dft_tools/triqs/lattice/tight_binding.cpp

/*******************************************************************************
 *
 * TRIQS: a Toolbox for Research in Interacting Quantum Systems
 *
 * Copyright (C) 2011 by M. Ferrero, O. Parcollet
 *
 * TRIQS is free software: you can redistribute it and/or modify it under the
 * terms of the GNU General Public License as published by the Free Software
 * Foundation, either version 3 of the License, or (at your option) any later
 * version.
 *
 * TRIQS is distributed in the hope that it will be useful, but WITHOUT ANY
 * WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
 * FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
 * details.
 *
 * You should have received a copy of the GNU General Public License along with
 * TRIQS. If not, see <http://www.gnu.org/licenses/>.
 *
 ******************************************************************************/
#include "tight_binding.hpp"
#include <triqs/arrays/algorithms.hpp>
#include <triqs/arrays/linalg/eigenelements.hpp>
#include "grid_generator.hpp"
namespace triqs {
namespace lattice {

 using namespace arrays;

 tight_binding::tight_binding(bravais_lattice const& bl, std::vector<std::vector<long>> all_disp,
                              std::vector<matrix<dcomplex>> all_matrices)
    : bl_(bl), all_disp(std::move(all_disp)), all_matrices(std::move(all_matrices)) {

  // checking inputs
  if (all_disp.size() != all_matrices.size()) TRIQS_RUNTIME_ERROR << " Number of displacements != Number of matrices";
  for (int i = 0; i < all_disp.size(); ++i) {
   if (all_disp[i].size() != bl_.dim())
    TRIQS_RUNTIME_ERROR << "displacement of incorrect size : got " << all_disp[i].size() << "instead of " << bl_.dim();
   if (first_dim(all_matrices[i]) != n_bands())
    TRIQS_RUNTIME_ERROR << "the first dim matrix is of size " << first_dim(all_matrices[i]) << " instead of " << n_bands();
   if (second_dim(all_matrices[i]) != n_bands())
    TRIQS_RUNTIME_ERROR << "the second dim matrix is of size " << second_dim(all_matrices[i]) << " instead of " << n_bands();
  }
 }

 //------------------------------------------------------
 array<dcomplex, 3> hopping_stack(tight_binding const& TB, arrays::array_const_view<double, 2> k_stack) {
  auto TK = fourier(TB);
  array<dcomplex, 3> res(TB.n_bands(), TB.n_bands(), k_stack.shape(1));
  for (int i = 0; i < k_stack.shape(1); ++i) res(range(), range(), i) = TK(k_stack(range(), i));
  return res;
 }

 //------------------------------------------------------
 array<double, 2> energies_on_bz_path(tight_binding const& TB, k_t const& K1, k_t const& K2, int n_pts) {
  auto TK = fourier(TB);
  int norb = TB.lattice().n_orbitals();
  int ndim = TB.lattice().dim();
  array<double, 2> eval(norb, n_pts);
  k_t dk = (K2 - K1) / double(n_pts), k = K1;
  for (int i = 0; i < n_pts; ++i, k += dk) {
   eval(range(), i) = linalg::eigenvalues(TK(k(range(0, ndim)))());
  }
  return eval;
 }

 //------------------------------------------------------
 array<dcomplex, 3> energy_matrix_on_bz_path(tight_binding const& TB, k_t const& K1, k_t const& K2, int n_pts) {
  auto TK = fourier(TB);
  int norb = TB.lattice().n_orbitals();
  int ndim = TB.lattice().dim();
  array<dcomplex, 3> eval(norb,norb,n_pts);
  k_t dk = (K2 - K1) / double(n_pts), k = K1;
  for (int i = 0; i < n_pts; ++i, k += dk) {
   eval(range(),range(),i) = TK(k(range(0, ndim)))();
  }
  return eval;
 }

 //------------------------------------------------------
 array<double, 2> energies_on_bz_grid(tight_binding const& TB, int n_pts) {

  auto TK = fourier(TB);
  int norb = TB.lattice().n_orbitals();
  int ndim = TB.lattice().dim();
  grid_generator grid(ndim, n_pts);
  array<double, 2> eval(norb, grid.size());
  for (; grid; ++grid) {
   eval(range(), grid.index()) = linalg::eigenvalues(TK((*grid)(range(0, ndim)))());
  }
  return eval;
 }

 //------------------------------------------------------

 std::pair<array<double, 1>, array<double, 2>> dos(tight_binding const& TB, int nkpts, int neps) {

  // The fourier transform of TK
  auto TK = fourier(TB);

  // loop on the BZ
  int ndim = TB.lattice().dim();
  int norb = TB.lattice().n_orbitals();
  grid_generator grid(ndim, nkpts);
  array<double, 1> tempeval(norb);
  array<dcomplex, 3> evec(norb, norb, grid.size());
  array<double, 2> eval(norb, grid.size());
  if (norb == 1)
   for (; grid; ++grid) {
    double ee = real(TK((*grid)(range(0, ndim)))(0, 0));
    eval(0, grid.index()) = ee;
    evec(0, 0, grid.index()) = 1;
   }
  else
   for (; grid; ++grid) {
    // cerr<<" index = "<<grid.index()<<endl;
    array_view<double, 1> eval_sl = eval(range(), grid.index());
    array_view<dcomplex, 2> evec_sl = evec(range(), range(), grid.index());
    std::tie(eval_sl, evec_sl) = linalg::eigenelements(TK((*grid)(range(0, ndim)))); //,  true);
    // cerr<< " point "<< *grid <<  " value "<< eval_sl<< endl; //" "<< (*grid) (range(0,ndim)) << endl;
   }

  // define the epsilon mesh, etc.
  array<double, 1> epsilon(neps);
  double epsmax = max_element(eval);
  double epsmin = min_element(eval);
  double deps = (epsmax - epsmin) / neps;
  // for (int i =0; i< neps; ++i) epsilon(i)= epsmin+i/(neps-1.0)*(epsmax-epsmin);
  for (int i = 0; i < neps; ++i) epsilon(i) = epsmin + (i + 0.5) * deps;

  // bin the eigenvalues according to their energy
  // NOTE: a is defined as an integer. it is the index for the DOS.
  // REPORT <<"Starting Binning ...."<<endl;
  array<double, 2> rho(neps, norb);
  rho() = 0;
  for (int l = 0; l < norb; l++) {
   for (int j = 0; j < grid.size(); j++) {
    int a = int((eval(l, j) - epsmin) / deps);
    if (a == int(neps)) a = a - 1;
    for (int k = 0; k < norb; k++) {
     rho(a, l) += real(conj(evec(l, k, j)) * evec(l, k, j));
    }
   }
  }
  rho /= grid.size() * deps;
  return std::make_pair(epsilon, rho);
 }

 //----------------------------------------------------------------------------------

 std::pair<array<double, 1>, array<double, 1>> dos_patch(tight_binding const& TB, const array<double, 2>& triangles, int neps,
                                                         int ndiv) {
  // WARNING: This version only works for a single band Hamiltonian in 2 dimensions!!!!
  // triangles is an array of points defining the triangles of the patch
  // neps in the number of bins in energy
  // ndiv in the number of divisions used to divide the triangles

  // int ndim=TB.lattice().dim();
  // int norb=TB.lattice().n_orbitals();
  int ntri = triangles.shape(0) / 3;
  array<double, 1> dos(neps);

  // Check consistency
  int ndim = TB.lattice().dim();
  // int norb=TB.lattice().n_orbitals();
  if (ndim != 2) TRIQS_RUNTIME_ERROR << "dos_patch : dimension 2 only !";
  if (triangles.shape(1) != ndim)
   TRIQS_RUNTIME_ERROR << "dos_patch : the second dimension of the 'triangle' array in not " << ndim;

  // Every triangle has ndiv*ndiv k points
  int nk = ntri * ndiv * ndiv;
  int k_index = 0;
  double epsmax = -100000, epsmin = 100000;
  array<dcomplex, 2> thop(1, 1);
  array<double, 1> energ(nk), weight(nk);

  // a, b, c are the corners of the triangle
  // g the center of gravity taken from a
  array<double, 1> a(ndim), b(ndim), c(ndim), g(ndim), rv(ndim);
  int pt = 0;
  double s, t;

  // The fourier transform of TK
  auto TK = fourier(TB);

  // loop over the triangles
  for (int tri = 0; tri < ntri; tri++) {
   a = triangles(pt, range());
   pt++;
   b = triangles(pt, range());
   pt++;
   c = triangles(pt, range());
   pt++;
   g = ((a + b + c) / 3.0 - a) / double(ndiv);

   // the area around a k point might be different from one triangle to the other
   // so I use it to weight the sum in the dos
   double area = abs(0.5 * ((b(0) - a(0)) * (c(1) - a(1)) - (b(1) - a(1)) * (c(0) - a(0))) / (ndiv * ndiv));

   for (int i = 0; i < ndiv; i++) {
    s = i / double(ndiv);
    for (int j = 0; j < ndiv - i; j++) {
     t = j / double(ndiv);
     for (int k = 0; k < 2; k++) {

      rv = a + s * (b - a) + t * (c - a) + (k + 1.0) * g;

      if (k == 0 || j < ndiv - i - 1) {

       energ(k_index) = real(TK(rv)(0, 0));
       // compute(rv);
       // energ(k_index) = real(tk_for_eval(1,1)); //tk_for_eval is Fortran array
       weight(k_index) = area;
       if (energ(k_index) > epsmax) epsmax = energ(k_index);
       if (energ(k_index) < epsmin) epsmin = energ(k_index);
       k_index++;
      }
     }
    }
   }
  }
  // check consistency
  assert(k_index == nk);

  // define the epsilon mesh, etc.
  array<double, 1> epsilon(neps);
  double deps = (epsmax - epsmin) / neps;
  for (int i = 0; i < neps; ++i) epsilon(i) = epsmin + i / (neps - 1.0) * (epsmax - epsmin);

  // bin the eigenvalues according to their energy
  int ind;
  double totalweight(0.0);
  dos() = 0.0;
  for (int j = 0; j < nk; j++) {
   ind = int((energ(j) - epsmin) / deps);
   if (ind == int(neps)) ind--;
   dos(ind) += weight(j);
   totalweight += weight(j);
  }
  dos /= deps; // Normalize the DOS
  return {std::move(epsilon), std::move(dos)};
 }
}
}