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)
Iter Function value Gradient norm 0 3.064853e+01 8.031630e+00 * time: 0.0034520626068115234 1 2.872422e+01 3.927619e+00 * time: 0.01581597328186035 2 1.928271e+01 4.008898e+00 * time: 0.032022953033447266 3 1.405867e+01 2.676105e+00 * time: 0.11735105514526367 4 1.232810e+01 3.506580e+00 * time: 0.12948989868164062 5 1.224701e+01 2.975797e+00 * time: 0.1411600112915039 6 1.007523e+01 1.629148e+00 * time: 0.15286493301391602 7 8.892682e+00 1.648603e+00 * time: 0.16461801528930664 8 8.438044e+00 1.325825e+00 * time: 0.17624807357788086 9 8.233540e+00 7.573126e-01 * time: 0.18779206275939941 10 8.166378e+00 9.814119e-01 * time: 0.19676589965820312 11 8.083050e+00 5.237860e-01 * time: 0.20556092262268066 12 8.069300e+00 7.612078e-01 * time: 0.21447205543518066 13 8.044603e+00 5.990265e-01 * time: 0.22327399253845215 14 7.986142e+00 4.366388e-01 * time: 0.23208904266357422 15 7.922525e+00 4.692781e-01 * time: 0.24103212356567383 16 7.897134e+00 3.585648e-01 * time: 0.2503659725189209 17 7.873650e+00 3.831230e-01 * time: 0.25938892364501953 18 7.869721e+00 5.088469e-01 * time: 0.2683219909667969 19 7.858000e+00 4.313498e-01 * time: 0.2777550220489502 20 7.834704e+00 2.994196e-01 * time: 0.28693199157714844 21 7.815244e+00 3.418351e-01 * time: 0.296065092086792 22 7.801572e+00 2.418865e-01 * time: 0.30562710762023926 23 7.789103e+00 3.094291e-01 * time: 0.3179469108581543 24 7.776449e+00 1.570042e-01 * time: 0.33028697967529297 25 7.763614e+00 1.599361e-01 * time: 0.33951401710510254 26 7.756710e+00 2.514865e-01 * time: 0.3488318920135498 27 7.756056e+00 2.415029e-01 * time: 0.3585491180419922 28 7.749525e+00 3.524544e-01 * time: 0.3683810234069824 29 7.740761e+00 1.920991e-01 * time: 0.38157010078430176 30 7.733919e+00 2.491474e-01 * time: 0.3912670612335205 31 7.722286e+00 1.710827e-01 * time: 0.40096092224121094 32 7.718729e+00 2.739167e-01 * time: 0.4107019901275635 33 7.715922e+00 2.116974e-01 * time: 0.4203629493713379 34 7.708390e+00 1.547255e-01 * time: 0.4302220344543457 35 7.707042e+00 3.066302e-01 * time: 0.44003796577453613 36 7.701642e+00 2.172196e-01 * time: 0.45300793647766113 37 7.696708e+00 2.410515e-01 * time: 0.4658939838409424 38 7.691069e+00 1.642459e-01 * time: 0.4787099361419678 39 7.687370e+00 1.320677e-01 * time: 0.48846912384033203 40 7.684869e+00 1.052816e-01 * time: 0.4983189105987549 41 7.684100e+00 1.585718e-01 * time: 0.5083110332489014 42 7.682800e+00 1.423136e-01 * time: 0.5182199478149414 43 7.679797e+00 1.042916e-01 * time: 0.5310111045837402 44 7.677765e+00 1.356282e-01 * time: 0.5407040119171143 45 7.673509e+00 1.401235e-01 * time: 0.6075088977813721 46 7.670254e+00 8.285267e-02 * time: 0.6192500591278076 47 7.666169e+00 1.283240e-01 * time: 0.6281208992004395 48 7.665128e+00 1.666791e-01 * time: 0.6369531154632568 49 7.662586e+00 1.195202e-01 * time: 0.645766019821167 50 7.661456e+00 1.738547e-01 * time: 0.6545538902282715 51 7.659635e+00 1.463413e-01 * time: 0.6636600494384766 52 7.658185e+00 1.270792e-01 * time: 0.6725120544433594 53 7.653346e+00 9.461236e-02 * time: 0.6812670230865479 54 7.649535e+00 1.246124e-01 * time: 0.6900649070739746 55 7.649355e+00 1.773132e-01 * time: 0.6987340450286865 56 7.648321e+00 1.248029e-01 * time: 0.7076430320739746 57 7.646454e+00 1.338393e-01 * time: 0.7164640426635742 58 7.643501e+00 1.048567e-01 * time: 0.7253448963165283 59 7.641085e+00 8.072875e-02 * time: 0.7370100021362305 60 7.638532e+00 7.148484e-02 * time: 0.7457621097564697 61 7.638208e+00 1.403919e-01 * time: 0.754709005355835 62 7.637273e+00 1.084078e-01 * time: 0.7639811038970947 63 7.637092e+00 1.406007e-01 * time: 0.7732589244842529 64 7.635933e+00 1.082790e-01 * time: 0.782426118850708 65 7.634022e+00 8.774640e-02 * time: 0.7916619777679443 66 7.632019e+00 7.344738e-02 * time: 0.8038809299468994 67 7.629759e+00 5.504603e-02 * time: 0.8161571025848389 68 7.629059e+00 7.657035e-02 * time: 0.8254680633544922 69 7.628610e+00 7.601238e-02 * time: 0.8349390029907227 70 7.627950e+00 6.829508e-02 * time: 0.8445971012115479 71 7.627402e+00 5.458241e-02 * time: 0.8544449806213379 72 7.627078e+00 4.669210e-02 * time: 0.8642499446868896 73 7.626630e+00 4.010358e-02 * time: 0.8773970603942871 74 7.626158e+00 6.665720e-02 * time: 0.8873980045318604 75 7.625656e+00 4.493045e-02 * time: 0.900360107421875 76 7.625317e+00 4.144523e-02 * time: 0.9102649688720703 77 7.624953e+00 3.902635e-02 * time: 0.9199678897857666 78 7.624549e+00 2.875387e-02 * time: 0.929764986038208 79 7.624247e+00 4.469483e-02 * time: 0.9394590854644775 80 7.624067e+00 5.950494e-02 * time: 0.9490890502929688 81 7.623787e+00 4.068272e-02 * time: 0.9619810581207275 82 7.623463e+00 3.531159e-02 * time: 0.975074052810669 83 7.623174e+00 6.333561e-02 * time: 0.984699010848999 84 7.622899e+00 6.057601e-02 * time: 0.9943931102752686 85 7.622876e+00 9.952892e-02 * time: 1.0040130615234375 86 7.622760e+00 5.706444e-02 * time: 1.0138049125671387 87 7.622095e+00 5.028931e-02 * time: 1.0235490798950195 88 7.621818e+00 4.752382e-02 * time: 1.0822250843048096 89 7.620863e+00 4.527760e-02 * time: 1.0912880897521973 90 7.620535e+00 8.786624e-02 * time: 1.1001861095428467 91 7.619746e+00 5.142631e-02 * time: 1.1117639541625977 92 7.619159e+00 4.623085e-02 * time: 1.1204509735107422 93 7.618688e+00 6.119446e-02 * time: 1.1291890144348145 94 7.618070e+00 5.331830e-02 * time: 1.140815019607544 95 7.617729e+00 6.318177e-02 * time: 1.1496551036834717 96 7.617086e+00 7.101925e-02 * time: 1.1613030433654785 97 7.616505e+00 5.422527e-02 * time: 1.1730530261993408 98 7.616166e+00 6.203030e-02 * time: 1.1817739009857178 99 7.615711e+00 4.276814e-02 * time: 1.1934270858764648 100 7.615343e+00 5.922140e-02 * time: 1.202347993850708 101 7.615040e+00 5.645076e-02 * time: 1.2110869884490967 102 7.614801e+00 4.679424e-02 * time: 1.2198710441589355 103 7.614424e+00 6.272480e-02 * time: 1.22900390625 104 7.614331e+00 6.228596e-02 * time: 1.238158941268921 105 7.614002e+00 4.768302e-02 * time: 1.2511720657348633 106 7.613527e+00 4.600170e-02 * time: 1.2608489990234375 107 7.613518e+00 6.340237e-02 * time: 1.2703170776367188 108 7.613226e+00 4.098458e-02 * time: 1.280163049697876 109 7.613144e+00 6.575054e-02 * time: 1.289659023284912 110 7.613001e+00 7.402516e-02 * time: 1.298935890197754 111 7.612605e+00 2.947488e-02 * time: 1.308255910873413 112 7.612277e+00 4.520349e-02 * time: 1.3178589344024658 113 7.612211e+00 5.992683e-02 * time: 1.3277499675750732 114 7.611955e+00 3.864056e-02 * time: 1.3374249935150146 115 7.611775e+00 4.342243e-02 * time: 1.3472249507904053 116 7.611519e+00 3.591867e-02 * time: 1.3600969314575195 117 7.611217e+00 3.598254e-02 * time: 1.369823932647705 118 7.610815e+00 2.419189e-02 * time: 1.38264799118042 119 7.610613e+00 2.835290e-02 * time: 1.3926301002502441 120 7.610478e+00 3.254908e-02 * time: 1.4054429531097412 121 7.610234e+00 1.491802e-02 * time: 1.4181859493255615 122 7.610122e+00 1.770463e-02 * time: 1.430866003036499 123 7.610034e+00 2.270566e-02 * time: 1.4404819011688232 124 7.609917e+00 1.833947e-02 * time: 1.4535059928894043 125 7.609865e+00 4.327749e-02 * time: 1.4633779525756836 126 7.609669e+00 1.623283e-02 * time: 1.473134994506836 127 7.609610e+00 4.239019e-02 * time: 1.482896089553833 128 7.609396e+00 3.937529e-02 * time: 1.492691993713379 129 7.609164e+00 3.238064e-02 * time: 1.5024070739746094 130 7.608860e+00 2.409427e-02 * time: 1.546694040298462 131 7.608820e+00 4.490076e-02 * time: 1.5554580688476562 132 7.608593e+00 4.870038e-02 * time: 1.56427001953125 133 7.608397e+00 3.562430e-02 * time: 1.5730130672454834 134 7.608189e+00 3.055247e-02 * time: 1.5846021175384521 135 7.608116e+00 5.698404e-02 * time: 1.5933520793914795 136 7.608024e+00 4.765325e-02 * time: 1.6022100448608398 137 7.607750e+00 4.310979e-02 * time: 1.6111729145050049 138 7.607425e+00 2.691758e-02 * time: 1.6228001117706299 139 7.607346e+00 4.195626e-02 * time: 1.631688117980957 140 7.607143e+00 4.325139e-02 * time: 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2.128131e-02 * time: 1.8253540992736816 159 7.604818e+00 1.708001e-02 * time: 1.8380520343780518 160 7.604804e+00 2.025791e-02 * time: 1.847733974456787 161 7.604745e+00 1.430979e-02 * time: 1.8574531078338623 162 7.604735e+00 1.822069e-02 * time: 1.8672130107879639 163 7.604693e+00 1.270004e-02 * time: 1.8799901008605957 164 7.604633e+00 1.523765e-02 * time: 1.8897581100463867 165 7.604627e+00 1.676233e-02 * time: 1.8997249603271484 166 7.604589e+00 1.532376e-02 * time: 1.909562110900879 167 7.604579e+00 1.800365e-02 * time: 1.9193251132965088 168 7.604554e+00 1.213322e-02 * time: 1.9291059970855713 169 7.604523e+00 1.189675e-02 * time: 1.9389209747314453 170 7.604501e+00 1.378389e-02 * time: 1.9489331245422363 171 7.604466e+00 8.967032e-03 * time: 1.9618489742279053 172 7.604447e+00 1.643008e-02 * time: 2.0028281211853027 173 7.604417e+00 9.474755e-03 * time: 2.014698028564453 174 7.604405e+00 1.378552e-02 * time: 2.0235280990600586 175 7.604393e+00 1.322419e-02 * time: 2.032357931137085 176 7.604384e+00 1.681210e-02 * time: 2.0411179065704346 177 7.604361e+00 1.550413e-02 * time: 2.0499320030212402 178 7.604349e+00 1.334120e-02 * time: 2.0586941242218018 179 7.604330e+00 1.580802e-02 * time: 2.0675010681152344 180 7.604291e+00 1.215444e-02 * time: 2.0762939453125 181 7.604270e+00 1.886301e-02 * time: 2.084980010986328 182 7.604248e+00 1.732085e-02 * time: 2.0937130451202393 183 7.604194e+00 1.556568e-02 * time: 2.105380058288574 184 7.604163e+00 1.463800e-02 * time: 2.114206075668335 185 7.604133e+00 1.775350e-02 * time: 2.1258420944213867 186 7.604099e+00 1.001496e-02 * time: 2.1375460624694824 187 7.604066e+00 7.685736e-03 * time: 2.149385929107666 188 7.604047e+00 8.924950e-03 * time: 2.158590078353882 189 7.604029e+00 6.323236e-03 * time: 2.170898914337158 190 7.604024e+00 9.338043e-03 * time: 2.1800179481506348 191 7.604012e+00 1.136305e-02 * time: 2.189276933670044 192 7.603992e+00 1.029057e-02 * time: 2.198496103286743 193 7.603987e+00 1.091054e-02 * time: 2.207705020904541 194 7.603969e+00 1.199465e-02 * time: 2.216926097869873 195 7.603939e+00 1.348590e-02 * time: 2.2261669635772705 196 7.603932e+00 8.823553e-03 * time: 2.2356419563293457 197 7.603895e+00 8.155785e-03 * time: 2.2454020977020264 198 7.603884e+00 2.372553e-02 * time: 2.255002021789551 199 7.603838e+00 1.636150e-02 * time: 2.2647290229797363 200 7.603828e+00 1.162792e-02 * time: 2.2744860649108887 201 7.603823e+00 1.491861e-02 * time: 2.284147024154663 202 7.603798e+00 1.282482e-02 * time: 2.2970879077911377 203 7.603790e+00 1.670479e-02 * time: 2.3068559169769287 204 7.603744e+00 1.519446e-02 * time: 2.319643020629883 205 7.603740e+00 1.964022e-02 * time: 2.329369068145752 206 7.603699e+00 2.262660e-02 * time: 2.339282989501953 207 7.603640e+00 1.417637e-02 * time: 2.35246205329895 208 7.603600e+00 1.605070e-02 * time: 2.362337112426758 209 7.603551e+00 1.112042e-02 * time: 2.3751261234283447 210 7.603540e+00 1.697648e-02 * time: 2.38496994972229 211 7.603507e+00 1.247357e-02 * time: 2.3978030681610107 212 7.603480e+00 1.139893e-02 * time: 2.410754919052124 213 7.603454e+00 1.328567e-02 * time: 2.4205009937286377 214 7.603410e+00 1.564419e-02 * time: 2.4644908905029297 215 7.603395e+00 2.244253e-02 * time: 2.4734909534454346 216 7.603358e+00 1.187735e-02 * time: 2.4852240085601807 217 7.603338e+00 1.192945e-02 * time: 2.4940450191497803 218 7.603304e+00 1.301269e-02 * time: 2.502798080444336 219 7.603294e+00 1.648888e-02 * time: 2.5116209983825684 220 7.603283e+00 1.490564e-02 * time: 2.520282030105591 221 7.603255e+00 1.490893e-02 * time: 2.5318639278411865 222 7.603232e+00 9.751160e-03 * time: 2.5433859825134277 223 7.603219e+00 1.170834e-02 * time: 2.5521600246429443 224 7.603192e+00 1.277754e-02 * time: 2.563755989074707 225 7.603177e+00 1.174417e-02 * time: 2.5725018978118896 226 7.603168e+00 1.407575e-02 * time: 2.5813310146331787 227 7.603158e+00 9.410011e-03 * time: 2.5901060104370117 228 7.603148e+00 1.438652e-02 * time: 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