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,

In [1]:
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:

In [2]:
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

In [3]:
C = 1.0
α = 2;

... and with this build the model

In [4]:
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:

In [5]:
basis = PlaneWaveBasis(model, Ecut=500, kgrid=(1, 1, 1))
scfres = direct_minimization(basis, tol=1e-8) # This is a constrained preconditioned LBFGS
scfres.energies
Iter     Function value   Gradient norm 
     0     1.676296e+02     1.569304e+02
 * time: 0.0005040168762207031
     1     1.569729e+02     1.172376e+02
 * time: 0.002096891403198242
     2     1.154506e+02     1.278276e+02
 * time: 0.004133939743041992
     3     6.027551e+01     9.631214e+01
 * time: 0.006357908248901367
     4     5.432612e+01     9.923688e+01
 * time: 0.0083160400390625
     5     4.936006e+01     8.955212e+01
 * time: 0.010009050369262695
     6     3.116001e+01     7.705015e+01
 * time: 0.011693954467773438
     7     5.906720e+00     1.845137e+01
 * time: 0.01340484619140625
     8     3.246261e+00     5.803628e+00
 * time: 0.014837980270385742
     9     2.275165e+00     2.493780e+00
 * time: 0.016015052795410156
    10     1.880706e+00     2.604682e+00
 * time: 0.017187833786010742
    11     1.413685e+00     1.822961e+00
 * time: 0.018396854400634766
    12     1.283332e+00     1.781974e+00
 * time: 0.019362926483154297
    13     1.197557e+00     1.066727e+00
 * time: 0.02030801773071289
    14     1.179234e+00     8.486206e-01
 * time: 0.02121591567993164
    15     1.159120e+00     5.777676e-01
 * time: 0.02213001251220703
    16     1.149098e+00     2.456666e-01
 * time: 0.02305293083190918
    17     1.146033e+00     1.201687e-01
 * time: 0.023970842361450195
    18     1.144499e+00     7.030165e-02
 * time: 0.024890899658203125
    19     1.144197e+00     3.677684e-02
 * time: 0.025542020797729492
    20     1.144096e+00     3.173746e-02
 * time: 0.026453018188476562
    21     1.144067e+00     1.611968e-02
 * time: 0.027362823486328125
    22     1.144053e+00     1.210524e-02
 * time: 0.028301000595092773
    23     1.144049e+00     1.177349e-02
 * time: 0.028949975967407227
    24     1.144043e+00     7.191353e-03
 * time: 0.029866933822631836
    25     1.144039e+00     3.940658e-03
 * time: 0.030772924423217773
    26     1.144038e+00     2.952111e-03
 * time: 0.031685829162597656
    27     1.144037e+00     1.770095e-03
 * time: 0.032614946365356445
    28     1.144037e+00     9.939459e-04
 * time: 0.033531904220581055
    29     1.144037e+00     6.624174e-04
 * time: 0.034442901611328125
    30     1.144037e+00     5.483406e-04
 * time: 0.035347938537597656
    31     1.144037e+00     4.632020e-04
 * time: 0.03628182411193848
    32     1.144037e+00     3.189801e-04
 * time: 0.03719282150268555
    33     1.144037e+00     2.961620e-04
 * time: 0.03810000419616699
    34     1.144037e+00     1.536009e-04
 * time: 0.0390169620513916
    35     1.144037e+00     9.471999e-05
 * time: 0.03993082046508789
    36     1.144037e+00     5.617899e-05
 * time: 0.04095101356506348
    37     1.144037e+00     1.796171e-05
 * time: 0.04186201095581055
    38     1.144037e+00     1.343504e-05
 * time: 0.04284191131591797
    39     1.144037e+00     6.546055e-06
 * time: 0.04375100135803223
    40     1.144037e+00     4.383270e-06
 * time: 0.04468488693237305
    41     1.144037e+00     2.571939e-06
 * time: 0.0455930233001709
    42     1.144037e+00     1.601070e-06
 * time: 0.04649782180786133
    43     1.144037e+00     1.242006e-06
 * time: 0.04749894142150879
    44     1.144037e+00     7.492115e-07
 * time: 0.048429012298583984
    45     1.144037e+00     6.986179e-07
 * time: 0.049340009689331055
    46     1.144037e+00     5.973524e-07
 * time: 0.05034303665161133
    47     1.144037e+00     3.809266e-07
 * time: 0.051242828369140625
    48     1.144037e+00     2.567736e-07
 * time: 0.052201032638549805
Out[5]:
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:

In [6]:
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:

In [7]:
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:

In [8]:
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:

In [9]:
norm(scfres.ρ - abs2.(ψ))
Out[9]:
1.1225937441356955e-15

We summarize the ground state in a nice plot:

In [10]:
using Plots

p = plot(x, real.(ψ), label="real(ψ)")
plot!(p, x, imag.(ψ), label="imag(ψ)")
plot!(p, x, ρ, label="ρ")
Out[10]:

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.

In [11]:
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:

In [12]:
H = ham.blocks[1];

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

In [13]:
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:

In [14]:
norm(H * ψ11 - dot(ψ11, H * ψ11) * ψ11)
Out[14]:
3.322758294724426e-7

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

In [15]:
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))
Out[15]:
0.00022351464256441226