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     3.148120e+01     7.976782e+00
 * time: 0.0021741390228271484
     1     2.860931e+01     4.834721e+00
 * time: 0.006232023239135742
     2     1.878532e+01     4.231944e+00
 * time: 0.010436058044433594
     3     1.305985e+01     3.290575e+00
 * time: 0.014642000198364258
     4     1.154349e+01     3.492442e+00
 * time: 0.018026113510131836
     5     1.133521e+01     3.110641e+00
 * time: 0.021433115005493164
     6     9.973911e+00     1.461407e+00
 * time: 0.025832176208496094
     7     8.825201e+00     1.696388e+00
 * time: 0.029942035675048828
     8     8.340630e+00     1.257247e+00
 * time: 0.0341341495513916
     9     8.188388e+00     7.369264e-01
 * time: 0.08735418319702148
    10     8.181798e+00     1.369906e+00
 * time: 0.08998298645019531
    11     8.143823e+00     1.008025e+00
 * time: 0.09273409843444824
    12     8.070573e+00     5.562346e-01
 * time: 0.09541201591491699
    13     8.019478e+00     5.871768e-01
 * time: 0.09789013862609863
    14     7.987270e+00     5.179631e-01
 * time: 0.10050106048583984
    15     7.949139e+00     4.413989e-01
 * time: 0.10306119918823242
    16     7.921160e+00     2.473208e-01
 * time: 0.10559701919555664
    17     7.903649e+00     4.275275e-01
 * time: 0.1083061695098877
    18     7.887734e+00     2.483476e-01
 * time: 0.11151003837585449
    19     7.877275e+00     2.716477e-01
 * time: 0.11405205726623535
    20     7.863731e+00     3.190057e-01
 * time: 0.11655211448669434
    21     7.851747e+00     1.992245e-01
 * time: 0.12076020240783691
    22     7.845183e+00     2.990783e-01
 * time: 0.1255180835723877
    23     7.831563e+00     2.079099e-01
 * time: 0.12980914115905762
    24     7.819596e+00     2.182718e-01
 * time: 0.13413715362548828
    25     7.813778e+00     2.517873e-01
 * time: 0.13860416412353516
    26     7.803472e+00     2.572012e-01
 * time: 0.14226698875427246
    27     7.791507e+00     1.526725e-01
 * time: 0.14589715003967285
    28     7.783590e+00     2.070923e-01
 * time: 0.14956998825073242
    29     7.774926e+00     1.744098e-01
 * time: 0.1532430648803711
    30     7.771718e+00     2.111901e-01
 * time: 0.1575150489807129
    31     7.764113e+00     2.157635e-01
 * time: 0.16212916374206543
    32     7.757313e+00     1.946915e-01
 * time: 0.16663503646850586
    33     7.749997e+00     2.537610e-01
 * time: 0.17126011848449707
    34     7.746078e+00     2.994609e-01
 * time: 0.17576217651367188
    35     7.736647e+00     1.872230e-01
 * time: 0.18019604682922363
    36     7.733542e+00     2.962243e-01
 * time: 0.18457603454589844
    37     7.723686e+00     1.600278e-01
 * time: 0.18925213813781738
    38     7.716906e+00     1.818841e-01
 * time: 0.19369006156921387
    39     7.709998e+00     1.974226e-01
 * time: 0.19814014434814453
    40     7.703846e+00     2.382428e-01
 * time: 0.20284318923950195
    41     7.696816e+00     1.051032e-01
 * time: 0.20855212211608887
    42     7.694500e+00     2.571276e-01
 * time: 0.2129380702972412
    43     7.690573e+00     2.034471e-01
 * time: 0.23691201210021973
    44     7.685807e+00     2.182316e-01
 * time: 0.23952603340148926
    45     7.681004e+00     9.912048e-02
 * time: 0.24279117584228516
    46     7.676336e+00     1.797902e-01
 * time: 0.24538016319274902
    47     7.671407e+00     1.006924e-01
 * time: 0.24869799613952637
    48     7.666702e+00     1.548438e-01
 * time: 0.25131702423095703
    49     7.663790e+00     1.384494e-01
 * time: 0.2539050579071045
    50     7.660523e+00     1.402333e-01
 * time: 0.2564401626586914
    51     7.658122e+00     1.531875e-01
 * time: 0.258991003036499
    52     7.653058e+00     1.387979e-01
 * time: 0.2615470886230469
    53     7.649242e+00     1.176318e-01
 * time: 0.26500821113586426
    54     7.646228e+00     7.116394e-02
 * time: 0.268233060836792
    55     7.642844e+00     9.009411e-02
 * time: 0.2708010673522949
    56     7.641380e+00     1.094158e-01
 * time: 0.27336812019348145
    57     7.641047e+00     1.180147e-01
 * time: 0.2758660316467285
    58     7.640899e+00     8.176400e-02
 * time: 0.27842020988464355
    59     7.640045e+00     1.683780e-01
 * time: 0.28124117851257324
    60     7.638125e+00     8.808581e-02
 * time: 0.28443312644958496
    61     7.636426e+00     1.047422e-01
 * time: 0.2869300842285156
    62     7.635526e+00     8.817420e-02
 * time: 0.28956103324890137
    63     7.633994e+00     5.655445e-02
 * time: 0.29207301139831543
    64     7.633591e+00     8.076289e-02
 * time: 0.2963531017303467
    65     7.632499e+00     5.568526e-02
 * time: 0.30078601837158203
    66     7.631722e+00     4.675207e-02
 * time: 0.30649304389953613
    67     7.631436e+00     9.169248e-02
 * time: 0.31104016304016113
    68     7.630670e+00     7.257136e-02
 * time: 0.31676602363586426
    69     7.629986e+00     6.682596e-02
 * time: 0.32233405113220215
    70     7.629326e+00     5.659019e-02
 * time: 0.3282191753387451
    71     7.629080e+00     5.506472e-02
 * time: 0.3325660228729248
    72     7.628601e+00     6.706456e-02
 * time: 0.3370530605316162
    73     7.627928e+00     3.153190e-02
 * time: 0.34305405616760254
    74     7.627421e+00     4.915726e-02
 * time: 0.3477320671081543
    75     7.627369e+00     5.874008e-02
 * time: 0.3653261661529541
    76     7.626745e+00     5.288012e-02
 * time: 0.3679921627044678
    77     7.626244e+00     6.666101e-02
 * time: 0.37061214447021484
    78     7.625692e+00     3.658142e-02
 * time: 0.3741281032562256
    79     7.625420e+00     5.861060e-02
 * time: 0.37681007385253906
    80     7.625175e+00     4.771474e-02
 * time: 0.37949514389038086
    81     7.624876e+00     3.696046e-02
 * time: 0.38205718994140625
    82     7.624821e+00     6.071859e-02
 * time: 0.38471198081970215
    83     7.624447e+00     3.546631e-02
 * time: 0.38793206214904785
    84     7.624115e+00     5.440242e-02
 * time: 0.3907740116119385
    85     7.624091e+00     7.003944e-02
 * time: 0.39345407485961914
    86     7.624034e+00     5.958998e-02
 * time: 0.3959810733795166
    87     7.623693e+00     5.046968e-02
 * time: 0.39850616455078125
    88     7.623268e+00     4.214160e-02
 * time: 0.4010441303253174
    89     7.622815e+00     4.709536e-02
 * time: 0.4035520553588867
    90     7.622685e+00     6.505086e-02
 * time: 0.40639305114746094
    91     7.622286e+00     4.334952e-02
 * time: 0.40963315963745117
    92     7.622043e+00     5.810219e-02
 * time: 0.41219210624694824
    93     7.621672e+00     4.556278e-02
 * time: 0.41547513008117676
    94     7.621499e+00     4.837781e-02
 * time: 0.4181091785430908
    95     7.621024e+00     4.616776e-02
 * time: 0.42082810401916504
    96     7.620702e+00     3.447455e-02
 * time: 0.42513203620910645
    97     7.620201e+00     2.457585e-02
 * time: 0.43077898025512695
    98     7.620025e+00     3.474768e-02
 * time: 0.4352850914001465
    99     7.619919e+00     3.446584e-02
 * time: 0.4399731159210205
   100     7.619796e+00     4.066059e-02
 * time: 0.44463419914245605
   101     7.619711e+00     3.385564e-02
 * time: 0.4492671489715576
   102     7.619422e+00     3.535886e-02
 * time: 0.4538850784301758
   103     7.619145e+00     2.939781e-02
 * time: 0.4594600200653076
   104     7.618927e+00     5.123870e-02
 * time: 0.4640212059020996
   105     7.618819e+00     3.919593e-02
 * time: 0.4678230285644531
   106     7.618702e+00     2.448087e-02
 * time: 0.47057008743286133
   107     7.618486e+00     2.572167e-02
 * time: 0.4736900329589844
   108     7.618325e+00     2.927185e-02
 * time: 0.48492908477783203
   109     7.618110e+00     4.595612e-02
 * time: 0.4875810146331787
   110     7.617931e+00     2.446269e-02
 * time: 0.49077320098876953
   111     7.617770e+00     2.808868e-02
 * time: 0.49405908584594727
   112     7.617557e+00     1.996933e-02
 * time: 0.4972701072692871
   113     7.617396e+00     1.993439e-02
 * time: 0.5000290870666504
   114     7.617339e+00     3.330451e-02
 * time: 0.5026090145111084
   115     7.617299e+00     4.919919e-02
 * time: 0.5051090717315674
   116     7.617074e+00     3.974564e-02
 * time: 0.5086400508880615
   117     7.616943e+00     5.094035e-02
 * time: 0.5113182067871094
   118     7.616784e+00     5.996913e-02
 * time: 0.5139870643615723
   119     7.616474e+00     3.445397e-02
 * time: 0.5172979831695557
   120     7.616265e+00     3.202310e-02
 * time: 0.5205211639404297
   121     7.616112e+00     2.351918e-02
 * time: 0.5237431526184082
   122     7.615987e+00     1.983246e-02
 * time: 0.5270512104034424
   123     7.615848e+00     2.571351e-02
 * time: 0.5297491550445557
   124     7.615661e+00     2.813978e-02
 * time: 0.5325310230255127
   125     7.615485e+00     2.614831e-02
 * time: 0.5357010364532471
   126     7.615394e+00     3.707523e-02
 * time: 0.5383419990539551
   127     7.615213e+00     2.412467e-02
 * time: 0.5418450832366943
   128     7.615074e+00     2.228796e-02
 * time: 0.5460531711578369
   129     7.614939e+00     1.987080e-02
 * time: 0.5510921478271484
   130     7.614826e+00     2.483946e-02
 * time: 0.5547940731048584
   131     7.614779e+00     3.045500e-02
 * time: 0.5574660301208496
   132     7.614654e+00     3.239148e-02
 * time: 0.5600371360778809
   133     7.614500e+00     1.972008e-02
 * time: 0.5637071132659912
   134     7.614399e+00     3.007318e-02
 * time: 0.5663430690765381
   135     7.614329e+00     2.113071e-02
 * time: 0.5690150260925293
   136     7.614264e+00     2.539635e-02
 * time: 0.5715830326080322
   137     7.614124e+00     1.723944e-02
 * time: 0.5748600959777832
   138     7.614026e+00     3.091121e-02
 * time: 0.5776140689849854
   139     7.613874e+00     3.204256e-02
 * time: 0.5889320373535156
   140     7.613865e+00     2.837453e-02
 * time: 0.591529130935669
   141     7.613794e+00     2.913205e-02
 * time: 0.5942111015319824
   142     7.613635e+00     1.961048e-02
 * time: 0.597520112991333
   143     7.613449e+00     2.332186e-02
 * time: 0.6000330448150635
   144     7.613411e+00     2.707925e-02
 * time: 0.6027050018310547
   145     7.613316e+00     2.222793e-02
 * time: 0.6053330898284912
   146     7.613127e+00     2.787146e-02
 * time: 0.6087100505828857
   147     7.613109e+00     2.205062e-02
 * time: 0.6113481521606445
   148     7.613059e+00     2.701923e-02
 * time: 0.6147921085357666
   149     7.613004e+00     2.499921e-02
 * time: 0.6174790859222412
   150     7.612939e+00     2.929434e-02
 * time: 0.6201090812683105
   151     7.612856e+00     2.152010e-02
 * time: 0.6235010623931885
   152     7.612782e+00     1.519923e-02
 * time: 0.6269180774688721
   153     7.612778e+00     2.687760e-02
 * time: 0.6295640468597412
   154     7.612689e+00     2.270307e-02
 * time: 0.6322581768035889
   155     7.612674e+00     2.574998e-02
 * time: 0.6348800659179688
   156     7.612614e+00     2.244497e-02
 * time: 0.6375851631164551
   157     7.612581e+00     2.725223e-02
 * time: 0.64034104347229
   158     7.612580e+00     2.602264e-02
 * time: 0.642963171005249
   159     7.612546e+00     3.133240e-02
 * time: 0.6456341743469238
   160     7.612474e+00     2.197814e-02
 * time: 0.6486630439758301
   161     7.612432e+00     1.938327e-02
 * time: 0.6528830528259277
   162     7.612367e+00     1.817147e-02
 * time: 0.6571981906890869
   163     7.612353e+00     2.780268e-02
 * time: 0.660865068435669
   164     7.612277e+00     1.841349e-02
 * time: 0.6642630100250244
   165     7.612221e+00     2.012426e-02
 * time: 0.6676101684570312
   166     7.612184e+00     1.866585e-02
 * time: 0.6711091995239258
   167     7.612158e+00     1.634046e-02
 * time: 0.673821210861206
   168     7.612153e+00     2.320378e-02
 * time: 0.6765201091766357
   169     7.612107e+00     1.624230e-02
 * time: 0.6798629760742188
   170     7.612064e+00     1.172636e-02
 * time: 0.6832690238952637
   171     7.612035e+00     2.047546e-02
 * time: 0.6946229934692383
   172     7.612019e+00     1.578750e-02
 * time: 0.6973211765289307
   173     7.611981e+00     1.632244e-02
 * time: 0.6999650001525879
   174     7.611929e+00     1.023703e-02
 * time: 0.7036080360412598
   175     7.611913e+00     1.406426e-02
 * time: 0.7062761783599854
   176     7.611877e+00     1.575528e-02
 * time: 0.7089290618896484
   177     7.611849e+00     1.741216e-02
 * time: 0.7115120887756348
   178     7.611818e+00     1.809885e-02
 * time: 0.7142801284790039
   179     7.611769e+00     1.009976e-02
 * time: 0.717764139175415
   180     7.611752e+00     1.870613e-02
 * time: 0.7204470634460449
   181     7.611719e+00     1.357782e-02
 * time: 0.7231659889221191
   182     7.611683e+00     1.890593e-02
 * time: 0.725836992263794
   183     7.611647e+00     1.296869e-02
 * time: 0.7291650772094727
   184     7.611603e+00     1.016939e-02
 * time: 0.7318680286407471
   185     7.611594e+00     2.030730e-02
 * time: 0.734745979309082
   186     7.611565e+00     1.921945e-02
 * time: 0.7374281883239746
   187     7.611562e+00     2.235582e-02
 * time: 0.740070104598999
   188     7.611529e+00     2.030852e-02
 * time: 0.7427651882171631
   189     7.611513e+00     2.291107e-02
 * time: 0.7454700469970703
   190     7.611467e+00     2.460136e-02
 * time: 0.748237133026123
   191     7.611432e+00     3.465308e-02
 * time: 0.7509911060333252
   192     7.611358e+00     2.909797e-02
 * time: 0.7543480396270752
   193     7.611278e+00     2.695200e-02
 * time: 0.7583701610565186
   194     7.611234e+00     3.069625e-02
 * time: 0.7625370025634766
   195     7.611189e+00     2.421526e-02
 * time: 0.7662761211395264
   196     7.611102e+00     1.703635e-02
 * time: 0.7697300910949707
   197     7.610992e+00     2.395691e-02
 * time: 0.7724390029907227
   198     7.610960e+00     3.960296e-02
 * time: 0.7751190662384033
   199     7.610933e+00     4.963861e-02
 * time: 0.7778241634368896
   200     7.610807e+00     3.442356e-02
 * time: 0.7813971042633057
   201     7.610700e+00     2.987762e-02
 * time: 0.784080982208252
   202     7.610486e+00     3.391001e-02
 * time: 0.7874701023101807
   203     7.610249e+00     2.099676e-02
 * time: 0.7908420562744141
   204     7.610182e+00     2.452088e-02
 * time: 0.8022661209106445
   205     7.610052e+00     2.823041e-02
 * time: 0.8050520420074463
   206     7.609936e+00     2.002126e-02
 * time: 0.8076980113983154
   207     7.609828e+00     2.505622e-02
 * time: 0.8112761974334717
   208     7.609678e+00     4.460734e-02
 * time: 0.8140101432800293
   209     7.609450e+00     3.118112e-02
 * time: 0.8174221515655518
   210     7.609296e+00     3.916590e-02
 * time: 0.8208181858062744
   211     7.609232e+00     4.030161e-02
 * time: 0.823523998260498
   212     7.609069e+00     4.536255e-02
 * time: 0.82625412940979
   213     7.608879e+00     4.431682e-02
 * time: 0.8297741413116455
   214     7.608644e+00     3.192863e-02
 * time: 0.8324601650238037
   215     7.608344e+00     3.273918e-02
 * time: 0.8350741863250732
   216     7.608053e+00     2.972049e-02
 * time: 0.8384401798248291
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 * time: 2.424889087677002
   684     7.601699e+00     1.898422e-04
 * time: 2.4279141426086426
   685     7.601699e+00     1.904448e-04
 * time: 2.4309589862823486
Out[5]: