# Gross-Pitaevskii equation in one dimension

In this example we will use DFTK to solve the Gross-Pitaevskii equation, and use this opportunity to explore a few internals.

## The model

The Gross-Pitaevskii equation (GPE) is a simple non-linear equation used to model bosonic systems in a mean-field approach. Denoting by $ψ$ the effective one-particle bosonic wave function, the time-independent GPE reads in atomic units:

\[ H ψ = \left(-\frac12 Δ + V + 2 C |ψ|^2\right) ψ = μ ψ \qquad \|ψ\|_{L^2} = 1\]

where $C$ provides the strength of the boson-boson coupling. It's in particular a favorite model of applied mathematicians because it has a structure simpler than but similar to that of DFT, and displays interesting behavior (especially in higher dimensions with magnetic fields, see Gross-Pitaevskii equation with magnetism).

We wish to model this equation in 1D using DFTK. First we set up the lattice. For a 1D case we supply two zero lattice vectors,

```
a = 10
lattice = a .* [[1 0 0.]; [0 0 0]; [0 0 0]];
```

which is special cased in DFTK to support 1D models.

For the potential term `V`

we just pick a harmonic potential. The real-space grid is in $[0,1)$ in fractional coordinates( see Lattices and lattice vectors), therefore:

`pot(x) = (x - a/2)^2;`

We setup each energy term in sequence: kinetic, potential and nonlinear term. For the non-linearity we use the `PowerNonlinearity(C, α)`

term of DFTK. This object introduces an energy term $C ∫ ρ(r)^α dr$ to the total energy functional, thus a potential term $α C ρ^{α-1}$. In our case we thus need the parameters

```
C = 1.0
α = 2;
```

... and with this build the model

```
using DFTK
using LinearAlgebra
n_electrons = 1 # Increase this for fun
terms = [Kinetic(),
ExternalFromReal(r -> pot(r[1])),
PowerNonlinearity(C, α),
]
model = Model(lattice; n_electrons=n_electrons, terms=terms,
spin_polarization=:spinless); # use "spinless electrons"
```

We discretize using a moderate Ecut (For 1D values up to `5000`

are completely fine) and run a direct minimization algorithm:

```
Ecut = 500
basis = PlaneWaveBasis(model, Ecut, kgrid=(1, 1, 1))
scfres = direct_minimization(basis, tol=1e-8) # This is a constrained preconditioned LBFGS
scfres.energies
```

Energy breakdown: Kinetic 0.2682057 ExternalFromReal 0.4707475 PowerNonlinearity 0.4050836 total 1.144036852755

## Internals

We use the opportunity to explore some of DFTK internals.

Extract the converged density and the obtained wave function:

```
ρ = real(scfres.ρ)[:, 1, 1, 1] # converged density, first spin component
ψ_fourier = scfres.ψ[1][:, 1]; # first kpoint, all G components, first eigenvector
```

Transform the wave function to real space and fix the phase:

```
ψ = G_to_r(basis, basis.kpoints[1], ψ_fourier)[:, 1, 1]
ψ /= (ψ[div(end, 2)] / abs(ψ[div(end, 2)]));
```

Check whether $ψ$ is normalised:

```
x = a * vec(first.(DFTK.r_vectors(basis)))
N = length(x)
dx = a / N # real-space grid spacing
@assert sum(abs2.(ψ)) * dx ≈ 1.0
```

The density is simply built from ψ:

`norm(scfres.ρ - abs2.(ψ))`

8.529998795425379e-16

We summarize the ground state in a nice plot:

```
using Plots
p = plot(x, real.(ψ), label="real(ψ)")
plot!(p, x, imag.(ψ), label="imag(ψ)")
plot!(p, x, ρ, label="ρ")
```

The `energy_hamiltonian`

function can be used to get the energy and effective Hamiltonian (derivative of the energy with respect to the density matrix) of a particular state (ψ, occupation). The density ρ associated to this state is precomputed and passed to the routine as an optimization.

```
E, ham = energy_hamiltonian(basis, scfres.ψ, scfres.occupation; ρ=scfres.ρ)
@assert E.total == scfres.energies.total
```

Now the Hamiltonian contains all the blocks corresponding to kpoints. Here, we just have one kpoint:

`H = ham.blocks[1];`

`H`

can be used as a linear operator (efficiently using FFTs), or converted to a dense matrix:

```
ψ11 = scfres.ψ[1][:, 1] # first kpoint, first eigenvector
Hmat = Array(H) # This is now just a plain Julia matrix,
# which we can compute and store in this simple 1D example
@assert norm(Hmat * ψ11 - H * ψ11) < 1e-10
```

Let's check that ψ11 is indeed an eigenstate:

`norm(H * ψ11 - dot(ψ11, H * ψ11) * ψ11)`

3.8678285122743665e-7

Build a finite-differences version of the GPE operator $H$, as a sanity check:

```
A = Array(Tridiagonal(-ones(N - 1), 2ones(N), -ones(N - 1)))
A[1, end] = A[end, 1] = -1
K = A / dx^2 / 2
V = Diagonal(pot.(x) + C .* α .* (ρ.^(α-1)))
H_findiff = K + V;
maximum(abs.(H_findiff*ψ - (dot(ψ, H_findiff*ψ) / dot(ψ, ψ)) * ψ))
```

0.00022337797377200804