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 external magnetic field).

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 pick a harmonic potential. We use the function ExternalFromReal which uses Cartesian coordinates ( see Lattices and lattice vectors).

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 LocalNonlinearity(f) term of DFTK, with f(ρ) = C ρ^α. 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])),
         LocalNonlinearity(ρ -> C * ρ^α),
]
model = Model(lattice; n_electrons, terms, spin_polarization=:spinless);  # spinless electrons

We discretize using a moderate Ecut (For 1D values up to 5000 are completely fine) and run a direct minimization algorithm:

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

Energy breakdown (in Ha):
    Kinetic             0.2682057 
    ExternalFromReal    0.4707475 
    LocalNonlinearity   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 k-point, all G components, first eigenvector

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

ψ = ifft(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))

8.979342025504384e-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 $k$-points. Here, we just have one $k$-point:

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 k-point, 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)

1.0811802705018139e-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))

0.00022341990873611954