Imaginary time and frequency meshes

Detailed Description

Mesh types for imaginary time and frequency domains.

Topics
	Matsubara frequencies
	Tools to represent and manipulate Matsubara frequencies.

Classes
class	triqs::mesh::chebyshev
	Chebyshev imaginary time mesh type. More...
class	triqs::mesh::dlr_imfreq
	Imaginary frequency discrete Lehmann representation (DLR) mesh type. More...
class	triqs::mesh::dlr_imtime
	Imaginary time discrete Lehmann representation (DLR) mesh type. More...
class	triqs::mesh::imfreq
	Imaginary frequency mesh type. More...
class	triqs::mesh::imtime
	Imaginary time mesh type. More...

Functions
auto	triqs::mesh::evaluate (chebyshev const &m, auto const &f, double tau)
	Barycentric interpolation of a function \( f \) at an imaginary time point \( \tau \in [0, \beta] \).
double	triqs::mesh::evaluate (dlr_imfreq const &m,...)=delete
	Evaluating a function \( f \) at a given Matsubara frequency \( i\omega_n \) is deleted for triqs::mesh::dlr_imfreq meshes.
double	triqs::mesh::evaluate (dlr_imtime const &m,...)=delete
	Evaluating a function \( f \) at a given imaginary time point \( \tau \in [0, \beta] \) is deleted for triqs::mesh::dlr_imtime meshes.
auto	triqs::mesh::evaluate (imfreq const &m, auto const &f, matsubara_freq const &iw)
	Evaluate a function \( f \) defined on a triqs::mesh::imfreq mesh at a given Matsubara frequency \(i\omega_n\).
auto	triqs::mesh::evaluate (imtime const &m, auto const &f, double tau)
	Linear interpolation of a function \( f \) defined on a triqs::mesh::imtime mesh at an imaginary time point \( \tau \in [0, \beta] \).
dlr_imtime	triqs::mesh::make_adjoint_mesh (dlr_imfreq const &m)
	Create the adjoint imaginary-time DLR mesh to a given imaginary-frequency DLR mesh.
dlr_imfreq	triqs::mesh::make_adjoint_mesh (dlr_imtime const &m)
	Create the adjoint imaginary-frequency DLR mesh to a given imaginary-time DLR mesh.
imtime	triqs::mesh::make_adjoint_mesh (imfreq const &m, long n_tau=-1)
	Create the adjoint imaginary-time mesh to a given imaginary-frequency mesh.
imfreq	triqs::mesh::make_adjoint_mesh (imtime const &m, long n_iw=-1)
	Create the adjoint imaginary-frequency mesh to a given imaginary-time mesh.

Function Documentation

evaluate() [1/3]

auto triqs::mesh::evaluate ( chebyshev const & m, auto const & f, double tau )

inline

#include <triqs/mesh/chebyshev.hpp>

Barycentric interpolation of a function \( f \) at an imaginary time point \( \tau \in [0, \beta] \).

Uses the barycentric interpolation formula:

\[ f(\tau) = \frac{\sum_{i=0}^{N-1} \frac{w_i}{x - x_i} f_i}{\sum_{i=0}^{N-1} \frac{w_i}{x - x_i}} \]

where \( x = 2\tau/\beta - 1 \) is the scaled coordinate, \( x_i \) are the Chebyshev points, \( w_i \) are the barycentric weights, and \( f_i = f(i) \) are the function values.

Parameters

m	triqs::mesh::chebyshev mesh.
f	Callable object \( f \) containing the function values \( f_n = f(\tau_n) \) at the mesh points.
tau	Imaginary time point \( \tau \in [0, \beta] \) at which to interpolate the function.

Returns

Barycentric interpolation of \( f(\tau) \).

Definition at line 459 of file chebyshev.hpp.

evaluate() [2/3]

auto triqs::mesh::evaluate	(	imfreq const &	m,
		auto const &	f,
		matsubara_freq const &	iw )

#include <triqs/mesh/imfreq.hpp>

Evaluate a function \( f \) defined on a triqs::mesh::imfreq mesh at a given Matsubara frequency \(i\omega_n\).

Parameters

m	triqs::mesh::imfreq mesh.
f	Callable object \( f \) containing the function values \( f_n = f(i\omega_n) \) at the mesh points.
iw	triqs::mesh::matsubara_freq \( i\omega_n \) at which to evaluate the function.

Returns

Function value \( f(i\omega_n) \).

Definition at line 470 of file imfreq.hpp.

evaluate() [3/3]

auto triqs::mesh::evaluate	(	imtime const &	m,
		auto const &	f,
		double	tau )

#include <triqs/mesh/imtime.hpp>

Linear interpolation of a function \( f \) defined on a triqs::mesh::imtime mesh at an imaginary time point \( \tau \in [0, \beta] \).

It simply calls the triqs::mesh::imtime::evaluate method of the mesh.

Parameters

m	triqs::mesh::imtime mesh.
f	Callable object \( f \) containing the function values \( f_n = f(\tau(n)) \) at the mesh points.
tau	Imaginary time point \( \tau \) at which to interpolate the function.

Returns

Linear interpolation of \( f(\tau) \).

Definition at line 178 of file imtime.hpp.

make_adjoint_mesh() [1/4]

dlr_imtime triqs::mesh::make_adjoint_mesh ( dlr_imfreq const & m )

inlinenodiscard

#include <triqs/mesh/adjoint.hpp>

Create the adjoint imaginary-time DLR mesh to a given imaginary-frequency DLR mesh.

It constructs the imaginary-time DLR mesh from the given imaginary-frequency DLR mesh.

Parameters

m	Input mesh.

Returns

Imaginary time DLR mesh.

Definition at line 92 of file adjoint.hpp.

make_adjoint_mesh() [2/4]

dlr_imfreq triqs::mesh::make_adjoint_mesh ( dlr_imtime const & m )

inlinenodiscard

#include <triqs/mesh/adjoint.hpp>

Create the adjoint imaginary-frequency DLR mesh to a given imaginary-time DLR mesh.

It constructs the imaginary-frequency DLR mesh from the given imaginary-time DLR mesh.

Parameters

m	Input mesh.

Returns

Imaginary frequency DLR mesh.

Definition at line 81 of file adjoint.hpp.

make_adjoint_mesh() [3/4]

imtime triqs::mesh::make_adjoint_mesh ( imfreq const & m, long n_tau = -1 )

inlinenodiscard

#include <triqs/mesh/adjoint.hpp>

Create the adjoint imaginary-time mesh to a given imaginary-frequency mesh.

If \( N = -1 \), the size of the imaginary time mesh is set to \( N = 6 (n_{\text{max}} + 1) + 1 \), where \( n_{\text{max}} \) is the largest positive Matsubara index in the given imaginary frequency mesh.

Parameters

m	Input mesh.
n_tau	Size of the imaginary time mesh.

Returns

Imaginary time mesh with the same \( \beta \) and particle statistics as the given imaginary frequency mesh and size \( N \).

Definition at line 67 of file adjoint.hpp.

make_adjoint_mesh() [4/4]

imfreq triqs::mesh::make_adjoint_mesh ( imtime const & m, long n_iw = -1 )

inlinenodiscard

#include <triqs/mesh/adjoint.hpp>

Create the adjoint imaginary-frequency mesh to a given imaginary-time mesh.

If \( N_{i\omega_n} = -1 \), the number of positive Matsubara frequencies is set to \( N_{i\omega_n} = N / 6 \), where \( N \) is the size of the given imaginary time mesh.

Parameters

m	Input mesh.
n_iw	Number of positive Matsubara frequencies, i.e. \( N_{i\omega_n} \).

Returns

Imaginary frequency mesh with the same \( \beta \) and particle statistics as the given imaginary time mesh and \( N_{i\omega_n} \) positive Matsubara frequencies.

Definition at line 50 of file adjoint.hpp.