Anyonic models

We solve the almost-bosonic anyon model of https://arxiv.org/pdf/1901.10739.pdf

In [1]:
using DFTK
using StaticArrays
using Plots

# Unit cell. Having one of the lattice vectors as zero means a 2D system
a = 14
lattice = a .* [[1 0 0.]; [0 1 0]; [0 0 0]];

# Confining scalar potential
pot(x, y, z) = ((x - a/2)^2 + (y - a/2)^2)

# Parameters
Ecut = 50
n_electrons = 1
β = 5

terms = [Kinetic(; scaling_factor=2),
         ExternalFromReal(X -> pot(X...)),
         Anyonic(1, β)
]
model = Model(lattice; n_electrons, terms, spin_polarization=:spinless)  # "spinless electrons"
basis = PlaneWaveBasis(model; Ecut, kgrid=(1, 1, 1))
scfres = direct_minimization(basis, tol=1e-14)  # Reduce tol for production
E = scfres.energies.total
s = 2
E11 = π/2 * (2(s+1)/s)^((s+2)/s) * (s/(s+2))^(2(s+1)/s) * E^((s+2)/s) / β
println("e(1,1) / (2π)= ", E11 / (2π))
display(heatmap(scfres.ρ[:, :, 1, 1], c=:blues))
Iter     Function value   Gradient norm 
     0     8.410347e+01     1.624468e+01
 * time: 0.003039836883544922
     1     6.368490e+01     1.253134e+01
 * time: 0.008939981460571289
     2     5.741571e+01     1.676497e+01
 * time: 0.022531986236572266
     3     4.275783e+01     1.139997e+01
 * time: 0.0408778190612793
     4     3.487506e+01     1.017819e+01
 * time: 0.05848383903503418
     5     1.490009e+01     3.157736e+00
 * time: 0.14609384536743164
     6     1.138340e+01     3.520194e+00
 * time: 0.15999794006347656
     7     9.900684e+00     2.191030e+00
 * time: 0.17305684089660645
     8     9.126685e+00     3.320207e+00
 * time: 0.18350481986999512
     9     8.811054e+00     3.575682e+00
 * time: 0.19388580322265625
    10     8.504275e+00     3.066914e+00
 * time: 0.20434999465942383
    11     8.062185e+00     1.735209e+00
 * time: 0.21474194526672363
    12     7.559192e+00     1.312268e+00
 * time: 0.22731399536132812
    13     7.172199e+00     1.502411e+00
 * time: 0.23790788650512695
    14     6.895554e+00     1.094797e+00
 * time: 0.24822092056274414
    15     6.719928e+00     8.923590e-01
 * time: 0.2929229736328125
    16     6.607344e+00     1.118035e+00
 * time: 0.3041698932647705
    17     6.502045e+00     7.087406e-01
 * time: 0.3154029846191406
    18     6.349434e+00     7.500807e-01
 * time: 0.32590389251708984
    19     6.119833e+00     7.781240e-01
 * time: 0.3365139961242676
    20     5.941530e+00     9.128185e-01
 * time: 0.3469219207763672
    21     5.830842e+00     6.289483e-01
 * time: 0.35698390007019043
    22     5.767541e+00     5.504176e-01
 * time: 0.36728882789611816
    23     5.757317e+00     1.224940e+00
 * time: 0.3751859664916992
    24     5.724518e+00     7.870377e-01
 * time: 0.3831508159637451
    25     5.714197e+00     6.780425e-01
 * time: 0.3911128044128418
    26     5.697801e+00     7.782283e-01
 * time: 0.42572784423828125
    27     5.677513e+00     5.120291e-01
 * time: 0.43662190437316895
    28     5.661575e+00     4.481369e-01
 * time: 0.4474318027496338
    29     5.650541e+00     6.530572e-01
 * time: 0.45535778999328613
    30     5.624435e+00     5.653876e-01
 * time: 0.46305394172668457
    31     5.608646e+00     3.829223e-01
 * time: 0.4734818935394287
    32     5.600703e+00     4.283360e-01
 * time: 0.4841279983520508
    33     5.592613e+00     3.799936e-01
 * time: 0.4950439929962158
    34     5.590145e+00     3.047189e-01
 * time: 0.5028908252716064
    35     5.581184e+00     2.546569e-01
 * time: 0.5114209651947021
    36     5.575281e+00     2.352586e-01
 * time: 0.5197818279266357
    37     5.571052e+00     2.531070e-01
 * time: 0.5276718139648438
    38     5.568984e+00     2.161152e-01
 * time: 0.5356478691101074
    39     5.567733e+00     1.375050e-01
 * time: 0.5613539218902588
    40     5.566603e+00     1.622589e-01
 * time: 0.569861888885498
    41     5.565739e+00     1.315509e-01
 * time: 0.5783689022064209
    42     5.564717e+00     1.945912e-01
 * time: 0.5863549709320068
    43     5.564227e+00     1.554160e-01
 * time: 0.5944509506225586
    44     5.563492e+00     1.274773e-01
 * time: 0.6049208641052246
    45     5.562688e+00     9.623832e-02
 * time: 0.6153128147125244
    46     5.562032e+00     1.202397e-01
 * time: 0.6230208873748779
    47     5.561544e+00     7.628104e-02
 * time: 0.6310157775878906
    48     5.561288e+00     8.709403e-02
 * time: 0.6388468742370605
    49     5.561095e+00     5.988173e-02
 * time: 0.6469218730926514
    50     5.560959e+00     4.685654e-02
 * time: 0.6572158336639404
    51     5.560826e+00     2.624886e-02
 * time: 0.6827578544616699
    52     5.560742e+00     4.890421e-02
 * time: 0.6913089752197266
    53     5.560731e+00     5.017957e-02
 * time: 0.6994597911834717
    54     5.560645e+00     3.419509e-02
 * time: 0.7072999477386475
    55     5.560596e+00     4.020835e-02
 * time: 0.7176787853240967
    56     5.560555e+00     3.075647e-02
 * time: 0.7279739379882812
    57     5.560543e+00     4.017573e-02
 * time: 0.7355008125305176
    58     5.560533e+00     2.090276e-02
 * time: 0.7430768013000488
    59     5.560515e+00     1.533790e-02
 * time: 0.7506668567657471
    60     5.560502e+00     1.263486e-02
 * time: 0.7585279941558838
    61     5.560487e+00     1.065857e-02
 * time: 0.7664659023284912
    62     5.560478e+00     9.467035e-03
 * time: 0.7745850086212158
    63     5.560473e+00     9.109492e-03
 * time: 0.7848029136657715
    64     5.560471e+00     5.783410e-03
 * time: 0.8103058338165283
    65     5.560469e+00     7.729054e-03
 * time: 0.8187828063964844
    66     5.560468e+00     7.873965e-03
 * time: 0.8270039558410645
    67     5.560468e+00     8.328648e-03
 * time: 0.8348469734191895
    68     5.560467e+00     4.205446e-03
 * time: 0.845383882522583
    69     5.560466e+00     6.067369e-03
 * time: 0.8558318614959717
    70     5.560465e+00     4.957374e-03
 * time: 0.863839864730835
    71     5.560464e+00     3.706725e-03
 * time: 0.8739898204803467
    72     5.560464e+00     2.196989e-03
 * time: 0.8816378116607666
    73     5.560463e+00     1.953301e-03
 * time: 0.891960859298706
    74     5.560463e+00     2.417199e-03
 * time: 0.899634838104248
    75     5.560463e+00     1.450785e-03
 * time: 0.90982985496521
    76     5.560463e+00     1.426738e-03
 * time: 0.9344789981842041
    77     5.560463e+00     1.438616e-03
 * time: 0.9455118179321289
    78     5.560463e+00     9.532110e-04
 * time: 0.9561269283294678
    79     5.560463e+00     1.237838e-03
 * time: 0.9640858173370361
    80     5.560463e+00     7.899421e-04
 * time: 0.9721689224243164
    81     5.560463e+00     7.256114e-04
 * time: 0.9830179214477539
    82     5.560463e+00     5.471578e-04
 * time: 0.9907839298248291
    83     5.560463e+00     2.145598e-04
 * time: 1.0010159015655518
    84     5.560463e+00     4.118338e-04
 * time: 1.008711814880371
    85     5.560463e+00     2.278516e-04
 * time: 1.016535997390747
    86     5.560463e+00     1.909337e-04
 * time: 1.026641845703125
    87     5.560463e+00     2.276338e-04
 * time: 1.036787986755371
e(1,1) / (2π)= 1.7391793874123116