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 3.067763e+01 8.144237e+00 * time: 0.001901865005493164 1 2.287284e+01 4.031038e+00 * time: 0.00860285758972168 2 1.603796e+01 3.810949e+00 * time: 0.014346837997436523 3 1.176708e+01 2.048021e+00 * time: 0.018249988555908203 4 9.845149e+00 1.509777e+00 * time: 0.02226400375366211 5 9.298430e+00 1.586762e+00 * time: 0.025421857833862305 6 8.951632e+00 7.527145e-01 * time: 0.028947830200195312 7 8.728261e+00 6.490648e-01 * time: 0.03245878219604492 8 8.523520e+00 7.172486e-01 * time: 0.03638887405395508 9 8.325071e+00 4.616136e-01 * time: 0.040390968322753906 10 8.223614e+00 4.459931e-01 * time: 0.044174909591674805 11 8.140842e+00 3.460492e-01 * time: 0.08905196189880371 12 8.117604e+00 6.704982e-01 * time: 0.09198999404907227 13 8.078690e+00 6.168289e-01 * time: 0.09460997581481934 14 8.048015e+00 5.007913e-01 * time: 0.09722781181335449 15 8.025201e+00 6.291057e-01 * time: 0.09972977638244629 16 7.964722e+00 4.012559e-01 * time: 0.102935791015625 17 7.946485e+00 7.036019e-01 * time: 0.10555100440979004 18 7.896823e+00 5.000649e-01 * time: 0.10815691947937012 19 7.857545e+00 5.312369e-01 * time: 0.11090993881225586 20 7.819183e+00 3.722608e-01 * time: 0.11350893974304199 21 7.815436e+00 5.901599e-01 * time: 0.11594080924987793 22 7.784831e+00 4.471065e-01 * time: 0.1190798282623291 23 7.753742e+00 3.391313e-01 * time: 0.12236499786376953 24 7.729238e+00 3.185656e-01 * time: 0.12483787536621094 25 7.723656e+00 5.285786e-01 * time: 0.12730884552001953 26 7.706413e+00 2.849959e-01 * time: 0.13060688972473145 27 7.696576e+00 3.588888e-01 * time: 0.13454699516296387 28 7.695533e+00 3.656795e-01 * time: 0.13738584518432617 29 7.687370e+00 2.819450e-01 * time: 0.14051485061645508 30 7.678839e+00 2.867153e-01 * time: 0.14397597312927246 31 7.673153e+00 2.894578e-01 * time: 0.14703679084777832 32 7.671417e+00 2.561779e-01 * time: 0.1507108211517334 33 7.663997e+00 1.967374e-01 * time: 0.15460777282714844 34 7.659219e+00 2.753159e-01 * time: 0.15850281715393066 35 7.657626e+00 2.810287e-01 * time: 0.1622908115386963 36 7.649913e+00 2.093605e-01 * time: 0.16711091995239258 37 7.646358e+00 1.872233e-01 * time: 0.17059087753295898 38 7.644971e+00 2.535843e-01 * time: 0.17325186729431152 39 7.639046e+00 1.356052e-01 * time: 0.17644691467285156 40 7.637764e+00 1.694389e-01 * time: 0.17927289009094238 41 7.636912e+00 1.713900e-01 * time: 0.1828138828277588 42 7.634924e+00 1.362879e-01 * time: 0.18677878379821777 43 7.632691e+00 6.653321e-02 * time: 0.19138288497924805 44 7.631874e+00 1.324745e-01 * time: 0.19516587257385254 45 7.631862e+00 1.321908e-01 * time: 0.19875097274780273 46 7.630641e+00 7.483434e-02 * time: 0.20266985893249512 47 7.629195e+00 6.948935e-02 * time: 0.2300269603729248 48 7.628776e+00 1.298703e-01 * time: 0.23297977447509766 49 7.627560e+00 8.030285e-02 * time: 0.23631691932678223 50 7.627159e+00 7.912940e-02 * time: 0.23898983001708984 51 7.627149e+00 8.516281e-02 * time: 0.24165797233581543 52 7.626326e+00 5.652772e-02 * time: 0.2449169158935547 53 7.625784e+00 7.533065e-02 * time: 0.24759793281555176 54 7.625593e+00 7.264850e-02 * time: 0.25032496452331543 55 7.625061e+00 9.728055e-02 * time: 0.25295495986938477 56 7.624381e+00 5.604104e-02 * time: 0.25612688064575195 57 7.624179e+00 6.381166e-02 * time: 0.2585029602050781 58 7.623875e+00 6.513670e-02 * time: 0.2608809471130371 59 7.623455e+00 5.759971e-02 * time: 0.26357388496398926 60 7.623016e+00 5.202802e-02 * time: 0.2660038471221924 61 7.622655e+00 5.194102e-02 * time: 0.2690269947052002 62 7.622159e+00 2.598533e-02 * time: 0.27213382720947266 63 7.621800e+00 3.634237e-02 * time: 0.2746858596801758 64 7.621432e+00 3.576771e-02 * time: 0.277756929397583 65 7.621412e+00 7.768333e-02 * time: 0.28020691871643066 66 7.621029e+00 5.179030e-02 * time: 0.2833437919616699 67 7.620756e+00 4.827403e-02 * time: 0.28603386878967285 68 7.620316e+00 5.649359e-02 * time: 0.2888457775115967 69 7.620153e+00 6.224495e-02 * time: 0.2922549247741699 70 7.619887e+00 5.549768e-02 * time: 0.2952549457550049 71 7.619399e+00 4.022220e-02 * time: 0.297717809677124 72 7.618829e+00 3.201794e-02 * time: 0.3001408576965332 73 7.618355e+00 3.894010e-02 * time: 0.30257081985473633 74 7.618283e+00 7.700909e-02 * time: 0.3054847717285156 75 7.617987e+00 5.298024e-02 * time: 0.3080928325653076 76 7.617617e+00 3.367163e-02 * time: 0.31144094467163086 77 7.617232e+00 3.865301e-02 * time: 0.31439900398254395 78 7.616895e+00 4.133729e-02 * time: 0.3172109127044678 79 7.616612e+00 4.986174e-02 * time: 0.3195679187774658 80 7.616185e+00 3.454561e-02 * time: 0.3226017951965332 81 7.615866e+00 3.516066e-02 * time: 0.3259298801422119 82 7.615545e+00 4.958953e-02 * time: 0.32828187942504883 83 7.615268e+00 4.656365e-02 * time: 0.34993886947631836 84 7.614801e+00 4.113635e-02 * time: 0.3532547950744629 85 7.614400e+00 3.849660e-02 * time: 0.3565859794616699 86 7.614014e+00 3.704992e-02 * time: 0.35906481742858887 87 7.613803e+00 2.896344e-02 * time: 0.36218690872192383 88 7.613575e+00 4.191638e-02 * time: 0.3645648956298828 89 7.613491e+00 4.177907e-02 * time: 0.3671407699584961 90 7.613347e+00 3.362270e-02 * time: 0.36943697929382324 91 7.613033e+00 4.187288e-02 * time: 0.37195396423339844 92 7.612599e+00 2.552502e-02 * time: 0.3743247985839844 93 7.612522e+00 4.448100e-02 * time: 0.3768479824066162 94 7.612387e+00 5.459840e-02 * time: 0.37909388542175293 95 7.612135e+00 3.287490e-02 * time: 0.3814258575439453 96 7.612087e+00 5.973921e-02 * time: 0.3838319778442383 97 7.611761e+00 4.355610e-02 * time: 0.3863668441772461 98 7.611580e+00 5.974710e-02 * time: 0.3890349864959717 99 7.611465e+00 6.540600e-02 * time: 0.3918299674987793 100 7.611429e+00 6.440547e-02 * time: 0.3944997787475586 101 7.611285e+00 4.426346e-02 * time: 0.3970217704772949 102 7.610642e+00 4.521453e-02 * time: 0.3992428779602051 103 7.610025e+00 4.111165e-02 * time: 0.40167880058288574 104 7.609973e+00 7.365035e-02 * time: 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