GF reality and hermiticity

Detailed Description

Reality and hermiticity checks and operations on Green's functions.

Predicates such as triqs::gfs::is_gf_real and triqs::gfs::is_gf_hermitian, together with related operations such as complex conjugation.

Functions
template<typename G> requires (is_gf_v<G>)
G::regular_type	triqs::gfs::conj (G const &g)
	Complex-conjugate a Green's function, returning a new Green's function.
template<typename G> requires (is_gf_v<G> or is_block_gf_v<G>)
G::regular_type::real_t	triqs::gfs::imag (G const &g)
	Take the imaginary part of a Green's function (no check), returning a new Green's function with a real target.
template<typename G> requires (is_gf_v<G> or is_block_gf_v<G>)
bool	triqs::gfs::is_gf_hermitian (G const &g, double tolerance=1.e-12)
	Test whether a Green's function satisfies the hermitian symmetry up to a tolerance \( \epsilon \).
template<typename G> requires (is_gf_v<G>)
bool	triqs::gfs::is_gf_real (G const &g, double tolerance=1.e-13)
	Test whether a Green's function is real up to a tolerance.
template<typename G>
bool	triqs::gfs::is_gf_real_in_tau (G const &g, double tolerance=1.e-12)
	Test whether a Matsubara Green's function corresponds to a real imaginary-time Green's function.
template<typename T, typename Layout>
gf< mesh::imfreq, T >	triqs::gfs::make_gf_from_real_gf (gf_const_view< mesh::imfreq, T, Layout > g)
	Build a full-mesh Matsubara Green's function from one defined on positive frequencies only.
template<typename G> requires (is_gf_v<G> or is_block_gf_v<G>)
G::regular_type	triqs::gfs::make_hermitian (G const &g)
	Symmetrize a Green's function so that it satisfies the hermitian symmetry.
template<typename G> requires (is_gf_v<G> or is_block_gf_v<G>)
G::regular_type	triqs::gfs::make_real_in_tau (G const &g)
	Symmetrize a Matsubara Green's function so that its imaginary-time partner is real-valued.
template<typename G> requires (is_gf_v<G> or is_block_gf_v<G>)
G::regular_type::real_t	triqs::gfs::real (G const &g)
	Take the real part of a Green's function (no check), returning a new Green's function with a real target.

Function Documentation

conj()

template<typename G>
requires (is_gf_v<G>)

G::regular_type triqs::gfs::conj

(

G const &

g

)

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

Complex-conjugate a Green's function, returning a new Green's function.

Template Parameters

G	The type of the Green's function.

Parameters

g	The Green's function.

Returns

A new Green's function holding the complex conjugate of g.

Definition at line 391 of file functions2.hpp.

imag()

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

G::regular_type::real_t triqs::gfs::imag

(

G const &

g

)

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

Take the imaginary part of a Green's function (no check), returning a new Green's function with a real target.

Template Parameters

G	Any Gf, BlockGf or Block2Gf type.

Parameters

g	The Green's function.

Returns

A Green's function holding the imaginary part of g.

Definition at line 355 of file functions2.hpp.

is_gf_hermitian()

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

bool triqs::gfs::is_gf_hermitian	(	G const &	g,
		double	tolerance = 1.e-12 )

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

Test whether a Green's function satisfies the hermitian symmetry up to a tolerance \( \epsilon \).

Depending on the mesh and target rank, one of the following relations is checked:

• \( G(i\omega) \approx \frac{1}{2} [ G(i\omega) + G^*(-i\omega) ] \)
• \( G(\tau) \approx \frac{1}{2} [ G(\tau) + G^*(\tau) ] \)
• \( G_{i,j}(i\omega) \approx \frac{1}{2} [ G_{i,j}(i\omega) + G_{j,i}^*(i\omega) ] \)
• \( G_{i,j}(\tau) \approx \frac{1}{2} [ G_{i,j}(\tau) + G_{j,i}^*(\tau) ] \)
• \( G_{i,j,k,l}(i\omega) \approx \frac{1}{2} [ G_{i,j,k,l}(i\omega)] + G_{k,l,i,j}^*(i\omega) ] \)
• \( G_{i,j,k,l}(\tau) \approx \frac{1}{2} [ G_{i,j,k,l}(\tau) + G_{k,l,i,j}(\tau) ] \)

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

Template Parameters

G	The type of the Green's function.

Parameters

g	The Green's function to check.
tolerance	Tolerance \( \epsilon \) for the check (default \( 10^{-12} \)).

Returns

True if the property holds at every point of the mesh.

Definition at line 123 of file imfreq.hpp.

is_gf_real()

template<typename G>
requires (is_gf_v<G>)

bool triqs::gfs::is_gf_real	(	G const &	g,
		double	tolerance = 1.e-13 )

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

Test whether a Green's function is real up to a tolerance.

Template Parameters

G	Any Green's function type.

Parameters

g	The Green's function.
tolerance	Tolerance threshold.

Returns

True if g is real up to tolerance.

Definition at line 312 of file functions2.hpp.

is_gf_real_in_tau()

template<typename G>

bool triqs::gfs::is_gf_real_in_tau	(	G const &	g,
		double	tolerance = 1.e-12 )

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

Test whether a Matsubara Green's function corresponds to a real imaginary-time Green's function.

The criterion checked, up to tolerance \( \epsilon \), is \( G_{i,j,\dots}(i\omega) \approx G_{i,j,\dots}^*(-i\omega) \) for every element of the target space and for every Matsubara frequency.

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

Template Parameters

G	The type of the Green's function.

Parameters

g	The Matsubara Green's function to check.
tolerance	Tolerance \( \epsilon \) for the check (default \( 10^{-12} \)).

Returns

True if the property holds at every point of the mesh.

Definition at line 211 of file imfreq.hpp.

make_gf_from_real_gf()

template<typename T, typename Layout>

gf< mesh::imfreq, T > triqs::gfs::make_gf_from_real_gf

(

gf_const_view< mesh::imfreq, T, Layout >

g

)

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

Build a full-mesh Matsubara Green's function from one defined on positive frequencies only.

Uses the relation \( G(-i\omega_n) = G(i\omega_n)^* \), valid for Green's functions with a real imaginary-time counterpart, to fill in the negative-frequency part of the mesh.

Template Parameters

T	Target type.
Layout	Layout type.

Parameters

g	Green's function defined on a positive-only Matsubara mesh.

Returns

The Green's function on the full (positive and negative) Matsubara mesh.

Definition at line 62 of file imfreq.hpp.

make_hermitian()

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

G::regular_type triqs::gfs::make_hermitian

(

G const &

g

)

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

Symmetrize a Green's function so that it satisfies the hermitian symmetry.

Depending on the mesh and target rank, one of the following transformations is applied:

• \( G(i\omega) \rightarrow \frac{1}{2} [ G(i\omega) + G^*(-i\omega) ] \)
• \( G(\tau) \rightarrow \frac{1}{2} [ G(\tau) + G^*(\tau) ] \)
• \( G_{i,j}(i\omega) \rightarrow \frac{1}{2} [ G_{i,j}(i\omega) + G_{j,i}^*(i\omega) ] \)
• \( G_{i,j}(\tau) \rightarrow \frac{1}{2} [ G_{i,j}(\tau) + G_{j,i}^*(\tau) ] \)
• \( G_{i,j,k,l}(i\omega) \rightarrow \frac{1}{2} [ G_{i,j,k,l}(i\omega)] + G_{k,l,i,j}^*(i\omega) ] \)
• \( G_{i,j,k,l}(\tau) \rightarrow \frac{1}{2} [ G_{i,j,k,l}(\tau) + G_{k,l,i,j}(\tau) ] \)

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

Template Parameters

G	The type of the Green's function.

Parameters

g	The Green's function to symmetrize.

Returns

The symmetrized Green's function.

Definition at line 251 of file imfreq.hpp.

make_real_in_tau()

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

G::regular_type triqs::gfs::make_real_in_tau

(

G const &

g

)

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

Symmetrize a Matsubara Green's function so that its imaginary-time partner is real-valued.

The transformation applied is \( G_{i,j,\dots}(i\omega) \rightarrow \frac{1}{2} [ G_{i,j,\dots}(i\omega) + G_{i,j,\dots}^*(-i\omega) ] \).

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

Template Parameters

G	The type of the Green's function.

Parameters

g	The Matsubara Green's function to symmetrize.

Returns

The symmetrized Green's function.

Definition at line 323 of file imfreq.hpp.

real()

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

G::regular_type::real_t triqs::gfs::real

(

G const &

g

)

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

Take the real part of a Green's function (no check), returning a new Green's function with a real target.

Template Parameters

G	Any Gf, BlockGf or Block2Gf type.

Parameters

g	The Green's function.

Returns

A Green's function holding the real part of g.

Definition at line 336 of file functions2.hpp.