High-Frequency moments of the Green’s function

The coefficients of the high-frequency expansion of a Green function (in short tail) play an essential role in acquiring accurate results for e.g. Fourier transforms and density calculations.

\[G(i\omega) \approx \sum_{n = 0}^N \frac{a_n}{(i\omega)^n}\]

The TRIQS library provides functionalities to determine these tail coefficiencts through a least-square fitting procedure.

API

We provide a variety of free functions in order to determine the tail coefficients. Given a Matsubara or Real-frequency Green function \(G(i\omega)\) with arbitrary G.target_shape() we have the following possibilities.

fit_tail(G) -> std::pair<array<dcomplex, Rank>, double>

  • This function will consider twenty percent of the frequencies (the largest by absolute value) for the least square fit. The largest order \(N\) for the fit is automatically determined based on the value of the largest frequency such that

    \(\|\omega_{max}\|^{1-N} > 10^{-16}\). The function will return a pair containing both the resulting high-frequency

    moment array and an error corresponding to the mean deviation from the data.

fit_tail(G, known_moments)

  • We can provide additional information to the fitting routine by passing an array with a set of known moments. The shape of this array should be of the form \((N + 1, i, j, ...)\) where the target_shape of G is given by \((i, j, ...)\).

fit_hermitian_tail(G_iw, [known_moments])

  • Matsubara Green functions fulfill the general relation \(G_{ij}(i\omega) = G_{ji}^*(-i\omega)\), which entails that the high-frequency coefficient matrices have to be hermitian. The fit_hermitian_tail function will enforce this property, but will otherwise take the same arguments as fit_tail.

Note

For convenience we prodive a free function make_zero_tail(G, max_order) that will, given the Green function G and the max_order, generate a tail-array of the proper shape initialized to 0.

Example

#include <triqs/gfs.hpp>
#include <triqs/mesh.hpp>

using namespace triqs::gfs;
using namespace triqs::mesh;

int main() {

  nda::clef::placeholder<0> iw_;

  double beta  = 10;
  auto iw_mesh = gf_mesh<imfreq>{beta, Fermion, 100};

  auto G = gf<imfreq, scalar_valued>{iw_mesh, make_shape()};
  G(iw_) << 1.0 / iw_ + 2.0 / iw_ / iw_ + 3.0 / iw_ / iw_ / iw_;

  // We provide both the zeroth and first order coefficient to the fit
  auto known_moments = array<dcomplex, 1>{0.0, 1.0};
  auto [tail, err]   = fit_tail(G, known_moments);

  std::cout << "tail: " << tail << "\nerror: " << err;
}

Adjusting the fit parameters (Advanced)

The experienced user can adjust the parameters of the fitting procedure directly:

G.mesh().set_tail_fit_parameters(tail_fraction, n_tail_max, expansion_order)

The fitting parameters are

  • tail_fraction (double, default = 0.2) A value between 0 and 1.0 denoting the fraction of the frequency mesh (starting from the outer frequencies) to be considered for the fit.

  • n_tail_max (integer, default = 30) The maximum number of frequency points to be considered for the fit on both the positive and negative side.

  • expansion_order (integer, optional) This parameter will fix the largest order to be considered for the fit.

Adjusting the fit window (Advanced)

We provide a free function to allow fitting on an adjusted Matsubara frequency window ranging from indices n_min to n_max (and [-n_max-1, -n_min-1] correspondingly)

fit_tail_on_window(G_iw, n_min, n_max, known_moments, n_tail_max) -> std::pair<array<dcomplex, R>, double>

The parameters known_moments and n_tail_max behave as described above. Note that a variant of this function called fit_hermitian_tail_on_window enforces hermiticity of the coefficient matrices.