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]:
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
Iter     Function value   Gradient norm 
     0     1.997247e+02     1.990093e+02
 * time: 0.0004818439483642578
     1     1.667603e+02     1.200257e+02
 * time: 0.0011508464813232422
     2     1.285001e+02     1.325640e+02
 * time: 0.0022268295288085938
     3     1.243423e+02     1.231108e+02
 * time: 0.003423929214477539
     4     8.649211e+01     1.207772e+02
 * time: 0.004788875579833984
     5     4.160952e+01     7.471932e+01
 * time: 0.006085872650146484
     6     1.570836e+01     3.726758e+01
 * time: 0.00716090202331543
     7     1.079337e+01     3.573691e+01
 * time: 0.007951021194458008
     8     5.886282e+00     1.826327e+01
 * time: 0.008810997009277344
     9     2.334631e+00     1.534672e+00
 * time: 0.009779930114746094
    10     2.213391e+00     4.461903e+00
 * time: 0.01041102409362793
    11     1.684217e+00     2.321502e+00
 * time: 0.011214017868041992
    12     1.461767e+00     3.713769e+00
 * time: 0.011807918548583984
    13     1.317962e+00     1.902135e+00
 * time: 0.012418031692504883
    14     1.269853e+00     1.091392e+00
 * time: 0.013106822967529297
    15     1.192025e+00     6.664404e-01
 * time: 0.013830900192260742
    16     1.156935e+00     3.388864e-01
 * time: 0.01458883285522461
    17     1.148640e+00     2.612526e-01
 * time: 0.015271902084350586
    18     1.145471e+00     2.996803e-01
 * time: 0.01587390899658203
    19     1.144421e+00     1.775842e-01
 * time: 0.016502857208251953
    20     1.144251e+00     5.194910e-02
 * time: 0.017027854919433594
    21     1.144174e+00     4.598001e-02
 * time: 0.017550945281982422
    22     1.144093e+00     3.709757e-02
 * time: 0.01812601089477539
    23     1.144063e+00     2.093214e-02
 * time: 0.01867389678955078
    24     1.144043e+00     7.320245e-03
 * time: 0.019235849380493164
    25     1.144041e+00     5.549434e-03
 * time: 0.019783973693847656
    26     1.144038e+00     5.757830e-03
 * time: 0.0203549861907959
    27     1.144037e+00     3.288540e-03
 * time: 0.021052837371826172
    28     1.144037e+00     1.895382e-03
 * time: 0.021615982055664062
    29     1.144037e+00     2.266862e-03
 * time: 0.022137880325317383
    30     1.144037e+00     1.813715e-03
 * time: 0.02275991439819336
    31     1.144037e+00     8.514685e-04
 * time: 0.02335500717163086
    32     1.144037e+00     3.814972e-04
 * time: 0.024024009704589844
    33     1.144037e+00     2.450873e-04
 * time: 0.024695873260498047
    34     1.144037e+00     1.428968e-04
 * time: 0.02536296844482422
    35     1.144037e+00     9.445069e-05
 * time: 0.02596592903137207
    36     1.144037e+00     8.213933e-05
 * time: 0.026525020599365234
    37     1.144037e+00     9.655909e-05
 * time: 0.027164936065673828
    38     1.144037e+00     8.344206e-05
 * time: 0.027835845947265625
    39     1.144037e+00     5.819665e-05
 * time: 0.02857184410095215
    40     1.144037e+00     1.857161e-05
 * time: 0.029340028762817383
    41     1.144037e+00     6.306767e-06
 * time: 0.03010082244873047
    42     1.144037e+00     6.215129e-06
 * time: 0.030841827392578125
    43     1.144037e+00     4.610874e-06
 * time: 0.03146982192993164
    44     1.144037e+00     4.565230e-06
 * time: 0.03209996223449707
    45     1.144037e+00     3.655589e-06
 * time: 0.032753944396972656
    46     1.144037e+00     2.403522e-06
 * time: 0.0334470272064209
    47     1.144037e+00     1.833121e-06
 * time: 0.03405880928039551
    48     1.144037e+00     8.700688e-07
 * time: 0.034626007080078125
    49     1.144037e+00     3.931458e-07
 * time: 0.03518199920654297
    50     1.144037e+00     2.198528e-07
 * time: 0.03572487831115723
    51     1.144037e+00     1.875905e-07
 * time: 0.03631091117858887
    52     1.144037e+00     1.184772e-07
 * time: 0.036966800689697266
    53     1.144037e+00     4.549511e-08
 * time: 0.03764200210571289
Out[5]:
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:

In [6]:
ρ = real(scfres.ρ.real)[:, 1, 1]  # converged density
ψ_fourier = scfres.ψ[1][:, 1];    # first kpoint, 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.real - abs2.(ψ))
Out[9]:
9.419154611360628e-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 kpoints. Here, we just have one kpoint:

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 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:

In [14]:
norm(H * ψ11 - dot(ψ11, H * ψ11) * ψ11)
Out[14]:
8.228242185673861e-8

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]:
0.00022342916312165066