100 +7.614349842332 -3.86 -1.86 9.44ms 101 +7.614208254409 -3.85 -1.90 9.42ms 102 +7.614149993960 -4.23 -2.10 7.20ms 103 +7.614106476178 -4.36 -1.80 7.17ms 104 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-2.32 7.10ms 130 +7.610482236219 -3.78 -1.95 7.11ms 131 +7.610463279868 -4.72 -2.22 7.11ms 132 +7.610322524939 -3.85 -1.78 9.40ms 133 +7.610225834150 -4.01 -2.28 7.13ms 134 +7.610102137674 -3.91 -2.01 10.3ms 135 +7.610037846012 -4.19 -2.14 7.92ms 136 +7.609872509824 -3.78 -1.65 10.5ms 137 +7.609789020764 -4.08 -2.27 7.97ms 138 +7.609731661903 -4.24 -2.08 7.91ms 139 +7.609569299689 -3.79 -1.97 7.96ms 140 +7.609493992098 -4.12 -1.98 10.5ms 141 +7.609400580727 -4.03 -2.25 7.95ms 142 +7.609273457434 -3.90 -1.81 20.7ms 143 +7.609220278471 -4.27 -2.21 7.41ms 144 +7.609170451145 -4.30 -2.04 7.21ms 145 +7.609072151318 -4.01 -2.30 7.21ms 146 +7.608958345316 -3.94 -1.95 9.46ms 147 +7.608943666798 -4.83 -2.24 7.20ms 148 +7.608925833335 -4.75 -2.04 7.19ms 149 +7.608762548655 -3.79 -1.68 9.50ms 150 +7.608706879287 -4.25 -2.13 7.27ms 151 +7.608617972318 -4.05 -2.32 9.48ms 152 +7.608463486537 -3.81 -2.11 7.14ms 153 +7.608368788649 -4.02 -2.17 9.37ms 154 +7.608253097404 -3.94 -1.92 9.40ms 155 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-2.16 7.23ms 181 +7.606208450478 -4.57 -1.98 7.41ms 182 +7.606124804598 -4.08 -2.09 10.5ms 183 +7.606052986123 -4.14 -2.29 10.5ms 184 +7.605951052422 -3.99 -1.74 10.5ms 185 +7.605850909186 -4.00 -1.75 10.5ms 186 +7.605729107365 -3.91 -2.24 7.95ms 187 +7.605608662172 -3.92 -1.73 10.5ms 188 +7.605450447773 -3.80 -1.76 10.5ms 189 +7.605373442927 -4.11 -2.34 18.4ms 190 +7.605331036797 -4.37 -2.10 7.36ms 191 +7.605263182081 -4.17 -1.75 7.33ms 192 +7.605183699077 -4.10 -2.06 7.24ms 193 +7.605054648944 -3.89 -1.74 9.45ms 194 +7.604999268601 -4.26 -2.06 7.19ms 195 +7.604834373346 -3.78 -1.93 9.51ms 196 +7.604719742155 -3.94 -2.13 9.49ms 197 +7.604600834555 -3.92 -1.90 7.19ms 198 +7.604464884506 -3.87 -2.04 9.41ms 199 +7.604326716369 -3.86 -1.77 9.40ms 200 +7.604162892228 -3.79 -2.15 7.15ms 201 +7.604047103667 -3.94 -1.80 9.42ms 202 +7.603944603173 -3.99 -2.04 7.23ms 203 +7.603790426028 -3.81 -1.68 9.48ms 204 +7.603659273504 -3.88 -1.96 9.39ms 205 +7.603515359544 -3.84 -1.53 9.38ms 206 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-3.86 7.05ms 997 +7.592354154760 -7.02 -3.81 7.16ms 998 +7.592354038393 -6.93 -3.46 9.54ms 999 +7.592353992348 -7.34 -3.91 7.96ms 1000 +7.592353970157 -7.65 -3.81 7.92ms ┌ Warning: DM not converged. └ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/scf/scf_callbacks.jl:60