2. Disentanglement of entangled manifold

In the second tutorial, we will run the disentanglement algorithm on the silicon valence + conduction bands. As usual, we need to

  1. generate the amn, mmn, and eig files by using Quantum ESPRESSO (QE)
  2. construct a Model for Wannier.jl, by reading the win, amn, mmn, and eig files
  3. run Wannier.jl disentangle on the Model to minimize the spread
  4. write the maximal localized gauge to a new amn file
Tip

This is a HTML version of the tutorial, you can download corresponding

Preparation

Load the package

using Wannier
using Printf  # for pretty print
Tip

Use the run.sh script which automate the scf, nscf, pw2wannier90 steps.

Model generation

We will use the read_w90 function to read the win, amn, mmn, and eig files, and construct a Model that abstracts the calculation

model = read_w90("si2")
lattice: Å
  a1:  0.00000  2.71527  2.71527
  a2:  2.71527  0.00000  2.71527
  a3:  2.71527  2.71527  0.00000

atoms: fractional
  Si:  0.00000  0.00000  0.00000
  Si:  0.25000  0.25000  0.25000

n_bands: 16
n_wann : 8
kgrid  : 4 4 4
n_kpts : 64
n_bvecs: 8

b-vectors:
         [bx, by, bz] / Å⁻¹                weight
  1       0.28925   -0.28925    0.28925    1.49401
  2      -0.28925   -0.28925   -0.28925    1.49401
  3      -0.28925    0.28925   -0.28925    1.49401
  4      -0.28925   -0.28925    0.28925    1.49401
  5       0.28925    0.28925   -0.28925    1.49401
  6      -0.28925    0.28925    0.28925    1.49401
  7       0.28925   -0.28925   -0.28925    1.49401
  8       0.28925    0.28925    0.28925    1.49401
Tip

The read_w90 function will parse the win file and set the frozen window for the Model according to the dis_froz_min and dis_froz_max parameters in the win file. However, you can also change these parameters by calling the set_frozen_win! function.

Disentanglement and maximal localization

The disentangle function will disentangle and maximally localize the spread functional, and returns the gauge matrices U,

U = disentangle(model)
16×8×64 Array{ComplexF64, 3}:
[:, :, 1] =
    0.111761+0.698219im     …   2.58153e-8+1.22317e-7im
 -4.87003e-8-1.43163e-8im         0.250007-0.0294491im
  3.98028e-8+7.13461e-9im        -0.252285-0.36804im
  5.07799e-8-1.11992e-10im       -0.485848+0.038466im
 -8.77988e-8+1.35684e-7im        -0.279255-0.197916im
 -1.54707e-7-4.39447e-7im   …      0.27269+0.346345im
  9.27033e-8+3.34284e-7im        -0.227774-0.369662im
  -0.0321752+0.706356im         1.96826e-7+4.36805e-7im
  2.22365e-9+7.77753e-9im        3.0155e-8-3.35082e-8im
  4.74728e-9+9.52079e-10im     -2.93517e-9+8.63692e-8im
  6.04808e-9-2.69522e-9im   …  -2.40791e-8+5.38945e-9im
  6.64407e-9-1.60265e-8im      -2.31374e-7-1.05431e-7im
  3.77989e-9+1.93567e-8im      -9.19568e-8-7.42529e-8im
 -4.87148e-9-2.80427e-10im      1.58719e-7+1.61337e-7im
 -0.00448668-0.00228008im       -3.3663e-9+1.68558e-9im
 6.34946e-10-8.35856e-10im  …  -3.72729e-8-1.77602e-8im

[:, :, 2] =
   -0.661154-0.197991im     0.0325423-0.0826608im  …   0.0130113+0.0878778im
    0.215855-0.383447im     -0.139837-0.287351im       -0.220069+0.231721im
  8.07755e-8-3.7246e-8im     0.247498+0.264292im       -0.439968-0.220343im
  9.67953e-8-3.22014e-8im    0.354636-0.276516im       -0.153893-0.25986im
  -0.0617892-0.455648im     -0.062591+0.292035im      -0.0960857-0.282789im
 -6.90833e-8-4.60069e-7im   -0.451142-0.267179im   …   -0.219797-0.35931im
  2.59152e-7-3.96048e-7im   0.0896819+0.160793im       0.0206672-0.361896im
    0.253347-0.226618im      0.317992+0.127168im        0.273153+0.206585im
 -2.60454e-9+5.00388e-8im  -0.0895461+0.0491383im     0.00364945+0.0191664im
 -1.00799e-7+1.479e-7im     0.0279981+0.0597544im     -0.0316425+0.115783im
  -0.0311393-0.0292906im   -0.0533976+0.0750946im  …    -0.08054-0.0447642im
  -0.0139832+0.0324625im   -0.0543361+0.0733565im     -0.0912503+0.00263846im
 -5.77148e-8+6.31325e-8im   0.0131604+0.0518483im      0.0113719+0.0669979im
 -9.01861e-9+5.99055e-9im   0.0348641-0.0371696im     -0.0027809+0.0288572im
 -9.97351e-8+6.00984e-8im   0.0211478+0.0505144im     0.00620119+0.0573059im
  5.59049e-9+3.9318e-8im   -0.0150015-0.0324432im  …   0.0139075+0.0275808im

[:, :, 3] =
    0.194735+0.665427im     …   -0.0225189-0.0769488im
    0.424525-0.193276im          -0.279245+0.127134im
 -3.94301e-9-5.37264e-8im         0.387526+0.288074im
  3.43764e-8+2.94624e-8im       -0.0253234+0.315482im
   -0.529944+0.0397836im          -0.26855+0.0201608im
 -3.65288e-7+1.14008e-7im   …   -0.0751052-0.486782im
 -8.26018e-8-1.21567e-7im          0.08397-0.165231im
  -0.0901249+0.0265775im         -0.259789+0.0766124im
 -8.65739e-9-1.3668e-8im       -6.89277e-8-8.32443e-8im
  4.10469e-8-1.39546e-8im       4.56205e-8-1.68473e-7im
  2.15434e-9-1.27165e-10im  …   8.85642e-9-1.1719e-8im
 -2.15567e-8-1.48861e-7im         0.222481+0.015572im
  1.42184e-8-8.75885e-8im        0.0801181+0.0149682im
   0.0713511-0.0719651im          0.205672-0.207444im
  1.53271e-8+8.42603e-9im       4.55633e-8+2.14888e-8im
  0.00524269+0.0125858im    …    0.0151125+0.036279im

;;; … 

[:, :, 62] =
  -0.0699355-0.661997im    …  -0.0487142+0.0268936im
   -0.205777+0.528887im           0.2158-0.0599727im
  -0.0930781+0.18439im         -0.273468-0.240602im
  -1.1529e-8-8.93419e-8im      -0.425952-0.261849im
    0.237539-0.170351im       -0.0891282-0.0339242im
   -0.183237-0.0632996im   …  -0.0215613-0.214618im
   -0.193365-0.128422im        -0.277801-0.254419im
 -1.76789e-7+3.17089e-7im       0.156194+0.39984im
  -0.0100242+0.0936122im       -0.251338+0.0988819im
 -7.93848e-8-1.64522e-7im      -0.185032-0.0692895im
 -9.76773e-8-1.07332e-7im  …   -0.153484+0.0144726im
   0.0444768-0.00579216im      0.0269545+0.0664283im
  -0.0168293-0.0547054im         0.10661-0.144434im
  0.00795924-0.0141399im       0.0186389+0.00215521im
 -6.20133e-8+1.30235e-8im     -2.7749e-5+0.0540993im
  -0.0122906+0.0257909im   …  -0.0526103+0.0557264im

[:, :, 63] =
       0.3079-0.598662im    …   -0.0687038+0.133584im
      -0.5526-0.127377im          0.188777+0.043514im
    0.0146735-0.133843im         0.0397664-0.362729im
   -1.8363e-8-3.12989e-8im        0.376624-0.328869im
     -0.24724-0.0974201im         0.100232+0.0394928im
   9.43668e-8-1.81287e-7im  …     0.357925+0.349127im
   -0.0556494+0.111995im         -0.100911+0.203086im
    -0.300382-0.115051im         -0.363839-0.139356im
  -5.00901e-8+1.02551e-9im     -1.38692e-7+3.55247e-9im
     0.081216+0.0793804im         0.147274+0.143944im
   -0.0285746+0.00544723im  …   -0.0534185+0.0101834im
   -0.0554396-0.0198563im        -0.180232-0.0645515im
   0.00904487+0.00439827im       0.0164016+0.00797549im
   0.00314076+0.00492826im      0.00569513+0.00893664im
  -2.54626e-8+1.15726e-8im     -2.69943e-7+1.57787e-7im
 -0.000159879-0.00318789im  …  -0.00486934-0.0971257im

[:, :, 64] =
    0.687305-0.0627372im    …  -0.0766937+0.0448319im
   -0.400027+0.183313im        -0.0580926-0.314245im
 -1.28304e-8+3.41617e-8im        0.128068-0.0855533im
 -7.28342e-8-1.76194e-8im      -0.0955143+0.548169im
   -0.163109+0.429917im           0.29437+0.0504904im
  2.40733e-7-1.73422e-8im   …   -0.185985+0.165522im
 -2.66338e-7-5.48803e-8im        0.490407-0.0795231im
    0.245784+0.234799im           0.28851+0.184532im
 -1.12221e-8-2.39453e-9im       0.0145893+0.00549528im
    7.622e-8-3.41044e-9im       -0.108438+0.0527799im
   0.0344254-0.0253473im    …   -0.071148-0.0585533im
   0.0151444+0.0319373im        0.0702677+0.0582753im
 -3.77952e-8+2.21599e-8im      0.00569708-0.0142884im
  4.06438e-8+7.86089e-8im      -0.0631211-0.0351796im
 -7.32305e-8+2.24036e-9im       0.0512993-0.014826im
 -2.01529e-8-5.14719e-10im  …   0.0374042-0.00510306im

The initial spread is

omega(model)
  WF     center [rx, ry, rz]/Š             spread/Ų
   1    -0.00000     0.00000     0.00000     1.33119
   2     0.00001     0.00011    -0.00000     1.92293
   3    -0.00000     0.00011     0.00011     1.92297
   4     0.00001    -0.00000     0.00011     1.92303
   5     1.35763     1.35763     1.35763     1.33119
   6     1.35762     1.35752     1.35763     1.92293
   7     1.35763     1.35752     1.35752     1.92297
   8     1.35762     1.35763     1.35752     1.92303
Sum spread: Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
   ΩI  =     9.69628
   Ω̃   =     4.50396
   ΩOD =     4.49427
   ΩD  =     0.00968
   Ω   =    14.20024

The final spread is

omega(model, U)
  WF     center [rx, ry, rz]/Š             spread/Ų
   1     0.00000     0.00000     0.00000     1.44670
   2     0.00000     0.00000    -0.00000     2.09677
   3    -0.00000    -0.00000     0.00000     2.09677
   4    -0.00000    -0.00000    -0.00000     2.09677
   5     1.35763     1.35763     1.35763     1.44670
   6     1.35763     1.35763     1.35763     2.09677
   7     1.35763     1.35763     1.35763     2.09677
   8     1.35763     1.35763     1.35763     2.09677
Sum spread: Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
   ΩI  =    10.05117
   Ω̃   =     5.42285
   ΩOD =     5.36573
   ΩD  =     0.05711
   Ω   =    15.47402
Note

The convergence thresholds is determined by the keyword arguments of disentangle, e.g., f_tol for the tolerance on spread, and g_tol for the tolerance on the norm of spread gradient, etc. You can use stricter thresholds to further minimize a bit the spread.

Save the new gauge

Again, we save the new gauge to an amn file, which can be used as the new initial guess for Wannier90, or reuse it in Wannier.jl.

write_amn("si2.dis.amn", U)
[ Info: Written to file: si2.dis.amn

Great! Now you have finished the disentanglement tutorial.

As you can see, the workflow is very similar to the previous tutorial: the Wannierization functions, max_localize and disentangle, accept a Model and some convergence thresholds, and return the gauge matrices. This interface are also adopted in other Wannierization algorithms, shown in later tutorials.

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