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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 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     2.015684e+02     1.451541e+02
 * time: 0.0006830692291259766
     1     1.842670e+02     1.117706e+02
 * time: 0.0014410018920898438
     2     1.381893e+02     1.274119e+02
 * time: 0.003370046615600586
     3     3.906053e+01     5.829388e+01
 * time: 0.005837202072143555
     4     1.500294e+01     1.056158e+01
 * time: 0.00799107551574707
     5     1.395365e+01     1.053121e+01
 * time: 0.008944034576416016
     6     8.274507e+00     2.477097e+01
 * time: 0.010503053665161133
     7     8.073489e+00     1.705653e+01
 * time: 0.011342048645019531
     8     5.723847e+00     9.970609e+00
 * time: 0.012673139572143555
     9     3.609341e+00     4.035644e+00
 * time: 0.013773202896118164
    10     2.120041e+00     2.933524e+00
 * time: 0.015034198760986328
    11     1.694492e+00     2.385546e+00
 * time: 0.01613020896911621
    12     1.354819e+00     9.856240e-01
 * time: 0.01720118522644043
    13     1.307600e+00     1.092034e+00
 * time: 0.01797008514404297
    14     1.258360e+00     8.472756e-01
 * time: 0.019118070602416992
    15     1.204909e+00     6.893778e-01
 * time: 0.02001500129699707
    16     1.174138e+00     5.458289e-01
 * time: 0.020847082138061523
    17     1.156438e+00     3.387400e-01
 * time: 0.021685123443603516
    18     1.148475e+00     2.537470e-01
 * time: 0.022644996643066406
    19     1.145530e+00     7.210905e-02
 * time: 0.023485183715820312
    20     1.144664e+00     7.525648e-02
 * time: 0.024079084396362305
    21     1.144361e+00     3.798331e-02
 * time: 0.02500009536743164
    22     1.144250e+00     4.237083e-02
 * time: 0.025930166244506836
    23     1.144154e+00     3.860363e-02
 * time: 0.02681708335876465
    24     1.144097e+00     2.468123e-02
 * time: 0.027675151824951172
    25     1.144055e+00     1.086695e-02
 * time: 0.02866816520690918
    26     1.144046e+00     6.443505e-03
 * time: 0.02957010269165039
    27     1.144040e+00     4.035874e-03
 * time: 0.030518054962158203
    28     1.144038e+00     2.810783e-03
 * time: 0.03145003318786621
    29     1.144037e+00     1.629475e-03
 * time: 0.03235316276550293
    30     1.144037e+00     1.123801e-03
 * time: 0.03327512741088867
    31     1.144037e+00     1.267942e-03
 * time: 0.03391313552856445
    32     1.144037e+00     3.290930e-04
 * time: 0.0345911979675293
    33     1.144037e+00     4.760476e-04
 * time: 0.0352480411529541
    34     1.144037e+00     4.453289e-04
 * time: 0.0362551212310791
    35     1.144037e+00     2.168422e-04
 * time: 0.037188053131103516
    36     1.144037e+00     1.035559e-04
 * time: 0.03815603256225586
    37     1.144037e+00     1.121138e-04
 * time: 0.04375410079956055
    38     1.144037e+00     6.797317e-05
 * time: 0.045060157775878906
    39     1.144037e+00     4.443128e-05
 * time: 0.04605722427368164
    40     1.144037e+00     2.640349e-05
 * time: 0.047057151794433594
    41     1.144037e+00     1.657018e-05
 * time: 0.04814505577087402
    42     1.144037e+00     1.428427e-05
 * time: 0.049221038818359375
    43     1.144037e+00     6.915570e-06
 * time: 0.05090022087097168
    44     1.144037e+00     2.914267e-06
 * time: 0.052781105041503906
    45     1.144037e+00     2.814019e-06
 * time: 0.057882070541381836
    46     1.144037e+00     1.664617e-06
 * time: 0.05901813507080078
    47     1.144037e+00     7.862169e-07
 * time: 0.06042814254760742
    48     1.144037e+00     6.914599e-07
 * time: 0.06145811080932617
    49     1.144037e+00     6.295958e-07
 * time: 0.062177181243896484
    50     1.144037e+00     3.682016e-07
 * time: 0.06310915946960449
    51     1.144037e+00     1.755326e-07
 * time: 0.06403803825378418
    52     1.144037e+00     8.182556e-08
 * time: 0.1143181324005127
    53     1.144037e+00     7.314198e-08
 * time: 0.11549711227416992

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]:

1.1177931243345485e-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]:

1.5505053735266486e-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]:

0.0002234216204111667