# 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

```
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[:, :, 1, 1], c=:blues)
```