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     1.765089e+02     1.182079e+02
 * time: 0.0006511211395263672
     1     1.708323e+02     9.753269e+01
 * time: 0.0019850730895996094
     2     1.256698e+02     9.437160e+01
 * time: 0.004372119903564453
     3     1.076115e+02     8.272244e+01
 * time: 0.007039070129394531
     4     4.933072e+01     6.689338e+01
 * time: 0.009698152542114258
     5     3.165150e+01     4.701766e+01
 * time: 0.012061119079589844
     6     1.017234e+01     9.863353e+00
 * time: 0.01418614387512207
     7     1.013702e+01     1.736950e+01
 * time: 0.014991998672485352
     8     6.807519e+00     7.913527e+00
 * time: 0.016422033309936523
     9     3.305763e+00     3.922085e+00
 * time: 0.018199920654296875
    10     2.527361e+00     9.239986e+00
 * time: 0.019628047943115234
    11     1.819008e+00     4.401821e+00
 * time: 0.021079063415527344
    12     1.653479e+00     2.715344e+00
 * time: 0.022576093673706055
    13     1.491605e+00     1.672313e+00
 * time: 0.02370309829711914
    14     1.436884e+00     2.405348e+00
 * time: 0.024821996688842773
    15     1.311629e+00     1.844782e+00
 * time: 0.026051998138427734
    16     1.232842e+00     1.205995e+00
 * time: 0.027163982391357422
    17     1.226469e+00     7.379742e-01
 * time: 0.02796006202697754
    18     1.185173e+00     5.690920e-01
 * time: 0.02908015251159668
    19     1.151745e+00     2.702299e-01
 * time: 0.030213117599487305
    20     1.146948e+00     2.121817e-01
 * time: 0.03132009506225586
    21     1.144795e+00     1.357340e-01
 * time: 0.03242802619934082
    22     1.144300e+00     3.863231e-02
 * time: 0.03355813026428223
    23     1.144113e+00     1.533640e-02
 * time: 0.03465700149536133
    24     1.144099e+00     1.431367e-02
 * time: 0.03576207160949707
    25     1.144072e+00     1.131954e-02
 * time: 0.03686714172363281
    26     1.144058e+00     1.158915e-02
 * time: 0.03799796104431152
    27     1.144042e+00     5.220822e-03
 * time: 0.03909802436828613
    28     1.144039e+00     3.325529e-03
 * time: 0.040216922760009766
    29     1.144038e+00     3.065392e-03
 * time: 0.041355133056640625
    30     1.144037e+00     1.240680e-03
 * time: 0.04250812530517578
    31     1.144037e+00     1.036939e-03
 * time: 0.04366898536682129
    32     1.144037e+00     9.482615e-04
 * time: 0.04481196403503418
    33     1.144037e+00     7.319115e-04
 * time: 0.0459599494934082
    34     1.144037e+00     3.303911e-04
 * time: 0.047070980072021484
    35     1.144037e+00     2.160884e-04
 * time: 0.04818296432495117
    36     1.144037e+00     1.803334e-04
 * time: 0.049317121505737305
    37     1.144037e+00     1.028342e-04
 * time: 0.05043292045593262
    38     1.144037e+00     7.265890e-05
 * time: 0.051553964614868164
    39     1.144037e+00     4.889936e-05
 * time: 0.052670955657958984
    40     1.144037e+00     2.569856e-05
 * time: 0.05381011962890625
    41     1.144037e+00     1.637467e-05
 * time: 0.05491495132446289
    42     1.144037e+00     1.474808e-05
 * time: 0.05602693557739258
    43     1.144037e+00     9.318347e-06
 * time: 0.057138919830322266
    44     1.144037e+00     5.161005e-06
 * time: 0.05825495719909668
    45     1.144037e+00     3.368800e-06
 * time: 0.05934596061706543
    46     1.144037e+00     2.132506e-06
 * time: 0.06044912338256836
    47     1.144037e+00     1.623603e-06
 * time: 0.061231136322021484
    48     1.144037e+00     1.731472e-06
 * time: 0.06238102912902832
    49     1.144037e+00     8.606646e-07
 * time: 0.06348395347595215
    50     1.144037e+00     4.209374e-07
 * time: 0.0645899772644043
    51     1.144037e+00     3.540226e-07
 * time: 0.06574797630310059
    52     1.144037e+00     1.692595e-07
 * time: 0.06686806678771973
    53     1.144037e+00     2.020204e-07
 * time: 0.06765007972717285
    54     1.144037e+00     9.579822e-08
 * time: 0.06843113899230957
    55     1.144037e+00     1.073623e-07
 * time: 0.06955504417419434
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 = 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]:
1.161130389502071e-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]:
1.321459555244377e-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]:
0.00022342332015932995