# 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
* time: 0.6576230525970459
218     7.597049e+00     3.379008e-02
* time: 0.6597909927368164
219     7.596952e+00     3.413711e-02
* time: 0.6618199348449707
220     7.596793e+00     2.281244e-02
* time: 0.6645081043243408
221     7.596735e+00     4.283948e-02
* time: 0.6665489673614502
222     7.596677e+00     3.536720e-02
* time: 0.6685330867767334
223     7.596594e+00     3.330253e-02
* time: 0.6704990863800049
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Out[5]: