GF tail fitting

Detailed Description

Fitting and manipulation of the high-frequency tail of a Green's function.

Tools to extract the high-frequency moments of a Green's function (triqs::gfs::fit_tail, triqs::gfs::fit_hermitian_tail) and to construct or substitute tails.

Functions
template<int N = 0, typename BG, typename A = std::vector<typename BG::g_t::data_t::regular_type>> requires (is_block_gf_v<BG, 1>)
std::pair< std::vector< typename BG::g_t::data_t::regular_type >, double >	triqs::gfs::fit_hermitian_tail (BG const &bg, A const &known_moments={})
	Fit the high-frequency tail of a block Green's function, imposing hermitian symmetry block by block.
template<int N = 0, typename G, typename A = typename G::data_t> requires (is_gf_v<G>)
std::pair< typename A::regular_type, double >	triqs::gfs::fit_hermitian_tail (G const &g, A const &known_moments={})
	Fit the high-frequency tail of a Green's function, imposing hermitian symmetry on the fitted moments.
template<template< typename, typename... > typename G, typename T>
auto	triqs::gfs::fit_hermitian_tail_on_window (G< mesh::imfreq, T > const &g, int n_min, int n_max, array_const_view< dcomplex, 3 > known_moments, int n_tail_max, int expansion_order)
	Fit the high-frequency tail on a restricted window, imposing hermitian moment matrices.
template<int N = 0, typename BG, typename BA = std::vector<typename BG::g_t::data_t::regular_type>> requires (is_block_gf_v<BG, 1>)
std::pair< std::vector< typename BG::g_t::data_t::regular_type >, double >	triqs::gfs::fit_tail (BG const &bg, BA const &known_moments={})
	Fit the high-frequency tail of a block Green's function using a least-squares procedure.
template<int N = 0, typename G, typename A = typename G::const_view_type::data_t> requires (is_gf_v<G>)
std::pair< typename A::regular_type, double >	triqs::gfs::fit_tail (G const &g, A const &known_moments={})
	Fit the high-frequency tail of a Green's function using a least-squares procedure.
template<template< typename, typename, typename... > typename G, typename T>
auto	triqs::gfs::fit_tail_on_window (G< mesh::imfreq, T > const &g, int n_min, int n_max, array_const_view< dcomplex, 3 > known_moments, int n_tail_max, int expansion_order)
	Fit the high-frequency tail of a Matsubara Green's function on a restricted frequency window.
array< dcomplex, 3 >	triqs::gfs::get_tail (gf_const_view< mesh::legendre > gl)
	Extract the leading high-frequency moments of a Legendre Green's function.
template<int N = 0, typename G>
auto	triqs::gfs::make_zero_tail (G const &g, int n_moments=10)
	Create a zero-initialized tail object for a given Green function object.
template<typename T>
void	triqs::gfs::replace_by_tail (gf_view< mesh::imfreq, T > g, array_const_view< dcomplex, 1+T::rank > tail, int n_min)
	Overwrite the high-frequency tail of a Matsubara Green's function.
template<typename T>
void	triqs::gfs::replace_by_tail_in_fit_window (gf_view< mesh::imfreq, T > g, array_const_view< dcomplex, 1+T::rank > tail)
	Overwrite the high-frequency portion of a Matsubara Green's function with the tail expansion.

Function Documentation

fit_hermitian_tail() [1/2]

template<int N = 0, typename BG, typename A = std::vector<typename BG::g_t::data_t::regular_type>>
requires (is_block_gf_v<BG, 1>)

std::pair< std::vector< typename BG::g_t::data_t::regular_type >, double > triqs::gfs::fit_hermitian_tail	(	BG const &	bg,
		A const &	known_moments = {} )

#include <triqs/gfs/functions/functions2.hpp>

Fit the high-frequency tail of a block Green's function, imposing hermitian symmetry block by block.

The symmetry constraint is \( G_{i,j}(i\omega) = G_{j,i}^*(-i\omega) \).

Each block is fitted independently with the same symmetry constraint. The returned error is the maximum across blocks.

Template Parameters

N	Position of the frequency mesh in a product mesh (default \( 0 \)).
BG	The type of the block Green's function.
A	The type of the per-block known-moment array.

Parameters

bg	The block Green's function whose tail is to be fitted.
known_moments	Per-block array of known high-frequency moments.

Returns

A pair containing the per-block fitted tail moments and the worst-block fitting error.

Definition at line 161 of file functions2.hpp.

fit_hermitian_tail() [2/2]

template<int N = 0, typename G, typename A = typename G::data_t>
requires (is_gf_v<G>)

std::pair< typename A::regular_type, double > triqs::gfs::fit_hermitian_tail	(	G const &	g,
		A const &	known_moments = {} )

#include <triqs/gfs/functions/functions2.hpp>

Fit the high-frequency tail of a Green's function, imposing hermitian symmetry on the fitted moments.

The symmetry constraint is \( G_{i,j}(i\omega) = G_{j,i}^*(-i\omega) \).

Template Parameters

N	Position of the frequency mesh in a product mesh (default \( 0 \)).
G	The type of the Green's function.
A	The type of the array of known high-frequency moments.

Parameters

g	The Green's function whose tail is to be fitted.
known_moments	Array of known high-frequency moments to constrain the fit.

Returns

A pair containing the fitted tail moments and the fitting error.

Definition at line 127 of file functions2.hpp.

fit_hermitian_tail_on_window()

template<template< typename, typename... > typename G, typename T>

auto triqs::gfs::fit_hermitian_tail_on_window	(	G< mesh::imfreq, T > const &	g,
		int	n_min,
		int	n_max,
		array_const_view< dcomplex, 3 >	known_moments,
		int	n_tail_max,
		int	expansion_order )

#include <triqs/gfs/functions/imfreq.hpp>

Fit the high-frequency tail on a restricted window, imposing hermitian moment matrices.

Behaves like fit_tail_on_window but enforces the symmetry \( G_{i,j}(i\omega) = G_{j,i}^*(-i\omega) \) on the fitted moments.

Template Parameters

G	The Green's function container template.
T	The target type of the Green's function.

Parameters

g	The Matsubara Green's function whose tail is to be fitted.
n_min	Minimum Matsubara index of the fit window.
n_max	Maximum Matsubara index of the fit window ( \( -1 \) means use the last index of the mesh).
known_moments	Array of known high-frequency moments to constrain the fit.
n_tail_max	Maximum frequency index used internally by the tail fitter.
expansion_order	Order of the tail expansion to fit.

Returns

A pair containing the fitted tail moments and the fitting error.

Definition at line 442 of file imfreq.hpp.

fit_tail() [1/2]

template<int N = 0, typename BG, typename BA = std::vector<typename BG::g_t::data_t::regular_type>>
requires (is_block_gf_v<BG, 1>)

std::pair< std::vector< typename BG::g_t::data_t::regular_type >, double > triqs::gfs::fit_tail	(	BG const &	bg,
		BA const &	known_moments = {} )

#include <triqs/gfs/functions/functions2.hpp>

Fit the high-frequency tail of a block Green's function using a least-squares procedure.

Each block is fitted independently using fit_tail. The returned error is the maximum across blocks.

Template Parameters

N	Position of the frequency mesh in a product mesh (default \( 0 \)).
BG	The type of the block Green's function.
BA	The type of the per-block known-moment array.

Parameters

bg	The block Green's function whose tail is to be fitted.
known_moments	Per-block array of known high-frequency moments.

Returns

A pair containing the per-block fitted tail moments and the worst-block fitting error.

Definition at line 100 of file functions2.hpp.

fit_tail() [2/2]

template<int N = 0, typename G, typename A = typename G::const_view_type::data_t>
requires (is_gf_v<G>)

std::pair< typename A::regular_type, double > triqs::gfs::fit_tail	(	G const &	g,
		A const &	known_moments = {} )

#include <triqs/gfs/functions/functions2.hpp>

Fit the high-frequency tail of a Green's function using a least-squares procedure.

The result is the set of expansion moments that best reproduces the high-frequency behavior of \( G \) on the configured tail-fit window. Known moments, when provided, are treated as exact constraints on the fit.

Template Parameters

N	Position of the frequency mesh in a product mesh (default \( 0 \)).
G	The type of the Green's function.
A	The type of the array of known high-frequency moments.

Parameters

g	The Green's function whose tail is to be fitted.
known_moments	Array of known high-frequency moments to constrain the fit.

Returns

A pair containing the fitted tail moments and the fitting error.

Definition at line 79 of file functions2.hpp.

fit_tail_on_window()

template<template< typename, typename, typename... > typename G, typename T>

auto triqs::gfs::fit_tail_on_window	(	G< mesh::imfreq, T > const &	g,
		int	n_min,
		int	n_max,
		array_const_view< dcomplex, 3 >	known_moments,
		int	n_tail_max,
		int	expansion_order )

#include <triqs/gfs/functions/imfreq.hpp>

Fit the high-frequency tail of a Matsubara Green's function on a restricted frequency window.

The fit is performed on the window \( [n_{\min}, n_{\max}] \) of the Matsubara mesh ( \( n_{\max} = -1 \) selects the last index of the mesh). The tail fitter is configured from n_tail_max and expansion_order, and the fit is delegated to fit_tail.

Template Parameters

G	The Green's function container template.
T	The target type of the Green's function.

Parameters

g	The Matsubara Green's function whose tail is to be fitted.
n_min	Minimum Matsubara index of the fit window.
n_max	Maximum Matsubara index of the fit window ( \( -1 \) means use the last index of the mesh).
known_moments	Array of known high-frequency moments to constrain the fit.
n_tail_max	Maximum frequency index used internally by the tail fitter.
expansion_order	Order of the tail expansion to fit.

Returns

A pair containing the fitted tail moments and the fitting error.

Definition at line 415 of file imfreq.hpp.

get_tail()

array< dcomplex, 3 > triqs::gfs::get_tail

(

gf_const_view< mesh::legendre >

gl

)

#include <triqs/gfs/functions/legendre.hpp>

Extract the leading high-frequency moments of a Legendre Green's function.

Returns the array of moments \( c_n \) such that the corresponding Matsubara Green's function behaves as \( G(i\omega) \sim \sum_n c_n / (i\omega)^n \) at large frequency.

Parameters

gl	The Legendre Green's function.

Returns

An array of high-frequency moments.

make_zero_tail()

template<int N = 0, typename G>

auto triqs::gfs::make_zero_tail	(	G const &	g,
		int	n_moments = 10 )

#include <triqs/gfs/functions/functions2.hpp>

Create a zero-initialized tail object for a given Green function object.

Template Parameters

N	The mesh position of the frequency or time mesh [default: 0].
G	The type of the Green function (gf, gf_view, block_gf, ...).

Parameters

g	The Green function object to create the tail object for.
n_moments	The number of high-frequency moments to provide (including the zeroth moment).

Definition at line 191 of file functions2.hpp.

replace_by_tail()

template<typename T>

void triqs::gfs::replace_by_tail	(	gf_view< mesh::imfreq, T >	g,
		array_const_view< dcomplex, 1+T::rank >	tail,
		int	n_min )

#include <triqs/gfs/functions/imfreq.hpp>

Overwrite the high-frequency tail of a Matsubara Green's function.

For every Matsubara index with \( |n| \geq n_{\min} \), the value of the Green's function is replaced by the tail expansion evaluated at that frequency. Values at lower indices are left unchanged.

Template Parameters

T	The target type of the Green's function.

Parameters

g	The Matsubara Green's function to modify in place.
tail	The high-frequency moments used to build the tail.
n_min	Minimum absolute Matsubara index from which to apply the tail.

Definition at line 374 of file imfreq.hpp.

replace_by_tail_in_fit_window()

template<typename T>

void triqs::gfs::replace_by_tail_in_fit_window	(	gf_view< mesh::imfreq, T >	g,
		array_const_view< dcomplex, 1+T::rank >	tail )

#include <triqs/gfs/functions/imfreq.hpp>

Overwrite the high-frequency portion of a Matsubara Green's function with the tail expansion.

The cutoff \( n_{\min} \) is first set automatically from the tail-fit window of the mesh. Then the function delegates to replace_by_tail.

Template Parameters

T	The target type of the Green's function.

Parameters

g	The Matsubara Green's function to modify in place.
tail	The high-frequency moments used to build the tail.

Definition at line 390 of file imfreq.hpp.