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     3.159302e+01     7.954240e+00
 * time: 0.002009868621826172
     1     2.780263e+01     4.201989e+00
 * time: 0.008098840713500977
     2     1.844724e+01     3.927732e+00
 * time: 0.013222932815551758
     3     1.179513e+01     2.401116e+00
 * time: 0.01809096336364746
     4     1.176026e+01     3.197918e+00
 * time: 0.021857023239135742
     5     1.067411e+01     2.006407e+00
 * time: 0.025758028030395508
     6     9.876183e+00     1.644597e+00
 * time: 0.08055496215820312
     7     9.267052e+00     1.386918e+00
 * time: 0.08441281318664551
     8     8.693555e+00     1.211421e+00
 * time: 0.08846282958984375
     9     8.313659e+00     5.763612e-01
 * time: 0.09302282333374023
    10     8.111231e+00     3.628822e-01
 * time: 0.09696197509765625
    11     8.070203e+00     8.309101e-01
 * time: 0.10015082359313965
    12     8.013909e+00     6.662059e-01
 * time: 0.10308980941772461
    13     7.964668e+00     5.244057e-01
 * time: 0.10687994956970215
    14     7.918505e+00     6.537759e-01
 * time: 0.11037087440490723
    15     7.882190e+00     6.176930e-01
 * time: 0.11362695693969727
    16     7.861291e+00     5.792358e-01
 * time: 0.11765003204345703
    17     7.857775e+00     8.097500e-01
 * time: 0.12124991416931152
    18     7.848050e+00     5.014490e-01
 * time: 0.12543487548828125
    19     7.840127e+00     5.779733e-01
 * time: 0.1297929286956787
    20     7.824555e+00     4.517695e-01
 * time: 0.13374900817871094
    21     7.794490e+00     3.513582e-01
 * time: 0.13894295692443848
    22     7.757036e+00     3.081699e-01
 * time: 0.14368200302124023
    23     7.734985e+00     3.045205e-01
 * time: 0.14804697036743164
    24     7.718664e+00     2.388876e-01
 * time: 0.15254592895507812
    25     7.705537e+00     2.366791e-01
 * time: 0.15892481803894043
    26     7.692254e+00     1.739177e-01
 * time: 0.16576004028320312
    27     7.683319e+00     1.584658e-01
 * time: 0.1727449893951416
    28     7.677054e+00     1.507266e-01
 * time: 0.17956280708312988
    29     7.672841e+00     1.055739e-01
 * time: 0.1869640350341797
    30     7.668999e+00     9.682580e-02
 * time: 0.19370388984680176
    31     7.668828e+00     2.076956e-01
 * time: 0.19881796836853027
    32     7.666279e+00     1.110749e-01
 * time: 0.20399904251098633
    33     7.662265e+00     8.720288e-02
 * time: 0.20971393585205078
    34     7.657904e+00     8.038331e-02
 * time: 0.21512484550476074
    35     7.654586e+00     7.594841e-02
 * time: 0.2205049991607666
    36     7.651832e+00     1.043558e-01
 * time: 0.2258739471435547
    37     7.651782e+00     1.814430e-01
 * time: 0.23126792907714844
    38     7.651068e+00     1.309087e-01
 * time: 0.23656702041625977
    39     7.648804e+00     8.965576e-02
 * time: 0.2410259246826172
    40     7.647562e+00     1.303120e-01
 * time: 0.27387499809265137
    41     7.646463e+00     1.290807e-01
 * time: 0.2770049571990967
    42     7.644676e+00     6.908459e-02
 * time: 0.2811448574066162
    43     7.643080e+00     6.639827e-02
 * time: 0.28412604331970215
    44     7.642016e+00     9.022525e-02
 * time: 0.2871739864349365
    45     7.641164e+00     7.792273e-02
 * time: 0.29041194915771484
    46     7.640939e+00     7.791173e-02
 * time: 0.294003963470459
    47     7.640468e+00     6.759223e-02
 * time: 0.29709887504577637
    48     7.639373e+00     6.334034e-02
 * time: 0.301088809967041
    49     7.639336e+00     1.203100e-01
 * time: 0.3041689395904541
    50     7.638557e+00     7.425392e-02
 * time: 0.3071858882904053
    51     7.638444e+00     1.192827e-01
 * time: 0.3139069080352783
    52     7.637401e+00     6.461574e-02
 * time: 0.3180990219116211
    53     7.636460e+00     6.890534e-02
 * time: 0.32116198539733887
    54     7.636328e+00     1.082896e-01
 * time: 0.32561802864074707
    55     7.636240e+00     9.020901e-02
 * time: 0.33007192611694336
    56     7.635659e+00     8.720103e-02
 * time: 0.33474183082580566
    57     7.634665e+00     7.511869e-02
 * time: 0.3385770320892334
    58     7.634052e+00     1.004545e-01
 * time: 0.3418450355529785
    59     7.633051e+00     9.960058e-02
 * time: 0.3453710079193115
    60     7.631484e+00     7.833726e-02
 * time: 0.34929490089416504
    61     7.631166e+00     1.032684e-01
 * time: 0.3528900146484375
    62     7.630135e+00     6.785583e-02
 * time: 0.35856199264526367
    63     7.629366e+00     7.404326e-02
 * time: 0.3621039390563965
    64     7.628062e+00     5.742880e-02
 * time: 0.3652470111846924
    65     7.626807e+00     5.870567e-02
 * time: 0.3690769672393799
    66     7.625676e+00     4.600444e-02
 * time: 0.3733029365539551
    67     7.625311e+00     5.860455e-02
 * time: 0.37666893005371094
    68     7.625202e+00     8.044384e-02
 * time: 0.37981486320495605
    69     7.624505e+00     7.451637e-02
 * time: 0.3832108974456787
    70     7.623780e+00     5.906647e-02
 * time: 0.38716697692871094
    71     7.623195e+00     6.250966e-02
 * time: 0.3905489444732666
    72     7.622122e+00     5.222588e-02
 * time: 0.3939328193664551
    73     7.621521e+00     4.825247e-02
 * time: 0.3981039524078369
    74     7.621401e+00     6.166178e-02
 * time: 0.4238569736480713
    75     7.621026e+00     7.264602e-02
 * time: 0.42705202102661133
    76     7.620596e+00     4.512260e-02
 * time: 0.4309418201446533
    77     7.620242e+00     4.078696e-02
 * time: 0.4354228973388672
    78     7.619790e+00     6.124635e-02
 * time: 0.4388718605041504
    79     7.619088e+00     2.700327e-02
 * time: 0.4429128170013428
    80     7.618657e+00     3.571521e-02
 * time: 0.44611597061157227
    81     7.618261e+00     3.572477e-02
 * time: 0.44927287101745605
    82     7.617977e+00     3.869710e-02
 * time: 0.4524409770965576
    83     7.617864e+00     5.379989e-02
 * time: 0.4558110237121582
    84     7.617623e+00     3.964675e-02
 * time: 0.45918703079223633
    85     7.617367e+00     5.980302e-02
 * time: 0.46233701705932617
    86     7.617175e+00     5.003426e-02
 * time: 0.4653019905090332
    87     7.616841e+00     3.962685e-02
 * time: 0.4692239761352539
    88     7.616767e+00     6.209490e-02
 * time: 0.4726858139038086
    89     7.616384e+00     4.306300e-02
 * time: 0.4757578372955322
    90     7.616216e+00     5.521380e-02
 * time: 0.4792978763580322
    91     7.615980e+00     4.927200e-02
 * time: 0.48257899284362793
    92     7.615740e+00     3.535453e-02
 * time: 0.48575401306152344
    93     7.615676e+00     5.091764e-02
 * time: 0.4891040325164795
    94     7.615635e+00     5.922005e-02
 * time: 0.4926269054412842
    95     7.615382e+00     3.365309e-02
 * time: 0.49715185165405273
    96     7.615225e+00     4.059725e-02
 * time: 0.5007200241088867
    97     7.614919e+00     3.376194e-02
 * time: 0.5052328109741211
    98     7.614778e+00     6.411183e-02
 * time: 0.5083239078521729
    99     7.614484e+00     3.135854e-02
 * time: 0.513124942779541
   100     7.614367e+00     3.038978e-02
 * time: 0.5160679817199707
   101     7.614215e+00     3.256369e-02
 * time: 0.5201749801635742
   102     7.614127e+00     3.432930e-02
 * time: 0.5234129428863525
   103     7.614009e+00     3.052841e-02
 * time: 0.526453971862793
   104     7.613897e+00     4.097112e-02
 * time: 0.5294768810272217
   105     7.613811e+00     3.884824e-02
 * time: 0.5325348377227783
   106     7.613644e+00     3.899195e-02
 * time: 0.5359549522399902
   107     7.613528e+00     3.106242e-02
 * time: 0.5393610000610352
   108     7.613366e+00     3.288238e-02
 * time: 0.5520198345184326
   109     7.613356e+00     4.421898e-02
 * time: 0.555513858795166
   110     7.613263e+00     3.957131e-02
 * time: 0.5582869052886963
   111     7.613135e+00     4.413959e-02
 * time: 0.5611448287963867
   112     7.613004e+00     2.652923e-02
 * time: 0.5642340183258057
   113     7.612972e+00     4.891870e-02
 * time: 0.5671529769897461
   114     7.612966e+00     3.703895e-02
 * time: 0.5701429843902588
   115     7.612914e+00     4.892153e-02
 * time: 0.5734758377075195
   116     7.612712e+00     2.989786e-02
 * time: 0.5783460140228271
   117     7.612522e+00     4.287837e-02
 * time: 0.5811920166015625
   118     7.612293e+00     2.162952e-02
 * time: 0.5850579738616943
   119     7.612208e+00     3.290756e-02
 * time: 0.5880539417266846
   120     7.612017e+00     2.568855e-02
 * time: 0.5917940139770508
   121     7.611898e+00     2.371954e-02
 * time: 0.5965328216552734
   122     7.611759e+00     1.890072e-02
 * time: 0.6005599498748779
   123     7.611732e+00     3.422119e-02
 * time: 0.6039919853210449
   124     7.611525e+00     3.120028e-02
 * time: 0.6076958179473877
   125     7.611297e+00     2.547554e-02
 * time: 0.6114048957824707
   126     7.611059e+00     2.431897e-02
 * time: 0.6146559715270996
   127     7.610825e+00     3.625206e-02
 * time: 0.6173849105834961
   128     7.610526e+00     3.781822e-02
 * time: 0.6200888156890869
   129     7.610417e+00     4.893061e-02
 * time: 0.6231808662414551
   130     7.610247e+00     3.874295e-02
 * time: 0.6271228790283203
   131     7.610010e+00     2.800011e-02
 * time: 0.6302328109741211
   132     7.609868e+00     3.540337e-02
 * time: 0.6333780288696289
   133     7.609769e+00     3.460240e-02
 * time: 0.6366488933563232
   134     7.609707e+00     3.322852e-02
 * time: 0.639807939529419
   135     7.609559e+00     3.685286e-02
 * time: 0.6437070369720459
   136     7.609443e+00     2.940410e-02
 * time: 0.6479809284210205
   137     7.609228e+00     2.176114e-02
 * time: 0.6512179374694824
   138     7.609060e+00     1.422011e-02
 * time: 0.6555449962615967
   139     7.608874e+00     3.141634e-02
 * time: 0.6586589813232422
   140     7.608725e+00     1.458045e-02
 * time: 0.6625618934631348
   141     7.608650e+00     2.319127e-02
 * time: 0.6657910346984863
   142     7.608526e+00     2.548996e-02
 * time: 0.6800198554992676
   143     7.608524e+00     4.327768e-02
 * time: 0.6832330226898193
   144     7.608452e+00     4.522959e-02
 * time: 0.6863839626312256
   145     7.608287e+00     2.708023e-02
 * time: 0.6904528141021729
   146     7.608241e+00     6.658936e-02
 * time: 0.6933979988098145
   147     7.608022e+00     3.544047e-02
 * time: 0.6973960399627686
   148     7.607789e+00     3.740394e-02
 * time: 0.7004148960113525
   149     7.607629e+00     3.391425e-02
 * time: 0.7035799026489258
   150     7.607389e+00     1.588451e-02
 * time: 0.7068350315093994
   151     7.607292e+00     4.355173e-02
 * time: 0.7101099491119385
   152     7.607093e+00     2.850758e-02
 * time: 0.7141568660736084
   153     7.606998e+00     3.101312e-02
 * time: 0.71744704246521
   154     7.606855e+00     2.594400e-02
 * time: 0.7215099334716797
   155     7.606778e+00     3.635407e-02
 * time: 0.7245399951934814
   156     7.606642e+00     3.450853e-02
 * time: 0.7277638912200928
   157     7.606606e+00     4.402602e-02
 * time: 0.7312829494476318
   158     7.606565e+00     4.094055e-02
 * time: 0.7343778610229492
   159     7.606487e+00     3.704301e-02
 * time: 0.7376339435577393
   160     7.606447e+00     3.864993e-02
 * time: 0.7407138347625732
   161     7.606254e+00     3.042315e-02
 * time: 0.7438130378723145
   162     7.606229e+00     4.218392e-02
 * time: 0.7468860149383545
   163     7.606210e+00     4.565703e-02
 * time: 0.7499790191650391
   164     7.606082e+00     3.075004e-02
 * time: 0.7540948390960693
   165     7.605982e+00     3.135226e-02
 * time: 0.7572829723358154
   166     7.605899e+00     2.052771e-02
 * time: 0.7611088752746582
   167     7.605870e+00     3.435161e-02
 * time: 0.7646918296813965
   168     7.605857e+00     2.972782e-02
 * time: 0.7677459716796875
   169     7.605777e+00     3.100541e-02
 * time: 0.7715909481048584
   170     7.605751e+00     3.656217e-02
 * time: 0.7746119499206543
   171     7.605658e+00     2.264689e-02
 * time: 0.7791159152984619
   172     7.605576e+00     2.423025e-02
 * time: 0.7824640274047852
   173     7.605512e+00     1.990331e-02
 * time: 0.7864558696746826
   174     7.605415e+00     1.215912e-02
 * time: 0.7904329299926758
   175     7.605395e+00     1.719230e-02
 * time: 0.7936129570007324
   176     7.605321e+00     1.330115e-02
 * time: 0.8070259094238281
   177     7.605261e+00     1.605919e-02
 * time: 0.8099079132080078
   178     7.605194e+00     1.836498e-02
 * time: 0.8136579990386963
   179     7.605116e+00     1.253004e-02
 * time: 0.8174679279327393
   180     7.605061e+00     1.889569e-02
 * time: 0.8203849792480469
   181     7.604989e+00     1.015175e-02
 * time: 0.8244638442993164
   182     7.604983e+00     1.964520e-02
 * time: 0.8276598453521729
   183     7.604942e+00     2.085835e-02
 * time: 0.8304719924926758
   184     7.604933e+00     2.871454e-02
 * time: 0.8337969779968262
   185     7.604889e+00     2.583029e-02
 * time: 0.8373029232025146
   186     7.604789e+00     1.900794e-02
 * time: 0.8414249420166016
   187     7.604787e+00     3.497923e-02
 * time: 0.8445630073547363
   188     7.604720e+00     2.842328e-02
 * time: 0.8480749130249023
   189     7.604665e+00     2.371401e-02
 * time: 0.851902961730957
   190     7.604607e+00     1.773186e-02
 * time: 0.8560309410095215
   191     7.604595e+00     2.132535e-02
 * time: 0.8593709468841553
   192     7.604542e+00     1.619714e-02
 * time: 0.8625249862670898
   193     7.604530e+00     1.977082e-02
 * time: 0.8656859397888184
   194     7.604480e+00     2.092404e-02
 * time: 0.8688819408416748
   195     7.604431e+00     1.956931e-02
 * time: 0.872553825378418
   196     7.604417e+00     2.387712e-02
 * time: 0.8758499622344971
   197     7.604361e+00     1.667128e-02
 * time: 0.8790678977966309
   198     7.604326e+00     1.974933e-02
 * time: 0.8821408748626709
   199     7.604290e+00     1.607898e-02
 * time: 0.8854408264160156
   200     7.604252e+00     1.818742e-02
 * time: 0.8895080089569092
   201     7.604226e+00     1.522552e-02
 * time: 0.8935248851776123
   202     7.604179e+00     1.174459e-02
 * time: 0.8975989818572998
   203     7.604165e+00     1.214319e-02
 * time: 0.90059494972229
   204     7.604129e+00     9.167667e-03
 * time: 0.9044218063354492
   205     7.604113e+00     1.133269e-02
 * time: 0.9074578285217285
   206     7.604106e+00     1.051147e-02
 * time: 0.9108128547668457
   207     7.604100e+00     2.228661e-02
 * time: 0.9138460159301758
   208     7.604097e+00     1.375321e-02
 * time: 0.9168038368225098
   209     7.604074e+00     1.935243e-02
 * time: 0.9298269748687744
   210     7.604052e+00     1.292436e-02
 * time: 0.9326198101043701
   211     7.604009e+00     1.016409e-02
 * time: 0.9358279705047607
   212     7.603977e+00     1.082781e-02
 * time: 0.9403009414672852
   213     7.603928e+00     1.321790e-02
 * time: 0.9434828758239746
   214     7.603912e+00     2.336417e-02
 * time: 0.9464948177337646
   215     7.603850e+00     1.048327e-02
 * time: 0.9497039318084717
   216     7.603811e+00     1.478139e-02
 * time: 0.9536178112030029
   217     7.603777e+00     1.097894e-02
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   685     7.600933e+00     5.617320e-04
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   687     7.600933e+00     6.341229e-04
 * time: 2.6933319568634033
   688     7.600933e+00     4.753666e-04
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   689     7.600933e+00     7.186152e-04
 * time: 2.7003040313720703
   690     7.600933e+00     3.902347e-04
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   691     7.600933e+00     5.295787e-04
 * time: 2.707223892211914
   692     7.600933e+00     6.316781e-04
 * time: 2.710361957550049
   693     7.600933e+00     3.645658e-04
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   694     7.600933e+00     2.301135e-04
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   695     7.600933e+00     3.117412e-04
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   696     7.600933e+00     4.562487e-04
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   697     7.600933e+00     2.322347e-04
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   698     7.600933e+00     3.596725e-04
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   699     7.600933e+00     3.003242e-04
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   700     7.600933e+00     3.720579e-04
 * time: 2.7366199493408203
   701     7.600933e+00     3.869649e-04
 * time: 2.7398879528045654
   702     7.600933e+00     6.380513e-04
 * time: 2.7426559925079346
   703     7.600933e+00     3.600823e-04
 * time: 2.7464640140533447
   704     7.600933e+00     3.737159e-04
 * time: 2.7496418952941895
   705     7.600933e+00     5.017782e-04
 * time: 2.752591848373413
   706     7.600933e+00     2.915398e-04
 * time: 2.7561700344085693
   707     7.600933e+00     3.284832e-04
 * time: 2.759058952331543
   708     7.600933e+00     3.471458e-04
 * time: 2.76190185546875
Out[5]: