We solve the almost-bosonic anyon model of https://arxiv.org/pdf/1901.10739.pdf
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;
# Collect all the terms, build and run the model
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))
heatmap(scfres.ρ[:, :, 1, 1], c=:blues)
Iter Function value Gradient norm 0 8.209483e+01 1.693345e+01 * time: 0.003222942352294922 1 6.286831e+01 1.134613e+01 * time: 0.009351968765258789 2 5.624295e+01 1.598555e+01 * time: 0.022549867630004883 3 4.266253e+01 1.113962e+01 * time: 0.04113483428955078 4 3.542143e+01 1.081219e+01 * time: 0.05989694595336914 5 1.522505e+01 2.954300e+00 * time: 0.0786588191986084 6 1.273767e+01 3.787509e+00 * time: 0.14101195335388184 7 1.034875e+01 2.592299e+00 * time: 0.15454792976379395 8 9.040954e+00 4.116953e+00 * time: 0.1653578281402588 9 8.200656e+00 2.655012e+00 * time: 0.178696870803833 10 7.532824e+00 2.653283e+00 * time: 0.19208693504333496 11 6.914551e+00 1.805517e+00 * time: 0.205427885055542 12 6.868755e+00 3.450654e+00 * time: 0.21621179580688477 13 6.591945e+00 2.162202e+00 * time: 0.22715282440185547 14 6.122937e+00 1.661678e+00 * time: 0.2667689323425293 15 5.695538e+00 1.918827e+00 * time: 0.27773189544677734 16 5.372672e+00 1.465250e+00 * time: 0.2887399196624756 17 5.135286e+00 1.529023e+00 * time: 0.2995929718017578 18 4.985276e+00 8.599688e-01 * time: 0.31037378311157227 19 4.911035e+00 8.713404e-01 * time: 0.32114386558532715 20 4.848684e+00 3.808398e-01 * time: 0.33200788497924805 21 4.824886e+00 5.627872e-01 * time: 0.342756986618042 22 4.805354e+00 3.318072e-01 * time: 0.35362696647644043 23 4.793341e+00 6.023915e-01 * time: 0.38312387466430664 24 4.778300e+00 3.950741e-01 * time: 0.3939669132232666 25 4.765894e+00 3.385150e-01 * time: 0.40473294258117676 26 4.752738e+00 3.875306e-01 * time: 0.41297292709350586 27 4.742482e+00 2.241386e-01 * time: 0.42377686500549316 28 4.734147e+00 2.240833e-01 * time: 0.43469691276550293 29 4.725128e+00 2.248493e-01 * time: 0.4454197883605957 30 4.716832e+00 2.331933e-01 * time: 0.45629096031188965 31 4.709208e+00 1.888222e-01 * time: 0.4668388366699219 32 4.703200e+00 1.532798e-01 * time: 0.47745585441589355 33 4.695364e+00 2.664459e-01 * time: 0.49786376953125 34 4.688571e+00 2.002891e-01 * time: 0.5084738731384277 35 4.681985e+00 2.587897e-01 * time: 0.5191929340362549 36 4.675970e+00 1.529516e-01 * time: 0.5302338600158691 37 4.669853e+00 1.448113e-01 * time: 0.5411098003387451 38 4.665442e+00 1.486691e-01 * time: 0.5521149635314941 39 4.660491e+00 9.301940e-02 * time: 0.562694787979126 40 4.657968e+00 9.912207e-02 * time: 0.5733838081359863 41 4.656195e+00 8.582461e-02 * time: 0.5842158794403076 42 4.654953e+00 8.726706e-02 * time: 0.6047828197479248 43 4.653859e+00 6.002479e-02 * time: 0.6154289245605469 44 4.653014e+00 6.183910e-02 * time: 0.6262187957763672 45 4.652176e+00 6.328115e-02 * time: 0.6372637748718262 46 4.651685e+00 5.574722e-02 * time: 0.6480047702789307 47 4.651220e+00 4.594288e-02 * time: 0.6588377952575684 48 4.650771e+00 3.494131e-02 * time: 0.6695919036865234 49 4.650372e+00 3.889667e-02 * time: 0.6804468631744385 50 4.650077e+00 3.306406e-02 * time: 0.6912329196929932 51 4.649886e+00 2.803450e-02 * time: 0.7117729187011719 52 4.649699e+00 1.985481e-02 * time: 0.7225499153137207 53 4.649592e+00 2.403889e-02 * time: 0.733238935470581 54 4.649509e+00 2.096471e-02 * time: 0.743959903717041 55 4.649434e+00 1.779744e-02 * time: 0.7548589706420898 56 4.649377e+00 1.202830e-02 * time: 0.7658488750457764 57 4.649343e+00 1.284582e-02 * time: 0.776831865310669 58 4.649313e+00 1.332341e-02 * time: 0.787606954574585 59 4.649291e+00 9.195670e-03 * time: 0.7983269691467285 60 4.649276e+00 9.499483e-03 * time: 0.8189759254455566 61 4.649266e+00 7.657058e-03 * time: 0.8296558856964111 62 4.649256e+00 4.856072e-03 * time: 0.840364933013916 63 4.649249e+00 5.005833e-03 * time: 0.8510198593139648 64 4.649245e+00 3.950568e-03 * time: 0.8619859218597412 65 4.649241e+00 2.262305e-03 * time: 0.8727548122406006 66 4.649238e+00 3.105864e-03 * time: 0.883760929107666 67 4.649237e+00 2.454333e-03 * time: 0.8944427967071533 68 4.649236e+00 2.261856e-03 * time: 0.9053769111633301 69 4.649235e+00 2.159072e-03 * time: 0.9258239269256592 70 4.649234e+00 1.973001e-03 * time: 0.9366497993469238 71 4.649234e+00 2.014374e-03 * time: 0.9474809169769287 72 4.649233e+00 1.514604e-03 * time: 0.9582719802856445 73 4.649233e+00 1.214892e-03 * time: 0.969080924987793 74 4.649233e+00 9.434883e-04 * time: 0.9799518585205078 75 4.649232e+00 8.583170e-04 * time: 0.9908618927001953 76 4.649232e+00 8.354636e-04 * time: 1.0014917850494385 77 4.649232e+00 6.421308e-04 * time: 1.0121347904205322 78 4.649232e+00 5.965776e-04 * time: 1.0325639247894287 79 4.649232e+00 6.268132e-04 * time: 1.043409824371338 80 4.649232e+00 7.278144e-04 * time: 1.0540030002593994 81 4.649232e+00 5.722784e-04 * time: 1.0648109912872314 82 4.649232e+00 5.781954e-04 * time: 1.0755329132080078 83 4.649232e+00 4.765726e-04 * time: 1.0865747928619385 84 4.649232e+00 5.610144e-04 * time: 1.097606897354126 85 4.649231e+00 4.566525e-04 * time: 1.1084859371185303 86 4.649231e+00 3.759088e-04 * time: 1.1193978786468506 87 4.649231e+00 3.321314e-04 * time: 1.1402688026428223 88 4.649231e+00 3.248398e-04 * time: 1.1514167785644531 89 4.649231e+00 3.362427e-04 * time: 1.1624557971954346 90 4.649231e+00 2.975808e-04 * time: 1.1732618808746338 91 4.649231e+00 3.183932e-04 * time: 1.1842308044433594 92 4.649231e+00 2.626344e-04 * time: 1.1950299739837646 93 4.649231e+00 2.258527e-04 * time: 1.205826997756958 94 4.649231e+00 1.845068e-04 * time: 1.2165379524230957 95 4.649231e+00 1.604665e-04 * time: 1.2274589538574219 96 4.649231e+00 1.756582e-04 * time: 1.2379930019378662 97 4.649231e+00 1.070845e-04 * time: 1.2585229873657227 98 4.649231e+00 1.666360e-04 * time: 1.2692358493804932 99 4.649231e+00 9.945995e-05 * time: 1.2803468704223633 100 4.649231e+00 1.083455e-04 * time: 1.2910728454589844 101 4.649231e+00 1.575028e-04 * time: 1.3020977973937988 102 4.649231e+00 1.279942e-04 * time: 1.3128888607025146 103 4.649231e+00 1.613365e-04 * time: 1.323638916015625 104 4.649231e+00 1.018784e-04 * time: 1.3344249725341797 105 4.649231e+00 7.760829e-05 * time: 1.3454678058624268 106 4.649231e+00 1.997232e-04 * time: 1.3635668754577637 107 4.649231e+00 1.120386e-04 * time: 1.3745567798614502 108 4.649231e+00 1.617915e-04 * time: 1.3852307796478271 109 4.649231e+00 1.255093e-04 * time: 1.3958368301391602 110 4.649231e+00 1.051908e-04 * time: 1.4067068099975586 111 4.649231e+00 9.321034e-05 * time: 1.4174277782440186 112 4.649231e+00 6.582039e-05 * time: 1.4282758235931396 113 4.649231e+00 9.046414e-05 * time: 1.4392008781433105 114 4.649231e+00 7.129896e-05 * time: 1.4496798515319824 115 4.649231e+00 4.304140e-05 * time: 1.4699149131774902 116 4.649231e+00 3.010609e-05 * time: 1.4805779457092285 117 4.649231e+00 1.989701e-05 * time: 1.4914169311523438 118 4.649231e+00 2.117241e-05 * time: 1.5020818710327148 119 4.649231e+00 5.839264e-05 * time: 1.5101749897003174 120 4.649231e+00 3.680273e-05 * time: 1.521082878112793 121 4.649231e+00 3.672611e-05 * time: 1.5317928791046143 122 4.649231e+00 3.178531e-05 * time: 1.5426688194274902 123 4.649231e+00 3.765060e-05 * time: 1.5534298419952393 124 4.649231e+00 2.495741e-05 * time: 1.5740227699279785 125 4.649231e+00 2.040485e-05 * time: 1.5847418308258057 126 4.649231e+00 1.719288e-05 * time: 1.5955188274383545 127 4.649231e+00 1.298731e-05 * time: 1.6062068939208984 128 4.649231e+00 9.637198e-06 * time: 1.6170377731323242 129 4.649231e+00 6.912178e-06 * time: 1.6278159618377686 130 4.649231e+00 7.597055e-06 * time: 1.6384599208831787 131 4.649231e+00 7.237671e-06 * time: 1.6490209102630615 132 4.649231e+00 5.094302e-06 * time: 1.65982985496521 133 4.649231e+00 3.787071e-06 * time: 1.6801769733428955 134 4.649231e+00 3.269899e-06 * time: 1.691051959991455 135 4.649231e+00 3.607929e-06 * time: 1.7017848491668701 136 4.649231e+00 2.583156e-06 * time: 1.7124137878417969 137 4.649231e+00 2.686288e-06 * time: 1.7231128215789795 138 4.649231e+00 4.143543e-06 * time: 1.7336678504943848 139 4.649231e+00 3.092121e-06 * time: 1.7442259788513184 140 4.649231e+00 3.278746e-06 * time: 1.7547688484191895 141 4.649231e+00 1.971978e-06 * time: 1.7653989791870117 142 4.649231e+00 2.042079e-06 * time: 1.7763969898223877 143 4.649231e+00 1.476098e-06 * time: 1.796785831451416 144 4.649231e+00 2.461739e-06 * time: 1.8075878620147705 145 4.649231e+00 1.483098e-06 * time: 1.8182108402252197 146 4.649231e+00 1.095893e-06 * time: 1.8289299011230469 147 4.649231e+00 9.883785e-07 * time: 1.8395519256591797 148 4.649231e+00 1.107761e-06 * time: 1.8503057956695557 149 4.649231e+00 1.081220e-06 * time: 1.8610479831695557 150 4.649231e+00 6.737542e-07 * time: 1.8715698719024658 151 4.649231e+00 1.249127e-06 * time: 1.8820397853851318 152 4.649231e+00 8.000455e-07 * time: 1.9025158882141113 153 4.649231e+00 7.142990e-07 * time: 1.913435935974121 154 4.649231e+00 9.693776e-07 * time: 1.9242339134216309 155 4.649231e+00 6.215984e-07 * time: 1.935107946395874 156 4.649231e+00 1.153930e-06 * time: 1.943511962890625 157 4.649231e+00 6.575556e-07 * time: 1.9542629718780518 158 4.649231e+00 7.017550e-07 * time: 1.9650869369506836 159 4.649231e+00 1.345074e-06 * time: 1.975712776184082 160 4.649231e+00 5.901715e-07 * time: 1.986374855041504 161 4.649231e+00 4.935390e-07 * time: 2.006570816040039 162 4.649231e+00 4.122912e-07 * time: 2.0171117782592773 163 4.649231e+00 3.851991e-07 * time: 2.027930974960327 164 4.649231e+00 4.032574e-07 * time: 2.038801908493042 165 4.649231e+00 3.095923e-07 * time: 2.049362897872925 166 4.649231e+00 1.833493e-07 * time: 2.0603108406066895 e(1,1) / (2π)= 1.2158634835234436