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[:, :, 1, 1], c=:blues)
Iter     Function value   Gradient norm 
     0     3.092949e+01     7.144394e+00
 * time: 0.0018329620361328125
     1     3.039245e+01     4.509089e+00
 * time: 0.007828950881958008
     2     2.078743e+01     4.382530e+00
 * time: 0.014898061752319336
     3     1.507186e+01     3.588408e+00
 * time: 0.02097296714782715
     4     1.200423e+01     1.828797e+00
 * time: 0.027698993682861328
     5     1.147697e+01     2.192761e+00
 * time: 0.034364938735961914
     6     1.056654e+01     1.398145e+00
 * time: 0.039016008377075195
     7     9.477255e+00     1.622747e+00
 * time: 0.0434720516204834
     8     8.929882e+00     1.568717e+00
 * time: 0.047866106033325195
     9     8.678980e+00     1.114768e+00
 * time: 0.05199003219604492
    10     8.465320e+00     1.062911e+00
 * time: 0.055989980697631836
    11     8.287403e+00     1.022523e+00
 * time: 0.06016397476196289
    12     8.123438e+00     6.219178e-01
 * time: 0.06507611274719238
    13     8.016256e+00     4.476339e-01
 * time: 0.06985902786254883
    14     7.937866e+00     3.560243e-01
 * time: 0.07485198974609375
    15     7.901210e+00     6.028070e-01
 * time: 0.0787210464477539
    16     7.885194e+00     4.175437e-01
 * time: 0.08263301849365234
    17     7.877753e+00     7.210553e-01
 * time: 0.0866250991821289
    18     7.843726e+00     4.307479e-01
 * time: 0.09159207344055176
    19     7.810886e+00     4.612220e-01
 * time: 0.09560203552246094
    20     7.771616e+00     3.468957e-01
 * time: 0.09949588775634766
    21     7.753548e+00     3.208861e-01
 * time: 0.10350394248962402
    22     7.734905e+00     3.313122e-01
 * time: 0.10852289199829102
    23     7.718152e+00     2.835561e-01
 * time: 0.11344695091247559
    24     7.706856e+00     3.901759e-01
 * time: 0.11706089973449707
    25     7.701661e+00     3.109875e-01
 * time: 0.1208341121673584
    26     7.700990e+00     2.890605e-01
 * time: 0.15642595291137695
    27     7.695612e+00     2.386133e-01
 * time: 0.1594541072845459
    28     7.691636e+00     2.331825e-01
 * time: 0.16239690780639648
    29     7.683439e+00     1.674815e-01
 * time: 0.1654679775238037
    30     7.677170e+00     2.318238e-01
 * time: 0.16843295097351074
    31     7.672712e+00     1.914697e-01
 * time: 0.17133307456970215
    32     7.670791e+00     1.980980e-01
 * time: 0.17427396774291992
    33     7.668600e+00     1.394941e-01
 * time: 0.17716693878173828
    34     7.665276e+00     9.027116e-02
 * time: 0.18077397346496582
    35     7.664891e+00     2.126395e-01
 * time: 0.18364405632019043
    36     7.662996e+00     1.709069e-01
 * time: 0.18675804138183594
    37     7.661378e+00     1.907077e-01
 * time: 0.18974900245666504
    38     7.660759e+00     1.745404e-01
 * time: 0.19270896911621094
    39     7.660378e+00     1.150397e-01
 * time: 0.1957099437713623
    40     7.658113e+00     1.092295e-01
 * time: 0.19860291481018066
    41     7.656312e+00     1.714183e-01
 * time: 0.20154404640197754
    42     7.654793e+00     1.345714e-01
 * time: 0.20441889762878418
    43     7.651451e+00     1.158949e-01
 * time: 0.20821905136108398
    44     7.651082e+00     1.159841e-01
 * time: 0.21104693412780762
    45     7.649434e+00     1.281182e-01
 * time: 0.21421289443969727
    46     7.646652e+00     7.067507e-02
 * time: 0.2182319164276123
    47     7.644570e+00     7.518722e-02
 * time: 0.22190594673156738
    48     7.643033e+00     6.021664e-02
 * time: 0.22555088996887207
    49     7.641780e+00     8.117246e-02
 * time: 0.22931408882141113
    50     7.640602e+00     2.624572e-02
 * time: 0.23301196098327637
    51     7.640491e+00     6.791138e-02
 * time: 0.23584699630737305
    52     7.640415e+00     7.633086e-02
 * time: 0.2388620376586914
    53     7.639948e+00     7.220428e-02
 * time: 0.24179291725158691
    54     7.639241e+00     6.705693e-02
 * time: 0.24467706680297852
    55     7.638300e+00     9.608251e-02
 * time: 0.24758696556091309
    56     7.638262e+00     5.528776e-02
 * time: 0.2506740093231201
    57     7.637424e+00     3.926671e-02
 * time: 0.2789750099182129
    58     7.636904e+00     6.511189e-02
 * time: 0.28186798095703125
    59     7.636484e+00     6.006467e-02
 * time: 0.28484606742858887
    60     7.636103e+00     8.170569e-02
 * time: 0.28774189949035645
    61     7.635239e+00     6.511558e-02
 * time: 0.2906229496002197
    62     7.634345e+00     3.559750e-02
 * time: 0.29429197311401367
    63     7.633650e+00     3.705961e-02
 * time: 0.29814791679382324
    64     7.633483e+00     6.586718e-02
 * time: 0.3009200096130371
    65     7.632622e+00     4.458498e-02
 * time: 0.3036930561065674
    66     7.631810e+00     5.054105e-02
 * time: 0.3073410987854004
    67     7.631024e+00     3.047601e-02
 * time: 0.3111259937286377
    68     7.630530e+00     4.498980e-02
 * time: 0.313978910446167
    69     7.629999e+00     3.212314e-02
 * time: 0.31752490997314453
    70     7.629596e+00     3.507266e-02
 * time: 0.32129693031311035
    71     7.629121e+00     4.462318e-02
 * time: 0.32411789894104004
    72     7.628863e+00     4.731378e-02
 * time: 0.3269350528717041
    73     7.628691e+00     6.594313e-02
 * time: 0.32981109619140625
    74     7.628356e+00     2.989370e-02
 * time: 0.3334200382232666
    75     7.627832e+00     5.030390e-02
 * time: 0.3362081050872803
    76     7.627673e+00     3.972048e-02
 * time: 0.3391430377960205
    77     7.627116e+00     3.337398e-02
 * time: 0.3419530391693115
    78     7.626590e+00     3.622094e-02
 * time: 0.34461402893066406
    79     7.626237e+00     3.772358e-02
 * time: 0.34746694564819336
    80     7.626068e+00     5.355742e-02
 * time: 0.3503739833831787
    81     7.625921e+00     6.319780e-02
 * time: 0.35317492485046387
    82     7.625501e+00     5.445749e-02
 * time: 0.3560030460357666
    83     7.625219e+00     6.489779e-02
 * time: 0.35884690284729004
    84     7.624850e+00     6.726056e-02
 * time: 0.36181092262268066
    85     7.624435e+00     5.616170e-02
 * time: 0.36464595794677734
    86     7.624060e+00     5.613096e-02
 * time: 0.36748290061950684
    87     7.623566e+00     3.248903e-02
 * time: 0.388261079788208
    88     7.622762e+00     5.289825e-02
 * time: 0.3912529945373535
    89     7.622200e+00     3.355599e-02
 * time: 0.39495301246643066
    90     7.621552e+00     4.324419e-02
 * time: 0.3977320194244385
    91     7.621128e+00     4.930253e-02
 * time: 0.4014160633087158
    92     7.620677e+00     1.796776e-02
 * time: 0.4050929546356201
    93     7.620090e+00     2.794969e-02
 * time: 0.40794897079467773
    94     7.619696e+00     4.094311e-02
 * time: 0.41083788871765137
    95     7.619402e+00     4.664244e-02
 * time: 0.41356492042541504
    96     7.619316e+00     9.689157e-02
 * time: 0.4163360595703125
    97     7.618821e+00     4.791369e-02
 * time: 0.42000603675842285
    98     7.618453e+00     5.079010e-02
 * time: 0.42371392250061035
    99     7.617900e+00     4.724327e-02
 * time: 0.4265270233154297
   100     7.617323e+00     4.062095e-02
 * time: 0.42929601669311523
   101     7.616996e+00     3.213397e-02
 * time: 0.4322481155395508
   102     7.616759e+00     4.587416e-02
 * time: 0.435197114944458
   103     7.616481e+00     2.313925e-02
 * time: 0.4387509822845459
   104     7.616187e+00     3.874921e-02
 * time: 0.44160008430480957
   105     7.615872e+00     3.122982e-02
 * time: 0.44446492195129395
   106     7.615684e+00     3.908353e-02
 * time: 0.44733691215515137
   107     7.615346e+00     2.199401e-02
 * time: 0.45098304748535156
   108     7.615007e+00     2.563637e-02
 * time: 0.45381999015808105
   109     7.614907e+00     3.994865e-02
 * time: 0.45656609535217285
   110     7.614840e+00     5.328746e-02
 * time: 0.4593510627746582
   111     7.614720e+00     3.870166e-02
 * time: 0.46227002143859863
   112     7.614596e+00     5.527046e-02
 * time: 0.4651370048522949
   113     7.614375e+00     3.061071e-02
 * time: 0.467972993850708
   114     7.614209e+00     5.321926e-02
 * time: 0.4708380699157715
   115     7.613845e+00     2.474032e-02
 * time: 0.47449588775634766
   116     7.613610e+00     3.077614e-02
 * time: 0.4772670269012451
   117     7.613259e+00     3.313034e-02
 * time: 0.48003506660461426
   118     7.612879e+00     2.420330e-02
 * time: 0.49411606788635254
   119     7.612837e+00     4.552855e-02
 * time: 0.49697208404541016
   120     7.612749e+00     6.665934e-02
 * time: 0.4997549057006836
   121     7.612511e+00     4.065434e-02
 * time: 0.5034348964691162
   122     7.612304e+00     4.023747e-02
 * time: 0.5070860385894775
   123     7.612093e+00     4.559837e-02
 * time: 0.5099430084228516
   124     7.611757e+00     4.962074e-02
 * time: 0.5137150287628174
   125     7.611453e+00     1.860271e-02
 * time: 0.5172760486602783
   126     7.611201e+00     3.199354e-02
 * time: 0.5201330184936523
   127     7.611017e+00     3.525804e-02
 * time: 0.5230669975280762
   128     7.610882e+00     3.147620e-02
 * time: 0.5258979797363281
   129     7.610602e+00     3.271631e-02
 * time: 0.528764009475708
   130     7.610499e+00     5.527226e-02
 * time: 0.5317621231079102
   131     7.610371e+00     4.641534e-02
 * time: 0.534682035446167
   132     7.610278e+00     6.806592e-02
 * time: 0.5375819206237793
   133     7.610005e+00     3.757165e-02
 * time: 0.5405130386352539
   134     7.609653e+00     2.228037e-02
 * time: 0.5442090034484863
   135     7.609384e+00     2.447838e-02
 * time: 0.547144889831543
   136     7.609224e+00     2.839178e-02
 * time: 0.5500121116638184
   137     7.609101e+00     4.026911e-02
 * time: 0.5529220104217529
   138     7.608961e+00     1.848177e-02
 * time: 0.5566360950469971
   139     7.608831e+00     1.695676e-02
 * time: 0.5603399276733398
   140     7.608746e+00     2.388414e-02
 * time: 0.5633189678192139
   141     7.608660e+00     2.020082e-02
 * time: 0.566972017288208
   142     7.608637e+00     2.829618e-02
 * time: 0.5697650909423828
   143     7.608606e+00     3.268107e-02
 * time: 0.5726311206817627
   144     7.608496e+00     1.914120e-02
 * time: 0.5763719081878662
   145     7.608399e+00     2.836165e-02
 * time: 0.5792860984802246
   146     7.608280e+00     2.338237e-02
 * time: 0.5822598934173584
   147     7.608188e+00     2.627436e-02
 * time: 0.585211992263794
   148     7.608086e+00     1.180398e-02
 * time: 0.6003189086914062
   149     7.608006e+00     1.339341e-02
 * time: 0.6033039093017578
   150     7.607952e+00     2.705721e-02
 * time: 0.6062500476837158
   151     7.607904e+00     2.481473e-02
 * time: 0.6092839241027832
   152     7.607864e+00     1.970629e-02
 * time: 0.6121840476989746
   153     7.607781e+00     2.112683e-02
 * time: 0.6159629821777344
   154     7.607677e+00     8.457094e-03
 * time: 0.6197021007537842
   155     7.607624e+00     1.381003e-02
 * time: 0.6226639747619629
   156     7.607557e+00     1.273533e-02
 * time: 0.6257250308990479
   157     7.607503e+00     1.560984e-02
 * time: 0.6286380290985107
   158     7.607485e+00     1.506689e-02
 * time: 0.6315860748291016
   159     7.607429e+00     1.359827e-02
 * time: 0.6344900131225586
   160     7.607340e+00     1.604387e-02
 * time: 0.637347936630249
   161     7.607330e+00     2.319956e-02
 * time: 0.6402308940887451
   162     7.607325e+00     2.307167e-02
 * time: 0.6431660652160645
   163     7.607304e+00     2.358642e-02
 * time: 0.6462080478668213
   164     7.607173e+00     1.581495e-02
 * time: 0.6490380764007568
   165     7.607044e+00     1.904193e-02
 * time: 0.6518549919128418
   166     7.607010e+00     3.189679e-02
 * time: 0.6548349857330322
   167     7.606896e+00     2.190862e-02
 * time: 0.6585090160369873
   168     7.606753e+00     2.583350e-02
 * time: 0.6613221168518066
   169     7.606626e+00     1.627174e-02
 * time: 0.6650149822235107
   170     7.606575e+00     2.266247e-02
 * time: 0.6678969860076904
   171     7.606545e+00     2.153687e-02
 * time: 0.6706759929656982
   172     7.606456e+00     1.768502e-02
 * time: 0.6742970943450928
   173     7.606414e+00     2.668723e-02
 * time: 0.6770970821380615
   174     7.606333e+00     2.265223e-02
 * time: 0.6807210445404053
   175     7.606192e+00     2.872392e-02
 * time: 0.6835060119628906
   176     7.606140e+00     2.688193e-02
 * time: 0.6863601207733154
   177     7.606106e+00     1.954142e-02
 * time: 0.6892220973968506
   178     7.606010e+00     1.926650e-02
 * time: 0.6920359134674072
   179     7.605826e+00     3.136185e-02
 * time: 0.7062718868255615
   180     7.605707e+00     1.733690e-02
 * time: 0.7099380493164062
   181     7.605588e+00     1.650023e-02
 * time: 0.7135019302368164
   182     7.605495e+00     2.496859e-02
 * time: 0.716407060623169
   183     7.605404e+00     2.381374e-02
 * time: 0.719264030456543
   184     7.605377e+00     3.183078e-02
 * time: 0.722114086151123
   185     7.605298e+00     2.279632e-02
 * time: 0.7258520126342773
   186     7.605119e+00     2.538510e-02
 * time: 0.729482889175415
   187     7.605021e+00     2.862387e-02
 * time: 0.7323689460754395
   188     7.604896e+00     3.052032e-02
 * time: 0.7352859973907471
   189     7.604742e+00     1.964412e-02
 * time: 0.7390379905700684
   190     7.604688e+00     3.354532e-02
 * time: 0.741879940032959
   191     7.604669e+00     2.778443e-02
 * time: 0.7446689605712891
   192     7.604565e+00     2.636902e-02
 * time: 0.7485499382019043
   193     7.604426e+00     2.640297e-02
 * time: 0.751439094543457
   194     7.604288e+00     2.448490e-02
 * time: 0.7553260326385498
   195     7.604209e+00     2.910371e-02
 * time: 0.7583310604095459
   196     7.604078e+00     2.841721e-02
 * time: 0.7611560821533203
   197     7.603978e+00     2.278817e-02
 * time: 0.764847993850708
   198     7.603862e+00     2.605204e-02
 * time: 0.7677369117736816
   199     7.603851e+00     4.544651e-02
 * time: 0.7705249786376953
   200     7.603698e+00     2.478883e-02
 * time: 0.7734129428863525
   201     7.603662e+00     4.545850e-02
 * time: 0.7763159275054932
   202     7.603443e+00     3.030984e-02
 * time: 0.7799999713897705
   203     7.603293e+00     3.436837e-02
 * time: 0.7836019992828369
   204     7.603137e+00     2.810137e-02
 * time: 0.7865109443664551
   205     7.602982e+00     3.637739e-02
 * time: 0.7893979549407959
   206     7.602979e+00     3.396018e-02
 * time: 0.7922489643096924
   207     7.602875e+00     3.194552e-02
 * time: 0.7950499057769775
   208     7.602831e+00     3.224236e-02
 * time: 0.7980639934539795
   209     7.602682e+00     2.430060e-02
 * time: 0.8121390342712402
   210     7.602634e+00     3.961676e-02
 * time: 0.8150210380554199
   211     7.602402e+00     3.559408e-02
 * time: 0.8180301189422607
   212     7.602197e+00     2.968846e-02
 * time: 0.8216249942779541
   213     7.602050e+00     3.246387e-02
 * time: 0.8245069980621338
   214     7.601963e+00     3.666965e-02
 * time: 0.8275051116943359
   215     7.601887e+00     2.387274e-02
 * time: 0.8305950164794922
   216     7.601670e+00     2.413248e-02
 * time: 0.8343410491943359
   217     7.601430e+00     2.227738e-02
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Out[5]: