Gross-Pitaevskii equation with magnetism

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

In [1]:
using DFTK
using StaticArrays
using Plots

Unit cell. Having one of the lattice vectors as zero means a 2D system

In [2]:
a = 15
lattice = a .* [[1 0 0.]; [0 1 0]; [0 0 0]];

Confining scalar potential, and magnetic vector potential

In [3]:
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

In [4]:
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

In [5]:
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.real[:, :, 1], c=:blues)
Iter     Function value   Gradient norm 
     0     3.077016e+01     7.099970e+00
 * time: 0.0017120838165283203
     1     2.776218e+01     4.512457e+00
 * time: 0.0064890384674072266
     2     1.935778e+01     4.836689e+00
 * time: 0.01180410385131836
     3     1.343812e+01     3.448865e+00
 * time: 0.017058134078979492
     4     1.157216e+01     3.099052e+00
 * time: 0.02149510383605957
     5     1.066942e+01     2.204533e+00
 * time: 0.05390596389770508
     6     9.501471e+00     1.473333e+00
 * time: 0.05660605430603027
     7     8.777708e+00     1.091840e+00
 * time: 0.059127092361450195
     8     8.440348e+00     7.715083e-01
 * time: 0.06161093711853027
     9     8.403502e+00     1.330032e+00
 * time: 0.06386399269104004
    10     8.303916e+00     1.135657e+00
 * time: 0.06653308868408203
    11     8.271293e+00     1.160964e+00
 * time: 0.06931710243225098
    12     8.137761e+00     6.803516e-01
 * time: 0.07297110557556152
    13     8.048212e+00     6.851147e-01
 * time: 0.07725000381469727
    14     8.004141e+00     7.643399e-01
 * time: 0.08101511001586914
    15     7.962402e+00     4.475359e-01
 * time: 0.0842750072479248
    16     7.929738e+00     5.183875e-01
 * time: 0.08750295639038086
    17     7.902781e+00     3.430982e-01
 * time: 0.09162712097167969
    18     7.871907e+00     2.653334e-01
 * time: 0.09636998176574707
    19     7.846709e+00     4.816162e-01
 * time: 0.0997779369354248
    20     7.821301e+00     3.078363e-01
 * time: 0.10334396362304688
    21     7.797091e+00     4.357181e-01
 * time: 0.10680794715881348
    22     7.788851e+00     4.507833e-01
 * time: 0.11038804054260254
    23     7.767902e+00     2.896440e-01
 * time: 0.11412310600280762
    24     7.748439e+00     4.247666e-01
 * time: 0.11757612228393555
    25     7.744732e+00     4.381962e-01
 * time: 0.1211240291595459
    26     7.734628e+00     3.889163e-01
 * time: 0.12470698356628418
    27     7.711182e+00     2.092718e-01
 * time: 0.12864112854003906
    28     7.699710e+00     3.116059e-01
 * time: 0.13201093673706055
    29     7.686328e+00     2.396813e-01
 * time: 0.13657903671264648
    30     7.675046e+00     2.855162e-01
 * time: 0.14107799530029297
    31     7.663566e+00     1.737080e-01
 * time: 0.1461191177368164
    32     7.655602e+00     2.191812e-01
 * time: 0.14963793754577637
    33     7.649349e+00     1.262314e-01
 * time: 0.1541740894317627
    34     7.647846e+00     1.985115e-01
 * time: 0.15772700309753418
    35     7.645812e+00     1.488947e-01
 * time: 0.176285982131958
    36     7.643513e+00     1.658047e-01
 * time: 0.17835712432861328
    37     7.641865e+00     1.402109e-01
 * time: 0.1803750991821289
    38     7.638447e+00     1.417783e-01
 * time: 0.1824660301208496
    39     7.637949e+00     1.295858e-01
 * time: 0.18458199501037598
    40     7.635803e+00     7.828126e-02
 * time: 0.18709206581115723
    41     7.634192e+00     1.751718e-01
 * time: 0.18947505950927734
    42     7.633979e+00     1.848441e-01
 * time: 0.19156098365783691
    43     7.633042e+00     1.365978e-01
 * time: 0.193648099899292
    44     7.631218e+00     1.103198e-01
 * time: 0.19582104682922363
    45     7.628926e+00     6.951779e-02
 * time: 0.19847702980041504
    46     7.628550e+00     1.140155e-01
 * time: 0.2005629539489746
    47     7.628082e+00     1.216560e-01
 * time: 0.20261907577514648
    48     7.626822e+00     8.612412e-02
 * time: 0.20560693740844727
    49     7.625805e+00     9.981885e-02
 * time: 0.20760512351989746
    50     7.624409e+00     4.687745e-02
 * time: 0.21011710166931152
    51     7.623660e+00     7.903797e-02
 * time: 0.21327805519104004
    52     7.622912e+00     5.746376e-02
 * time: 0.2176680564880371
    53     7.621963e+00     4.712308e-02
 * time: 0.22241592407226562
    54     7.621056e+00     2.792622e-02
 * time: 0.22679710388183594
    55     7.620473e+00     4.247146e-02
 * time: 0.23029112815856934
    56     7.620007e+00     4.527969e-02
 * time: 0.23382210731506348
    57     7.619345e+00     4.876799e-02
 * time: 0.23890399932861328
    58     7.618601e+00     3.379615e-02
 * time: 0.24353694915771484
    59     7.617984e+00     3.279501e-02
 * time: 0.24718308448791504
    60     7.617432e+00     3.803242e-02
 * time: 0.2507619857788086
    61     7.616830e+00     3.485162e-02
 * time: 0.2542901039123535
    62     7.616441e+00     3.949164e-02
 * time: 0.25768303871154785
    63     7.615949e+00     5.935234e-02
 * time: 0.2600820064544678
    64     7.615870e+00     6.170036e-02
 * time: 0.2635040283203125
    65     7.615670e+00     5.540708e-02
 * time: 0.2667360305786133
    66     7.615193e+00     4.567557e-02
 * time: 0.27041196823120117
    67     7.614811e+00     4.141398e-02
 * time: 0.2846550941467285
    68     7.614615e+00     3.737054e-02
 * time: 0.2867860794067383
    69     7.614409e+00     4.051927e-02
 * time: 0.2888491153717041
    70     7.614267e+00     4.847423e-02
 * time: 0.29106807708740234
    71     7.614074e+00     3.389768e-02
 * time: 0.29305100440979004
    72     7.613861e+00     4.345371e-02
 * time: 0.2950451374053955
    73     7.613659e+00     3.658832e-02
 * time: 0.29713892936706543
    74     7.613401e+00     3.278005e-02
 * time: 0.29962992668151855
    75     7.613152e+00     2.286754e-02
 * time: 0.30224013328552246
    76     7.612922e+00     2.451509e-02
 * time: 0.3047349452972412
    77     7.612815e+00     3.533614e-02
 * time: 0.3068680763244629
    78     7.612645e+00     4.091742e-02
 * time: 0.30910801887512207
    79     7.612290e+00     2.756706e-02
 * time: 0.3118460178375244
    80     7.612222e+00     4.471561e-02
 * time: 0.31412696838378906
    81     7.611944e+00     3.815673e-02
 * time: 0.31670212745666504
    82     7.611844e+00     5.912756e-02
 * time: 0.3186929225921631
    83     7.611519e+00     5.506328e-02
 * time: 0.32071995735168457
    84     7.611479e+00     5.425979e-02
 * time: 0.32273411750793457
    85     7.611125e+00     4.706506e-02
 * time: 0.32487010955810547
    86     7.610800e+00     3.669273e-02
 * time: 0.3270409107208252
    87     7.610594e+00     5.225045e-02
 * time: 0.3293769359588623
    88     7.610233e+00     2.602166e-02
 * time: 0.3322739601135254
    89     7.610010e+00     3.228904e-02
 * time: 0.33500194549560547
    90     7.609850e+00     3.296205e-02
 * time: 0.33707404136657715
    91     7.609641e+00     2.189554e-02
 * time: 0.3397400379180908
    92     7.609456e+00     2.068626e-02
 * time: 0.34183192253112793
    93     7.609412e+00     3.659626e-02
 * time: 0.34400510787963867
    94     7.609405e+00     4.413358e-02
 * time: 0.3465549945831299
    95     7.609303e+00     4.463400e-02
 * time: 0.3487880229949951
    96     7.609146e+00     2.616280e-02
 * time: 0.3507699966430664
    97     7.608947e+00     3.418183e-02
 * time: 0.3527860641479492
    98     7.608737e+00     1.961115e-02
 * time: 0.3562641143798828
    99     7.608591e+00     2.283681e-02
 * time: 0.36591100692749023
   100     7.608438e+00     1.791892e-02
 * time: 0.3681190013885498
   101     7.608436e+00     3.356030e-02
 * time: 0.3700859546661377
   102     7.608348e+00     2.633680e-02
 * time: 0.37213706970214844
   103     7.608270e+00     1.926483e-02
 * time: 0.3746640682220459
   104     7.608202e+00     3.104673e-02
 * time: 0.37706995010375977
   105     7.608122e+00     2.446319e-02
 * time: 0.3798251152038574
   106     7.608011e+00     1.843032e-02
 * time: 0.3823869228363037
   107     7.607988e+00     2.051295e-02
 * time: 0.38446712493896484
   108     7.607907e+00     1.743878e-02
 * time: 0.38643693923950195
   109     7.607831e+00     1.615467e-02
 * time: 0.38853001594543457
   110     7.607745e+00     1.930346e-02
 * time: 0.3907029628753662
   111     7.607686e+00     2.055955e-02
 * time: 0.39313411712646484
   112     7.607671e+00     3.002613e-02
 * time: 0.39538097381591797
   113     7.607616e+00     1.660431e-02
 * time: 0.39817214012145996
   114     7.607552e+00     2.128334e-02
 * time: 0.40025806427001953
   115     7.607517e+00     3.261214e-02
 * time: 0.40230393409729004
   116     7.607402e+00     2.078757e-02
 * time: 0.4043569564819336
   117     7.607360e+00     3.935413e-02
 * time: 0.4063839912414551
   118     7.607244e+00     3.024583e-02
 * time: 0.4095640182495117
   119     7.607225e+00     3.510511e-02
 * time: 0.4121730327606201
   120     7.607151e+00     2.509252e-02
 * time: 0.4148139953613281
   121     7.607045e+00     3.170160e-02
 * time: 0.4173390865325928
   122     7.606970e+00     2.794080e-02
 * time: 0.4197361469268799
   123     7.606821e+00     1.501891e-02
 * time: 0.4225120544433594
   124     7.606755e+00     2.582995e-02
 * time: 0.42508411407470703
   125     7.606684e+00     2.400049e-02
 * time: 0.4277489185333252
   126     7.606675e+00     3.718530e-02
 * time: 0.42980408668518066
   127     7.606649e+00     4.930170e-02
 * time: 0.4318821430206299
   128     7.606530e+00     2.751900e-02
 * time: 0.4346311092376709
   129     7.606441e+00     2.970860e-02
 * time: 0.4366309642791748
   130     7.606345e+00     3.258805e-02
 * time: 0.43886804580688477
   131     7.606250e+00     1.762523e-02
 * time: 0.4495360851287842
   132     7.606151e+00     2.278745e-02
 * time: 0.45201802253723145
   133     7.606064e+00     2.015622e-02
 * time: 0.45459699630737305
   134     7.605979e+00     1.867804e-02
 * time: 0.45714902877807617
   135     7.605948e+00     2.860041e-02
 * time: 0.45933008193969727
   136     7.605940e+00     2.437282e-02
 * time: 0.4613020420074463
   137     7.605872e+00     2.039272e-02
 * time: 0.46381402015686035
   138     7.605825e+00     2.123576e-02
 * time: 0.46578001976013184
   139     7.605743e+00     2.260424e-02
 * time: 0.4677579402923584
   140     7.605690e+00     3.143599e-02
 * time: 0.4700291156768799
   141     7.605598e+00     2.632557e-02
 * time: 0.47216796875
   142     7.605590e+00     2.514000e-02
 * time: 0.47412705421447754
   143     7.605487e+00     1.748873e-02
 * time: 0.4761049747467041
   144     7.605442e+00     2.370791e-02
 * time: 0.47817301750183105
   145     7.605382e+00     2.691387e-02
 * time: 0.48021912574768066
   146     7.605341e+00     3.098558e-02
 * time: 0.48218202590942383
   147     7.605276e+00     2.237810e-02
 * time: 0.48415112495422363
   148     7.605253e+00     3.362893e-02
 * time: 0.48645806312561035
   149     7.605224e+00     3.060594e-02
 * time: 0.4885129928588867
   150     7.605159e+00     2.065927e-02
 * time: 0.4910399913787842
   151     7.605049e+00     2.260923e-02
 * time: 0.49353814125061035
   152     7.604983e+00     2.513777e-02
 * time: 0.4956369400024414
   153     7.604896e+00     1.567406e-02
 * time: 0.49812912940979004
   154     7.604817e+00     2.453672e-02
 * time: 0.5001239776611328
   155     7.604817e+00     2.821076e-02
 * time: 0.5024199485778809
   156     7.604726e+00     2.414880e-02
 * time: 0.504418134689331
   157     7.604664e+00     2.765053e-02
 * time: 0.5063910484313965
   158     7.604546e+00     2.351364e-02
 * time: 0.5083940029144287
   159     7.604431e+00     2.820630e-02
 * time: 0.5108821392059326
   160     7.604306e+00     1.975110e-02
 * time: 0.5133979320526123
   161     7.604277e+00     2.645090e-02
 * time: 0.515388011932373
   162     7.604220e+00     2.124467e-02
 * time: 0.5182621479034424
   163     7.604175e+00     2.928145e-02
 * time: 0.5276060104370117
   164     7.604103e+00     1.935455e-02
 * time: 0.5297539234161377
   165     7.604036e+00     2.533088e-02
 * time: 0.5320439338684082
   166     7.604007e+00     2.913689e-02
 * time: 0.5343999862670898
   167     7.603907e+00     2.529067e-02
 * time: 0.536628007888794
   168     7.603797e+00     3.070156e-02
 * time: 0.5388450622558594
   169     7.603688e+00     4.111796e-02
 * time: 0.5411150455474854
   170     7.603658e+00     2.461042e-02
 * time: 0.5431830883026123
   171     7.603571e+00     3.763564e-02
 * time: 0.5453081130981445
   172     7.603514e+00     3.744338e-02
 * time: 0.5476419925689697
   173     7.603379e+00     2.850032e-02
 * time: 0.5506680011749268
   174     7.603278e+00     4.535200e-02
 * time: 0.5528790950775146
   175     7.603258e+00     4.605009e-02
 * time: 0.5550830364227295
   176     7.603199e+00     4.066571e-02
 * time: 0.5573430061340332
   177     7.603188e+00     4.199243e-02
 * time: 0.5593490600585938
   178     7.603044e+00     2.490256e-02
 * time: 0.5614480972290039
   179     7.602863e+00     3.425613e-02
 * time: 0.5636529922485352
   180     7.602672e+00     3.473836e-02
 * time: 0.5664839744567871
   181     7.602461e+00     2.091504e-02
 * time: 0.5693330764770508
   182     7.602338e+00     2.883649e-02
 * time: 0.5714800357818604
   183     7.602217e+00     1.835320e-02
 * time: 0.5743329524993896
   184     7.602159e+00     2.440067e-02
 * time: 0.576470136642456
   185     7.602111e+00     2.255788e-02
 * time: 0.5787930488586426
   186     7.602021e+00     2.013884e-02
 * time: 0.5816950798034668
   187     7.602002e+00     3.102767e-02
 * time: 0.5838420391082764
   188     7.601851e+00     2.595175e-02
 * time: 0.5867550373077393
   189     7.601720e+00     3.475127e-02
 * time: 0.5888419151306152
   190     7.601572e+00     3.361344e-02
 * time: 0.5908079147338867
   191     7.601409e+00     3.703097e-02
 * time: 0.5928809642791748
   192     7.601239e+00     3.116048e-02
 * time: 0.5958659648895264
   193     7.600998e+00     2.408272e-02
 * time: 0.5978460311889648
   194     7.600887e+00     3.952529e-02
 * time: 0.5998389720916748
   195     7.600661e+00     3.594162e-02
 * time: 0.6093449592590332
   196     7.600644e+00     5.121757e-02
 * time: 0.6117351055145264
   197     7.600526e+00     4.463163e-02
 * time: 0.6136970520019531
   198     7.600306e+00     5.538301e-02
 * time: 0.6162929534912109
   199     7.600089e+00     2.591094e-02
 * time: 0.6187429428100586
   200     7.599860e+00     5.801635e-02
 * time: 0.6207139492034912
   201     7.599602e+00     3.142447e-02
 * time: 0.6232099533081055
   202     7.599500e+00     3.666434e-02
 * time: 0.6251609325408936
   203     7.599313e+00     4.559024e-02
 * time: 0.6275241374969482
   204     7.599160e+00     4.020974e-02
 * time: 0.6294801235198975
   205     7.599032e+00     3.481220e-02
 * time: 0.6315009593963623
   206     7.599006e+00     3.146668e-02
 * time: 0.6334850788116455
   207     7.598930e+00     4.932272e-02
 * time: 0.6354529857635498
   208     7.598568e+00     4.059488e-02
 * time: 0.6373910903930664
   209     7.598431e+00     4.825930e-02
 * time: 0.6393489837646484
   210     7.598268e+00     4.749881e-02
 * time: 0.6414480209350586
   211     7.598073e+00     3.151981e-02
 * time: 0.6436889171600342
   212     7.597925e+00     3.625023e-02
 * time: 0.6457610130310059
   213     7.597797e+00     3.670013e-02
 * time: 0.6477789878845215
   214     7.597642e+00     3.577371e-02
 * time: 0.650346040725708
   215     7.597465e+00     2.485993e-02
 * time: 0.6529121398925781
   216     7.597389e+00     4.932192e-02
 * time: 0.6548850536346436
   217     7.597136e+00     3.155083e-02
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 * time: 1.8098890781402588
   681     7.592302e+00     7.443596e-04
 * time: 1.8119900226593018
   682     7.592302e+00     5.703205e-04
 * time: 1.814932107925415
   683     7.592302e+00     5.029727e-04
 * time: 1.817702054977417
   684     7.592302e+00     4.872329e-04
 * time: 1.8203880786895752
   685     7.592302e+00     6.226084e-04
 * time: 1.822883129119873
   686     7.592302e+00     3.679823e-04
 * time: 1.8253569602966309
   687     7.592302e+00     4.974733e-04
 * time: 1.8273289203643799
   688     7.592302e+00     4.966910e-04
 * time: 1.8295071125030518
   689     7.592302e+00     1.873319e-04
 * time: 1.8324079513549805
   690     7.592302e+00     4.438309e-04
 * time: 1.8343889713287354
   691     7.592302e+00     7.196288e-04
 * time: 1.8363730907440186
   692     7.592302e+00     3.588205e-04
 * time: 1.8383569717407227
   693     7.592302e+00     4.009830e-04
 * time: 1.8403539657592773
   694     7.592302e+00     2.760782e-04
 * time: 1.8423819541931152
Out[5]: