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,

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 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

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])),
         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:

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     1.664563e+02     1.245384e+02
 * time: 0.0007579326629638672
     1     1.580281e+02     1.135388e+02
 * time: 0.0023131370544433594
     2     1.198486e+02     1.287458e+02
 * time: 0.004064083099365234
     3     6.042523e+01     9.207999e+01
 * time: 0.005918025970458984
     4     5.016078e+01     9.286239e+01
 * time: 0.00757908821105957
     5     2.972430e+01     6.374134e+01
 * time: 0.009035110473632812
     6     6.081857e+00     1.610987e+01
 * time: 0.010477066040039062
     7     3.007466e+00     7.342660e+00
 * time: 0.011784076690673828
     8     1.938174e+00     7.306919e+00
 * time: 0.01295614242553711
     9     1.568287e+00     5.298099e+00
 * time: 0.014133930206298828
    10     1.263996e+00     2.558128e+00
 * time: 0.015151023864746094
    11     1.199731e+00     4.901372e-01
 * time: 0.0159299373626709
    12     1.179497e+00     6.225754e-01
 * time: 0.016742944717407227
    13     1.161225e+00     5.140684e-01
 * time: 0.017606019973754883
    14     1.148154e+00     2.737013e-01
 * time: 0.01850295066833496
    15     1.145688e+00     1.038930e-01
 * time: 0.019303083419799805
    16     1.144843e+00     1.288194e-01
 * time: 0.020127058029174805
    17     1.144426e+00     8.611239e-02
 * time: 0.020956039428710938
    18     1.144175e+00     2.451755e-02
 * time: 0.02182602882385254
    19     1.144169e+00     5.498763e-02
 * time: 0.022593021392822266
    20     1.144123e+00     3.892616e-02
 * time: 0.02345108985900879
    21     1.144085e+00     2.242200e-02
 * time: 0.024337053298950195
    22     1.144050e+00     1.772133e-02
 * time: 0.02522897720336914
    23     1.144044e+00     1.230024e-02
 * time: 0.026159048080444336
    24     1.144040e+00     4.246210e-03
 * time: 0.027143001556396484
    25     1.144038e+00     2.730571e-03
 * time: 0.027976036071777344
    26     1.144037e+00     2.240842e-03
 * time: 0.028825044631958008
    27     1.144037e+00     1.018842e-03
 * time: 0.029766082763671875
    28     1.144037e+00     9.278684e-04
 * time: 0.03064894676208496
    29     1.144037e+00     8.922247e-04
 * time: 0.031510114669799805
    30     1.144037e+00     8.167601e-04
 * time: 0.03245210647583008
    31     1.144037e+00     1.826841e-04
 * time: 0.03313612937927246
    32     1.144037e+00     1.456541e-04
 * time: 0.03401303291320801
    33     1.144037e+00     7.334269e-05
 * time: 0.03489208221435547
    34     1.144037e+00     4.135225e-05
 * time: 0.035787105560302734
    35     1.144037e+00     3.542041e-05
 * time: 0.0366971492767334
    36     1.144037e+00     2.301022e-05
 * time: 0.03755807876586914
    37     1.144037e+00     1.692301e-05
 * time: 0.03841996192932129
    38     1.144037e+00     1.301568e-05
 * time: 0.03928208351135254
    39     1.144037e+00     7.260509e-06
 * time: 0.03987598419189453
    40     1.144037e+00     5.555602e-06
 * time: 0.04073500633239746
    41     1.144037e+00     2.896837e-06
 * time: 0.04157304763793945
    42     1.144037e+00     2.316762e-06
 * time: 0.042504072189331055
    43     1.144037e+00     1.993169e-06
 * time: 0.04339408874511719
    44     1.144037e+00     1.722576e-06
 * time: 0.04422712326049805
    45     1.144037e+00     1.178663e-06
 * time: 0.04504704475402832
    46     1.144037e+00     1.008301e-06
 * time: 0.04584097862243652
Out[5]:
Energy breakdown (in Ha):
    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:

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 = 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:

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]:
7.770877854052157e-16

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]:
1.166509247683139e-6

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]:
0.000223573427600602