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     3.107540e+01     7.365105e+00
 * time: 0.0021550655364990234
     1     2.832922e+01     5.420036e+00
 * time: 0.009031057357788086
     2     1.902921e+01     5.182533e+00
 * time: 0.01669907569885254
     3     1.329596e+01     3.093076e+00
 * time: 0.02395796775817871
     4     1.080072e+01     2.132696e+00
 * time: 0.030076980590820312
     5     9.879542e+00     1.809222e+00
 * time: 0.03628897666931152
     6     9.051179e+00     1.421736e+00
 * time: 0.042201995849609375
     7     8.604693e+00     8.950588e-01
 * time: 0.048182010650634766
     8     8.389001e+00     5.294171e-01
 * time: 0.08923506736755371
     9     8.364583e+00     1.076011e+00
 * time: 0.09366798400878906
    10     8.293513e+00     7.205142e-01
 * time: 0.09738993644714355
    11     8.214306e+00     8.622818e-01
 * time: 0.10106301307678223
    12     8.168761e+00     9.687815e-01
 * time: 0.10471796989440918
    13     8.088904e+00     5.164650e-01
 * time: 0.10929703712463379
    14     8.056455e+00     9.690267e-01
 * time: 0.11301803588867188
    15     7.994777e+00     5.519575e-01
 * time: 0.11673593521118164
    16     7.951579e+00     5.587544e-01
 * time: 0.12045598030090332
    17     7.904620e+00     4.123606e-01
 * time: 0.12523412704467773
    18     7.893110e+00     6.273800e-01
 * time: 0.12903308868408203
    19     7.867269e+00     4.793957e-01
 * time: 0.13280510902404785
    20     7.867052e+00     6.520688e-01
 * time: 0.13739299774169922
    21     7.837965e+00     5.877329e-01
 * time: 0.14211297035217285
    22     7.807592e+00     5.062080e-01
 * time: 0.146867036819458
    23     7.806646e+00     6.478683e-01
 * time: 0.15161895751953125
    24     7.785684e+00     3.949976e-01
 * time: 0.1568460464477539
    25     7.750695e+00     3.127228e-01
 * time: 0.1631031036376953
    26     7.732096e+00     3.091383e-01
 * time: 0.16790413856506348
    27     7.723118e+00     3.431556e-01
 * time: 0.17270493507385254
    28     7.708838e+00     2.760018e-01
 * time: 0.17733311653137207
    29     7.695070e+00     1.855078e-01
 * time: 0.1833789348602295
    30     7.685093e+00     2.597307e-01
 * time: 0.18949103355407715
    31     7.671729e+00     1.415293e-01
 * time: 0.19579100608825684
    32     7.664509e+00     1.920841e-01
 * time: 0.20066094398498535
    33     7.658090e+00     1.528693e-01
 * time: 0.2056281566619873
    34     7.654741e+00     2.871110e-01
 * time: 0.21064209938049316
    35     7.650185e+00     1.444638e-01
 * time: 0.21690893173217773
    36     7.645376e+00     1.534617e-01
 * time: 0.22182011604309082
    37     7.640667e+00     1.324876e-01
 * time: 0.22824811935424805
    38     7.640119e+00     2.202956e-01
 * time: 0.2335340976715088
    39     7.638394e+00     1.392386e-01
 * time: 0.23862409591674805
    40     7.635471e+00     9.235842e-02
 * time: 0.2437291145324707
    41     7.634379e+00     1.607068e-01
 * time: 0.2680079936981201
    42     7.633266e+00     9.096718e-02
 * time: 0.27326011657714844
    43     7.632415e+00     1.382450e-01
 * time: 0.27713704109191895
    44     7.632346e+00     9.138581e-02
 * time: 0.28096508979797363
    45     7.631321e+00     6.920846e-02
 * time: 0.2847881317138672
    46     7.630623e+00     6.030462e-02
 * time: 0.288560152053833
    47     7.630181e+00     8.328194e-02
 * time: 0.2924530506134033
    48     7.629569e+00     9.800065e-02
 * time: 0.2963290214538574
    49     7.629194e+00     1.084551e-01
 * time: 0.3001229763031006
    50     7.628606e+00     6.971536e-02
 * time: 0.30401110649108887
    51     7.627748e+00     5.553708e-02
 * time: 0.3079080581665039
    52     7.627130e+00     8.007750e-02
 * time: 0.31185293197631836
    53     7.626466e+00     7.425145e-02
 * time: 0.3157320022583008
    54     7.625629e+00     6.297320e-02
 * time: 0.320544958114624
    55     7.624765e+00     4.275352e-02
 * time: 0.3254239559173584
    56     7.624398e+00     2.950372e-02
 * time: 0.3303050994873047
    57     7.624327e+00     6.777333e-02
 * time: 0.33417296409606934
    58     7.624028e+00     4.073272e-02
 * time: 0.3391270637512207
    59     7.623791e+00     3.701238e-02
 * time: 0.34400010108947754
    60     7.623666e+00     4.947461e-02
 * time: 0.34804606437683105
    61     7.623403e+00     3.964623e-02
 * time: 0.3530261516571045
    62     7.623111e+00     3.409254e-02
 * time: 0.3580019474029541
    63     7.622581e+00     3.822869e-02
 * time: 0.363116979598999
    64     7.622312e+00     4.046928e-02
 * time: 0.3681600093841553
    65     7.621909e+00     3.654855e-02
 * time: 0.37558794021606445
    66     7.621651e+00     7.307858e-02
 * time: 0.3830690383911133
    67     7.621304e+00     4.705988e-02
 * time: 0.38901615142822266
    68     7.620843e+00     3.643138e-02
 * time: 0.3963589668273926
    69     7.620564e+00     3.652456e-02
 * time: 0.40164899826049805
    70     7.620169e+00     5.302319e-02
 * time: 0.40696001052856445
    71     7.619755e+00     3.074383e-02
 * time: 0.41776609420776367
    72     7.619422e+00     2.531400e-02
 * time: 0.42750096321105957
    73     7.619196e+00     4.656699e-02
 * time: 0.43547701835632324
    74     7.618881e+00     4.457056e-02
 * time: 0.45738792419433594
    75     7.618596e+00     2.452552e-02
 * time: 0.46230411529541016
    76     7.618365e+00     3.922194e-02
 * time: 0.46607208251953125
    77     7.618086e+00     3.677140e-02
 * time: 0.470750093460083
    78     7.617767e+00     1.942138e-02
 * time: 0.4754490852355957
    79     7.617476e+00     3.247285e-02
 * time: 0.47919607162475586
    80     7.617093e+00     3.683518e-02
 * time: 0.48301100730895996
    81     7.616904e+00     2.881243e-02
 * time: 0.4867980480194092
    82     7.616666e+00     5.392086e-02
 * time: 0.49065303802490234
    83     7.616622e+00     6.427927e-02
 * time: 0.49456214904785156
    84     7.616195e+00     5.461086e-02
 * time: 0.4985051155090332
    85     7.615860e+00     3.011929e-02
 * time: 0.5033149719238281
    86     7.615354e+00     3.450115e-02
 * time: 0.5070829391479492
    87     7.614965e+00     4.274482e-02
 * time: 0.5108170509338379
    88     7.614690e+00     3.509695e-02
 * time: 0.515408992767334
    89     7.614582e+00     4.319979e-02
 * time: 0.5190141201019287
    90     7.614376e+00     5.006352e-02
 * time: 0.522589921951294
    91     7.614199e+00     4.972341e-02
 * time: 0.5262179374694824
    92     7.613961e+00     4.330566e-02
 * time: 0.5299079418182373
    93     7.613625e+00     4.505952e-02
 * time: 0.5334799289703369
    94     7.613238e+00     2.210281e-02
 * time: 0.539283037185669
    95     7.612850e+00     3.142382e-02
 * time: 0.5440859794616699
    96     7.612635e+00     3.082189e-02
 * time: 0.5488591194152832
    97     7.612276e+00     3.434491e-02
 * time: 0.5536329746246338
    98     7.612163e+00     4.547728e-02
 * time: 0.5584149360656738
    99     7.611882e+00     5.353603e-02
 * time: 0.5637280941009521
   100     7.611550e+00     3.385543e-02
 * time: 0.5717649459838867
   101     7.611448e+00     7.029064e-02
 * time: 0.5766451358795166
   102     7.611138e+00     4.917193e-02
 * time: 0.5829081535339355
   103     7.610564e+00     4.189295e-02
 * time: 0.5877931118011475
   104     7.610218e+00     5.975986e-02
 * time: 0.5947859287261963
   105     7.610062e+00     6.281731e-02
 * time: 0.6015720367431641
   106     7.609760e+00     5.439543e-02
 * time: 0.6066200733184814
   107     7.609345e+00     4.293755e-02
 * time: 0.6232011318206787
   108     7.609094e+00     7.381092e-02
 * time: 0.6268019676208496
   109     7.609000e+00     5.353021e-02
 * time: 0.6304259300231934
   110     7.608551e+00     4.944383e-02
 * time: 0.6339981555938721
   111     7.608366e+00     6.213509e-02
 * time: 0.637584924697876
   112     7.608130e+00     7.406605e-02
 * time: 0.641240119934082
   113     7.607792e+00     6.286223e-02
 * time: 0.6448979377746582
   114     7.607642e+00     6.913457e-02
 * time: 0.6485390663146973
   115     7.607390e+00     5.843240e-02
 * time: 0.6521470546722412
   116     7.607250e+00     6.623134e-02
 * time: 0.6560299396514893
   117     7.607244e+00     6.393042e-02
 * time: 0.6596510410308838
   118     7.606937e+00     5.876278e-02
 * time: 0.6632580757141113
   119     7.606664e+00     7.686747e-02
 * time: 0.6669330596923828
   120     7.606178e+00     4.329273e-02
 * time: 0.6714961528778076
   121     7.605789e+00     7.596819e-02
 * time: 0.6751980781555176
   122     7.605222e+00     4.704739e-02
 * time: 0.679840087890625
   123     7.604858e+00     3.698921e-02
 * time: 0.6834940910339355
   124     7.604528e+00     3.731729e-02
 * time: 0.6881649494171143
   125     7.604137e+00     2.392877e-02
 * time: 0.6927950382232666
   126     7.603930e+00     1.912216e-02
 * time: 0.697728157043457
   127     7.603839e+00     4.280346e-02
 * time: 0.702355146408081
   128     7.603554e+00     3.947048e-02
 * time: 0.7097029685974121
   129     7.603331e+00     2.725760e-02
 * time: 0.7170369625091553
   130     7.603120e+00     3.816653e-02
 * time: 0.7218899726867676
   131     7.603035e+00     4.933662e-02
 * time: 0.7266550064086914
   132     7.602555e+00     4.873558e-02
 * time: 0.7325890064239502
   133     7.602256e+00     2.635530e-02
 * time: 0.7396669387817383
   134     7.602110e+00     4.119356e-02
 * time: 0.7444961071014404
   135     7.601949e+00     3.607869e-02
 * time: 0.7493340969085693
   136     7.601706e+00     3.656263e-02
 * time: 0.7551231384277344
   137     7.601357e+00     3.039745e-02
 * time: 0.7612760066986084
   138     7.601119e+00     1.964239e-02
 * time: 0.7675600051879883
   139     7.600870e+00     2.901476e-02
 * time: 0.7830541133880615
   140     7.600775e+00     3.674219e-02
 * time: 0.786811113357544
   141     7.600632e+00     4.638907e-02
 * time: 0.7904150485992432
   142     7.600444e+00     2.054863e-02
 * time: 0.7949850559234619
   143     7.600318e+00     3.914595e-02
 * time: 0.7984800338745117
   144     7.600268e+00     4.536421e-02
 * time: 0.8020529747009277
   145     7.600017e+00     3.152636e-02
 * time: 0.8056440353393555
   146     7.599899e+00     6.392774e-02
 * time: 0.8091650009155273
   147     7.599570e+00     4.818001e-02
 * time: 0.813758134841919
   148     7.599491e+00     4.890954e-02
 * time: 0.8173789978027344
   149     7.599314e+00     3.968561e-02
 * time: 0.8220131397247314
   150     7.599121e+00     3.832940e-02
 * time: 0.8256590366363525
   151     7.598970e+00     3.176200e-02
 * time: 0.8303849697113037
   152     7.598830e+00     2.207130e-02
 * time: 0.8351240158081055
   153     7.598711e+00     2.305962e-02
 * time: 0.8386261463165283
   154     7.598634e+00     3.979465e-02
 * time: 0.8423380851745605
   155     7.598506e+00     2.270212e-02
 * time: 0.8470480442047119
   156     7.598386e+00     2.754944e-02
 * time: 0.8507421016693115
   157     7.598277e+00     2.642590e-02
 * time: 0.8544580936431885
   158     7.598207e+00     3.497605e-02
 * time: 0.8583359718322754
   159     7.598187e+00     4.334913e-02
 * time: 0.863023042678833
   160     7.598109e+00     3.310247e-02
 * time: 0.8682739734649658
   161     7.597954e+00     3.090851e-02
 * time: 0.8729259967803955
   162     7.597859e+00     3.860589e-02
 * time: 0.8772180080413818
   163     7.597851e+00     4.501223e-02
 * time: 0.8808670043945312
   164     7.597667e+00     3.647350e-02
 * time: 0.8844749927520752
   165     7.597508e+00     1.568076e-02
 * time: 0.8889851570129395
   166     7.597361e+00     2.878032e-02
 * time: 0.8926219940185547
   167     7.597236e+00     1.528677e-02
 * time: 0.8970820903778076
   168     7.597144e+00     2.886830e-02
 * time: 0.900670051574707
   169     7.597053e+00     2.064251e-02
 * time: 0.9051780700683594
   170     7.596940e+00     1.550360e-02
 * time: 0.9096930027008057
   171     7.596893e+00     2.662029e-02
 * time: 0.9243879318237305
   172     7.596815e+00     2.777277e-02
 * time: 0.9282679557800293
   173     7.596766e+00     4.347097e-02
 * time: 0.9318850040435791
   174     7.596594e+00     3.547622e-02
 * time: 0.9364049434661865
   175     7.596489e+00     4.567340e-02
 * time: 0.9399170875549316
   176     7.596333e+00     3.108210e-02
 * time: 0.9444811344146729
   177     7.596182e+00     2.865842e-02
 * time: 0.9481451511383057
   178     7.596181e+00     3.671855e-02
 * time: 0.9518289566040039
   179     7.596001e+00     2.819558e-02
 * time: 0.9554800987243652
   180     7.595844e+00     4.307962e-02
 * time: 0.9592130184173584
   181     7.595656e+00     2.947228e-02
 * time: 0.9639861583709717
   182     7.595409e+00     3.260256e-02
 * time: 0.9678270816802979
   183     7.595264e+00     2.890900e-02
 * time: 0.9726309776306152
   184     7.595229e+00     3.479443e-02
 * time: 0.9763169288635254
   185     7.595098e+00     4.370206e-02
 * time: 0.9800291061401367
   186     7.595019e+00     3.035468e-02
 * time: 0.9837391376495361
   187     7.594861e+00     3.160175e-02
 * time: 0.9873390197753906
   188     7.594790e+00     4.656035e-02
 * time: 0.9909660816192627
   189     7.594616e+00     3.060606e-02
 * time: 0.9954941272735596
   190     7.594457e+00     3.390058e-02
 * time: 0.9990060329437256
   191     7.594417e+00     5.418743e-02
 * time: 1.0024981498718262
   192     7.594262e+00     3.040336e-02
 * time: 1.0070369243621826
   193     7.594204e+00     3.392102e-02
 * time: 1.0106549263000488
   194     7.594180e+00     4.332347e-02
 * time: 1.0143370628356934
   195     7.594170e+00     4.440957e-02
 * time: 1.0180039405822754
   196     7.594039e+00     2.476913e-02
 * time: 1.0227410793304443
   197     7.594030e+00     2.562078e-02
 * time: 1.0264160633087158
   198     7.593953e+00     2.407675e-02
 * time: 1.0301730632781982
   199     7.593881e+00     2.856490e-02
 * time: 1.033869981765747
   200     7.593778e+00     2.413879e-02
 * time: 1.037519931793213
   201     7.593678e+00     1.788734e-02
 * time: 1.0422310829162598
   202     7.593605e+00     2.114120e-02
 * time: 1.045992136001587
   203     7.593602e+00     1.890415e-02
 * time: 1.0498390197753906
   204     7.593548e+00     1.530187e-02
 * time: 1.0658879280090332
   205     7.593498e+00     1.345330e-02
 * time: 1.070587158203125
   206     7.593449e+00     1.223288e-02
 * time: 1.0752160549163818
   207     7.593420e+00     1.617835e-02
 * time: 1.0787780284881592
   208     7.593412e+00     1.927417e-02
 * time: 1.0823769569396973
   209     7.593394e+00     2.136267e-02
 * time: 1.0860559940338135
   210     7.593352e+00     1.225094e-02
 * time: 1.0907459259033203
   211     7.593320e+00     1.286735e-02
 * time: 1.0943169593811035
   212     7.593306e+00     1.802514e-02
 * time: 1.0979199409484863
   213     7.593281e+00     2.040679e-02
 * time: 1.1015000343322754
   214     7.593240e+00     7.853626e-03
 * time: 1.1059761047363281
   215     7.593224e+00     1.528299e-02
 * time: 1.1095261573791504
   216     7.593193e+00     9.645594e-03
 * time: 1.113116979598999
   217     7.593170e+00     1.112982e-02
 * time: 1.1166870594024658
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   681     7.592300e+00     6.365099e-04
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   682     7.592300e+00     5.136595e-04
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   683     7.592300e+00     8.666012e-04
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   684     7.592300e+00     7.574047e-04
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   685     7.592300e+00     4.922392e-04
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   686     7.592300e+00     5.361759e-04
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   687     7.592300e+00     6.360257e-04
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   688     7.592300e+00     4.284642e-04
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   689     7.592300e+00     2.911199e-04
 * time: 3.1169869899749756
   690     7.592300e+00     3.370941e-04
 * time: 3.1205179691314697
   691     7.592300e+00     2.019571e-04
 * time: 3.125070095062256
   692     7.592300e+00     2.540708e-04
 * time: 3.1286330223083496
   693     7.592300e+00     3.138264e-04
 * time: 3.144050121307373
   694     7.592300e+00     3.927906e-04
 * time: 3.1478500366210938
   695     7.592300e+00     4.188998e-04
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   696     7.592300e+00     2.982373e-04
 * time: 3.155082941055298
   697     7.592300e+00     3.881722e-04
 * time: 3.1585681438446045
   698     7.592300e+00     3.083366e-04
 * time: 3.1623780727386475
   699     7.592300e+00     2.325073e-04
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   700     7.592300e+00     5.128841e-04
 * time: 3.1695971488952637
   701     7.592300e+00     4.464446e-04
 * time: 3.1731550693511963
   702     7.592300e+00     2.167851e-04
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   703     7.592300e+00     2.440500e-04
 * time: 3.1812660694122314
   704     7.592300e+00     3.416316e-04
 * time: 3.1848220825195312
   705     7.592300e+00     1.585320e-04
 * time: 3.189296007156372
   706     7.592300e+00     1.969492e-04
 * time: 3.1928350925445557
   707     7.592300e+00     2.664613e-04
 * time: 3.1963610649108887
   708     7.592300e+00     2.661792e-04
 * time: 3.200024127960205
   709     7.592300e+00     1.700777e-04
 * time: 3.204674005508423
   710     7.592300e+00     1.660844e-04
 * time: 3.2084460258483887
Out[5]: