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
using DFTK
using StaticArrays
using Plots
Unit cell. Having one of the lattice vectors as zero means a 2D system
a = 15
lattice = a .* [[1 0 0.]; [0 1 0]; [0 0 0]];
Confining scalar potential, and magnetic vector potential
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
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
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
217 7.607853e+00 3.100870e-02
* time: 0.8411750793457031
218 7.607648e+00 3.496789e-02
* time: 0.8447620868682861
219 7.607497e+00 2.572620e-02
* time: 0.8481550216674805
220 7.607365e+00 2.669537e-02
* time: 0.851499080657959
221 7.607231e+00 4.274631e-02
* time: 0.8541522026062012
222 7.607020e+00 2.798904e-02
* time: 0.8574941158294678
223 7.606865e+00 2.969221e-02
* time: 0.8602981567382812
224 7.606680e+00 2.520878e-02
* time: 0.8638100624084473
225 7.606672e+00 4.955680e-02
* time: 0.8670310974121094
226 7.606559e+00 3.827245e-02
* time: 0.8704161643981934
227 7.606450e+00 3.428008e-02
* time: 0.8734099864959717
228 7.606253e+00 2.355543e-02
* time: 0.876230001449585
229 7.606079e+00 1.852106e-02
* time: 0.8788330554962158
230 7.606008e+00 3.101181e-02
* time: 0.8814911842346191
231 7.605953e+00 2.070094e-02
* time: 0.8840770721435547
232 7.605917e+00 2.917829e-02
* time: 0.8866641521453857
233 7.605821e+00 3.039691e-02
* time: 0.8894500732421875
234 7.605783e+00 2.392698e-02
* time: 0.8921670913696289
235 7.605700e+00 2.815950e-02
* time: 0.8947710990905762
236 7.605576e+00 1.935715e-02
* time: 0.9069721698760986
237 7.605481e+00 1.931860e-02
* time: 0.9096810817718506
238 7.605422e+00 1.531388e-02
* time: 0.9129509925842285
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