# 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)
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
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9     8.678980e+00     1.114768e+00
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11     8.287403e+00     1.022523e+00
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* time: 0.06507611274719238
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* 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
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* time: 0.09949588775634766
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88     7.622762e+00     5.289825e-02
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Out[5]: