Detailed Description

Conversions between basis representations of a Green's function.

Conversions to and from compact basis representations such as Legendre polynomials and the discrete Lehmann representation (DLR).

Functions
void	triqs::gfs::enforce_discontinuity (gf_view< mesh::legendre > gl, nda::array_const_view< double, 2 > disc)
	Enforce a prescribed jump at \( \tau = 0 \) for a Legendre Green's function.
template<int = 0, typename G> requires (MemoryGf<G> or is_block_gf_v<G>)
double	triqs::gfs::find_w_max (G const &g, double eps=1e-10, bool symmetrize=true, double w_max_init=1.0, double w_max_max=200.0)
	Find a DLR energy cutoff \( \omega_{\mathrm{max}} \) that reproduces an imfreq Green's function within tolerance eps after a DLR round-trip.
template<int N = 0, int... Ns, typename G> requires (MemoryGf<G> or is_block_gf_v<G>)
auto	triqs::gfs::fit_gf_dlr (G const &g, double w_max, double eps, bool symmetrize=true)
	Fit an imaginary-time Green's function with a Discrete Lehmann Representation.
template<int N = 0, int... Ns, typename G> requires (MemoryGf<G> or is_block_gf_v<G>)
auto	triqs::gfs::fit_gf_dlr (G const &g, mesh::dlr const &dlr_mesh)
	Fit an imaginary-time Green's function on a given DLR coefficient mesh.
template<int N = 0, int... Ns, typename G> requires (MemoryGf<G> or is_block_gf_v<G>)
auto	triqs::gfs::make_gf_dlr (G const &g)
	Transform a DLR imaginary-time or DLR Matsubara Green's function to its DLR-coefficient representation.
template<int N = 0, int... Ns, typename G> requires (MemoryGf<G> or is_block_gf_v<G>)
auto	triqs::gfs::make_gf_dlr_imfreq (G const &g)
	Build a DLR Matsubara Green's function from a DLR-coefficient or DLR-imaginary-time input.
template<int N = 0, int... Ns, typename G> requires (MemoryGf<G> or is_block_gf_v<G>)
auto	triqs::gfs::make_gf_dlr_imfreq (G const &g, double w_max, double eps, bool symmetrize=true)
	Sample an imaginary frequency Green's function on the DLR Matsubara frequency nodes defined by (w_max, eps, symmetrize).
template<int N = 0, int... Ns, typename G> requires (MemoryGf<G> or is_block_gf_v<G>)
auto	triqs::gfs::make_gf_dlr_imtime (G const &g)
	Build a DLR imaginary-time Green's function from a DLR-coefficient or DLR-Matsubara input.
template<int N = 0, int... Ns, typename G> requires (MemoryGf<G> or is_block_gf_v<G>)
auto	triqs::gfs::make_gf_imfreq (G const &g, long n_iw)
	Build a uniform Matsubara Green's function from any DLR representation.
template<int N = 0, int... Ns, typename G> requires (MemoryGf<G> or is_block_gf_v<G>)
auto	triqs::gfs::make_gf_imtime (G const &g, long n_tau)
	Build a uniform imaginary-time Green's function from any DLR representation.
template<typename G> requires (MemoryGf<G> or is_block_gf_v<G>)
auto	triqs::gfs::tau_L2_norm (G const &g)
	Calculate the \( L^2 \) norm of a DLR Green's function.

Function Documentation

◆ enforce_discontinuity()

void triqs::gfs::enforce_discontinuity	(	gf_view< mesh::legendre >	gl,
		nda::array_const_view< double, 2 >	disc )

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

Enforce a prescribed jump at \( \tau = 0 \) for a Legendre Green's function.

The Legendre coefficients are adjusted in place so that the corresponding imaginary-time Green's function has the specified discontinuity \( G(0^+) - G(0^-) \) at \( \tau = 0 \) (which equals \( -1 \) for a fermionic propagator). Coefficients above the constrained subspace are left unchanged.

Parameters

gl	Legendre Green's function modified in place.
disc	Target discontinuity at \( \tau = 0 \).

Definition at line 57 of file legendre.cpp.

◆ find_w_max()

template<int = 0, typename G>
requires (MemoryGf<G> or is_block_gf_v<G>)

double triqs::gfs::find_w_max	(	G const &	g,
		double	eps = 1e-10,
		bool	symmetrize = true,
		double	w_max_init = 1.0,
		double	w_max_max = 200.0 )

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

Find a DLR energy cutoff \( \omega_{\mathrm{max}} \) that reproduces an imfreq Green's function within tolerance eps after a DLR round-trip.

Starting from w_max_init, the cutoff is grown by a factor of 1.5 per iteration. For each candidate \( \omega_{\mathrm{max}} \) the function

builds a triqs::mesh::dlr_imfreq mesh from ( \( \omega_{\mathrm{max}} \), eps, symmetrize),
samples g on that DLR mesh,
reconstructs the Gf on the original imfreq mesh of g via the DLR coefficients, and
measures the element-wise round-trip error
\[ e = \max_{i\omega_n, \mathrm{indices}} \left| g(i\omega_n) - g_{\mathrm{rec}}(i\omega_n) \right| \;. \]
The smallest \( \omega_{\mathrm{max}} \le \) w_max_max with \( e < \) eps is returned.

Candidate cutoffs whose DLR frequency range exceeds the range of the input imfreq mesh of g are skipped (they would produce an invalid DLR mesh).

For a block_gf, the error reported per iteration is the worst-case error across all blocks.

It throws an exception if w_max_init exceeds w_max_max, or if no candidate cutoff \( \le \) w_max_max achieves \( e < \) eps.

Template Parameters

G	An imfreq MemoryGf or block_gf type.

Parameters

g	Input Green's function on an imfreq mesh.
eps	Target DLR accuracy and round-trip error tolerance.
symmetrize	If true, use particle-hole symmetric DLR meshes.
w_max_init	Initial value of the DLR cutoff to try (must be \( \le \) w_max_max).
w_max_max	Maximum DLR cutoff to try before giving up.

Returns: The smallest \( \omega_{\mathrm{max}} \le \) w_max_max for which the round-trip error is below eps.

Definition at line 335 of file dlr.hpp.

◆ fit_gf_dlr() [1/2]

template<int N = 0, int... Ns, typename G>
requires (MemoryGf<G> or is_block_gf_v<G>)

auto triqs::gfs::fit_gf_dlr	(	G const &	g,
		double	w_max,
		double	eps,
		bool	symmetrize = true )

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

Fit an imaginary-time Green's function with a Discrete Lehmann Representation.

Builds a DLR coefficient Green's function by least-squares fitting the input Green's function (on a uniform imaginary-time mesh) on a DLR grid specified by the spectral cutoff \( \omega_{\max} \) and tolerance \( \epsilon \).

Template Parameters

N	Index of the mesh component to fit (default \( 0 \)).
Ns	Additional mesh component indices for product meshes.
G	The type of the input Green's function.

Parameters

g	The imaginary-time Green's function to fit.
w_max	Maximum real frequency captured by the DLR basis.
eps	Target accuracy of the DLR representation.
symmetrize	If true, the DLR grid is symmetric about \( \omega = 0 \).

Returns: A Green's function on the DLR coefficient mesh.

Definition at line 163 of file dlr.hpp.

◆ fit_gf_dlr() [2/2]

template<int N = 0, int... Ns, typename G>
requires (MemoryGf<G> or is_block_gf_v<G>)

auto triqs::gfs::fit_gf_dlr	(	G const &	g,
		mesh::dlr const &	dlr_mesh )

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

Fit an imaginary-time Green's function on a given DLR coefficient mesh.

Builds a DLR coefficient Green's function by least-squares fitting the input Green's function (on a uniform imaginary-time mesh) onto the supplied DLR grid dlr_mesh.

Template Parameters

N	Index of the mesh component to fit (default \( 0 \)).
Ns	Additional mesh component indices for product meshes.
G	The type of the input Green's function.

Parameters

g	The imaginary-time Green's function to fit.
dlr_mesh	The DLR coefficient mesh to fit onto.

Returns: A Green's function on the DLR coefficient mesh.

Definition at line 129 of file dlr.hpp.

◆ make_gf_dlr()

template<int N = 0, int... Ns, typename G>
requires (MemoryGf<G> or is_block_gf_v<G>)

auto triqs::gfs::make_gf_dlr ( G const & g )

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

Transform a DLR imaginary-time or DLR Matsubara Green's function to its DLR-coefficient representation.

The input must live on an imaginary-time or frequency DLR mesh. The output lives on the corresponding DLR (coefficient) mesh.

For block Green's functions the transform is applied block-wise.

Template Parameters

N	Index of the mesh component to transform (default \( 0 \)).
Ns	Additional mesh component indices for product meshes.
G	The type of the input Green's function.

Parameters

g	The DLR imaginary-time or DLR Matsubara Green's function.

Returns: The same Green's function expressed in DLR coefficients.

Definition at line 94 of file dlr.hpp.

◆ make_gf_dlr_imfreq() [1/2]

template<int N = 0, int... Ns, typename G>
requires (MemoryGf<G> or is_block_gf_v<G>)

auto triqs::gfs::make_gf_dlr_imfreq ( G const & g )

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

Build a DLR Matsubara Green's function from a DLR-coefficient or DLR-imaginary-time input.

The output lives on the Matsubara DLR mesh associated with the input's DLR grid. Block and product meshes are handled recursively.

Template Parameters

N	Index of the mesh component to transform (default \( 0 \)).
Ns	Additional mesh component indices for product meshes.
G	The type of the input Green's function.

Parameters

g	A Green's function on an imaginary-time or coefficient DLR mesh.

Returns: The Green's function evaluated on the Matsubara DLR mesh.

Definition at line 216 of file dlr.hpp.

◆ make_gf_dlr_imfreq() [2/2]

template<int N = 0, int... Ns, typename G>
requires (MemoryGf<G> or is_block_gf_v<G>)

auto triqs::gfs::make_gf_dlr_imfreq	(	G const &	g,
		double	w_max,
		double	eps,
		bool	symmetrize = true )

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

Sample an imaginary frequency Green's function on the DLR Matsubara frequency nodes defined by (w_max, eps, symmetrize).

Constructs a fresh triqs::mesh::dlr_imfreq mesh from w_max, eps and symmetrize (using the same beta and statistic as g) and returns a Gf on that mesh whose values are read off g at the corresponding Matsubara indices.

For a block_gf, the DLR mesh is built once and reused across blocks (the underlying cppdlr SVD is not cheap). For a product mesh, the operation is applied to the imfreq axis selected by the template parameter pack <N, Ns...>.

It throws an exception if the input imfreq mesh of g does not cover the Matsubara frequency range of the constructed DLR mesh.

Template Parameters

N	Index of the imfreq axis to transform when g has a product mesh. Must be 0 for non-product meshes.
Ns	Additional axis indices for nested product-mesh transformations.
G	A MemoryGf or block_gf type with an imfreq mesh on the selected axis.

Parameters

g	Input Green's function on an imfreq mesh (or product mesh containing one).
w_max	DLR energy cutoff used to build the dlr_imfreq mesh.
eps	DLR accuracy used to build the dlr_imfreq mesh.
symmetrize	If true, build a particle-hole symmetric DLR mesh.

Returns: A Gf on a freshly built dlr_imfreq mesh, sampled from g.

Definition at line 291 of file dlr.hpp.

◆ make_gf_dlr_imtime()

template<int N = 0, int... Ns, typename G>
requires (MemoryGf<G> or is_block_gf_v<G>)

auto triqs::gfs::make_gf_dlr_imtime ( G const & g )

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

Build a DLR imaginary-time Green's function from a DLR-coefficient or DLR-Matsubara input.

The output lives on the imaginary-time DLR mesh associated with the input's DLR grid. Block and product meshes are handled recursively.

Template Parameters

N	Index of the mesh component to transform (default \( 0 \)).
Ns	Additional mesh component indices for product meshes.
G	The type of the input Green's function.

Parameters

g	A Green's function on a Matsubara or coefficient DLR mesh.

Returns: The Green's function evaluated on the imaginary-time DLR mesh.

Definition at line 185 of file dlr.hpp.

◆ make_gf_imfreq()

template<int N = 0, int... Ns, typename G>
requires (MemoryGf<G> or is_block_gf_v<G>)

auto triqs::gfs::make_gf_imfreq	(	G const &	g,
		long	n_iw )

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

Build a uniform Matsubara Green's function from any DLR representation.

Evaluates the DLR Green's function on a regular Matsubara mesh with \( n_{i\omega} \) positive Matsubara frequencies. Input on imaginary-time or Matsubara DLR meshes are first converted to DLR coefficients internally.

Template Parameters

N	Index of the mesh component to transform (default \( 0 \)).
Ns	Additional mesh component indices for product meshes.
G	The type of the input Green's function.

Parameters

g	A Green's function on any DLR mesh.
n_iw	Number of positive Matsubara frequencies in the output mesh.

Returns: The Green's function evaluated on a uniform Matsubara mesh.

Definition at line 407 of file dlr.hpp.

◆ make_gf_imtime()

template<int N = 0, int... Ns, typename G>
requires (MemoryGf<G> or is_block_gf_v<G>)

auto triqs::gfs::make_gf_imtime	(	G const &	g,
		long	n_tau )

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

Build a uniform imaginary-time Green's function from any DLR representation.

Evaluates the DLR Green's function on a regular imaginary-time mesh of size \( n_{\tau} \). Inputs on imaginary-time or Matsubara DLR meshes are first converted to DLR coefficients internally.

Template Parameters

N	Index of the mesh component to transform (default \( 0 \)).
Ns	Additional mesh component indices for product meshes.
G	The type of the input Green's function.

Parameters

g	A Green's function on any DLR mesh.
n_tau	Number of points of the output imaginary-time mesh.

Returns: The Green's function evaluated on a uniform imaginary-time mesh.

Definition at line 375 of file dlr.hpp.

◆ tau_L2_norm()

template<typename G>
requires (MemoryGf<G> or is_block_gf_v<G>)

auto triqs::gfs::tau_L2_norm ( G const & g )

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

Calculate the \( L^2 \) norm of a DLR Green's function.

Template Parameters

G	The type of the input Green's function.

Parameters

g	A Green's function on any DLR mesh.

Returns: The \( L^2 \) norm of the input Green's function, either as a scalar (if target rank is 0) or as an array of norms for each element in the target domain (if target rank is greater than 0).

Definition at line 434 of file dlr.hpp.

Detailed Description

Functions

Function Documentation

◆ enforce_discontinuity()

◆ find_w_max()

◆ fit_gf_dlr() [1/2]

◆ fit_gf_dlr() [2/2]

◆ make_gf_dlr()

◆ make_gf_dlr_imfreq() [1/2]

◆ make_gf_dlr_imfreq() [2/2]

◆ make_gf_dlr_imtime()

◆ make_gf_imfreq()

◆ make_gf_imtime()

◆ tau_L2_norm()