triqs::gfs::imfreq

#include <triqs/mesh/imfreq.hpp>

Detailed Description

Imaginary frequency mesh type.

An imaginary frequency mesh is defined by the number of positive frequencies \( N_{i\omega_n} \geq 0 \), the inverse temperature \( \beta > 0 \), its particle statistics and an option (all_frequencies vs positive_frequencies_only).

An imaginary frequency mesh has the following properties:

• Each mesh point is identified by a unique index \( n \in \{ n_{\text{min}}, n_{\text{min}} + 1, \dots, n_{\text{max}} - 1, n_{\text{max}} \} \), where \( n_{\text{max}} = N_{i\omega_n} - 1 \) and

  \[ n_{\text{min}} = \begin{cases} -N_{i\omega_n} & \text{if fermionic statistics and all frequencies} \\ -N_{i\omega_n} + 1 & \text{if bosonic statistics and all frequencies} \\ 0 & \text{if positive frequencies only} \end{cases} \; . \]

• The size of the mesh is \( N = n_{\text{max}} - n_{\text{min}} + 1 \).
• An index \( n \) is mapped to the corresponding data index \( d \) by the function \( d(n) = n - n_{\text{min}} \). The inverse map is \( n(d) = d + n_{\text{min}} \).
• An index \( n \) is mapped to the corresponding value \( i\omega_n \equiv i\omega(n) \) by the function

  \[ i\omega_n = \begin{cases} i\frac{2n \pi}{\beta} & \text{if fermionic statistics} \\ i\frac{(2n + 1) \pi}{\beta} & \text{if bosonic statistics} \end{cases} \; . \]

Green's function containers that are based on an imaginary frequency mesh store the function values at the discrete frequency points \( i\omega_n \), i.e. \( f_n = f(i\omega(n)) \), and to evaluate the function at a specific Matsubara frequency \( i\omega_n \) with \( n_{\text{min}} \leq n \leq n_{\text{max}} \) the container simply returns the corresponding \( f_n \).

Definition at line 102 of file imfreq.hpp.

Inheritance diagram for triqs::gfs::imfreq:

Classes
class	mesh_point_t
	Mesh point of a triqs::mesh::imfreq mesh. More...

Public Types
using	data_index_t = long
	Data index type.
using	index_t = long
	Index type.
enum class	option
	Enum to specify which frequencies should be included in the mesh. More...
using	value_t = matsubara_freq
	Value type.

Public Member Functions
	imfreq ()=default
	Default constructor constructs an empty mesh.
	imfreq (double beta, statistic_enum statistic, energy_t w_max, option opt=option::all_frequencies)
	Construct an imaginary frequency mesh with a threshold for the largest positive Matsubara frequency on the mesh.
	imfreq (double beta, statistic_enum statistic, long n_iw=1025, option opt=option::all_frequencies)
	Construct an imaginary frequency mesh with \( N_{i\omega_n} \geq 0 \) positive Matsubara frequencies.
auto	begin () const
	Get an iterator to the beginning of the mesh.
double	beta () const noexcept
	Get the inverse temperature \( \beta \).
auto	cbegin () const
	Get a const iterator to the beginning of the mesh.
auto	cend () const
	Get a const iterator to the end of the mesh.
void	deserialize (auto &ar)
	Deserialize the mesh from a generic archive.
auto	end () const
	Get an iterator to the end of the mesh.
bool	eval_to_zero (index_t n) const
	Return true if the given Matsubara index \( n \) is not valid (see is_index_valid()).
bool	eval_to_zero (matsubara_freq iw) const
	Return true if the Matsubara index of the given Matsubara frequency \( i\omega_n \) is not valid.
bool	eval_to_zero (mesh_point_t mp) const
	Return true if the Matsubara index of the given mesh point is not valid (see is_index_valid()).
long	first_index () const
	Get the first Matsubara index, i.e. \( n_{\text{min}} \).
imfreq	get_positive_freq () const
	Get a new mesh with the same \( \beta \), particle statistics and \( N_{i\omega_n} \) but only positive frequencies.
bool	is_index_valid (index_t n) const
	Check if a Matsubara index \( n \) is valid.
long	last_index () const
	Get the last Matsubara index, i.e. \( n_{\text{max}} \).
uint64_t	mesh_hash () const noexcept
	Get the hash value of the mesh.
long	n_iw () const noexcept
	Get the number of positive Matsubara frequencies \( N_{i\omega_n} \).
mesh_point_t	operator() (long n) const
	Function call operator to access a mesh point by its Matsubara index \( n \in \{ n_{\text{min}}, \dots, n_{\text{max}} \} \).
bool	operator== (imfreq const &m) const
	Equal-to comparison operator compares \( \beta \), the particle statistics, \( N_{i\omega_n} \) and whether all or only positive frequencies are in the mesh.
mesh_point_t	operator[] (closest_mesh_point_t< value_t > const &cmp) const
	Subscript operator to access a mesh point by a Matsubara frequency \( i\omega_n \) contained in a triqs::mesh::closest_mesh_point_t.
mesh_point_t	operator[] (long d) const
	Subscript operator to access a mesh point by its data index \( d \in \{0, 1, \ldots, N-1\} \).
bool	positive_only () const
	Is the mesh restricted to positive Matsubara frequencies?
void	serialize (auto &ar) const
	Serialize the mesh to a generic archive.
long	size () const noexcept
	Get the size \( N \) of the mesh, i.e. the total number of mesh points.
statistic_enum	statistic () const noexcept
	Get the particle statistics.
data_index_t	to_data_index (closest_mesh_point_t< value_t > const &cmp) const
	Map a Matsubara frequency \( i\omega_n \) contained in a triqs::mesh::closest_mesh_point_t to its data index \( d(i\omega_n) \).
data_index_t	to_data_index (index_t n) const noexcept
	Map a Matsubara index \( n \in \{ n_{\text{min}}, \dots, n_{\text{max}} \} \) to its corresponding data index \( d(n) \).
data_index_t	to_data_index (matsubara_freq const &iw) const noexcept
	Map a Matsubara frequency \( i\omega_n \) to its data index \( d(i\omega_n) \).
index_t	to_index (closest_mesh_point_t< value_t > const &cmp) const
	Map a Matsubara frequency \( i\omega_n \) contained in a triqs::mesh::closest_mesh_point_t to its Matsubara index \( n(i\omega_n) \).
index_t	to_index (data_index_t d) const
	Map a data index \( d \in \{0, 1, \ldots, N-1\} \) to the corresponding Matsubara index \( n(d) \).
matsubara_freq	to_value (index_t n) const
	Map a Matsubara index \( n \in \{ n_{\text{min}}, \dots, n_{\text{max}} \} \) to its corresponding Matsubara frequency \( i\omega_n \).
std::complex< double >	w_max () const
	Get the complex value of the largest positive Matsubara frequency in the mesh.
Public Member Functions inherited from triqs::mesh::tail_fitter_handle
tail_fitter &	get_tail_fitter () const
	Get the triqs::mesh::tail_fitter object.
tail_fitter &	get_tail_fitter (double tail_fraction, int n_tail_max=tail_fitter::default_n_tail_max, std::optional< int > expansion_order={}) const
	Construct a new triqs::mesh::tail_fitter object with the given parameters and return it.
void	set_tail_fit_parameters (double tail_fraction, int n_tail_max=tail_fitter::default_n_tail_max, std::optional< int > expansion_order={}) const
	Set the pointer to a new tail-fitter object constructed with the given parameters.

Static Public Member Functions
static std::string	hdf5_format ()
	Get the HDF5 format tag.

Friends
void	h5_read (h5::group g, std::string name, imfreq &m)
	Read a triqs::mesh::imfreq mesh from HDF5.
void	h5_write (h5::group g, std::string name, imfreq const &m)
	Write a triqs::mesh::imfreq mesh to HDF5.
std::ostream &	operator<< (std::ostream &sout, imfreq const &m)
	Write a triqs::mesh::imfreq mesh to a std::ostream.

Member Enumeration Documentation

option

enum class triqs::mesh::imfreq::option

strong

Enum to specify which frequencies should be included in the mesh.

The following options are supported:

• all_frequencies: Include negative and positive frequencies.
• positive_frequencies_only: Include only positive frequencies.

Definition at line 167 of file imfreq.hpp.

Constructor & Destructor Documentation

imfreq() [1/2]

triqs::mesh::imfreq::imfreq ( double beta, statistic_enum statistic, long n_iw = 1025, option opt = option::all_frequencies )

inline

Construct an imaginary frequency mesh with \( N_{i\omega_n} \geq 0 \) positive Matsubara frequencies.

Parameters

beta	Inverse temperature \( \beta > 0 \).
statistic	Particle statistics.
n_iw	Number of positive Matsubara frequencies, i.e. \( N_{i\omega_n} \).
opt	Whether to use all or only the positive frequencies.

Definition at line 181 of file imfreq.hpp.

imfreq() [2/2]

triqs::mesh::imfreq::imfreq ( double beta, statistic_enum statistic, energy_t w_max, option opt = option::all_frequencies )

inline

Construct an imaginary frequency mesh with a threshold for the largest positive Matsubara frequency on the mesh.

The number of positive Matsubara frequencies is related to the threshold \( \omega_{\text{max}} \) by

\[ N_{i\omega_n} = \begin{cases} \left\lfloor \frac{\omega_{\text{max}} \beta / \pi - 1}{2} \right\rfloor & \text{if fermionic statistics} \\ \left\lfloor \frac{\omega_{\text{max}} \beta / \pi}{2} \right\rfloor & \text{if bosonic statistics} \\ \end{cases} \; . \]

Parameters

beta	Inverse temperature \( \beta > 0 \).
statistic	Particle statistics.
w_max	Threshold \( \omega_{\text{max}} > 0 \) for the largest positive Matsubara frequency.
opt	Whether to use all or only the positive frequencies.

Definition at line 206 of file imfreq.hpp.

Member Function Documentation

deserialize()

void triqs::mesh::imfreq::deserialize ( auto & ar )

inline

Deserialize the mesh from a generic archive.

Parameters

ar	Archive to deserialize from.

Definition at line 390 of file imfreq.hpp.

is_index_valid()

bool triqs::mesh::imfreq::is_index_valid ( index_t n ) const

inlinenodiscard

Check if a Matsubara index \( n \) is valid.

Parameters

n	Matsubara index \( n \) to check.

Returns

True if \( n_\text{min} \leq n \leq n_\text{max} \), false otherwise.

Definition at line 221 of file imfreq.hpp.

operator()()

mesh_point_t triqs::mesh::imfreq::operator() ( long n ) const

inlinenodiscard

Function call operator to access a mesh point by its Matsubara index \( n \in \{ n_{\text{min}}, \dots, n_{\text{max}} \} \).

Parameters

n	Matsubara index \( n \) of the mesh point.

Returns

mesh_point_t with the Matsubara index \( n \), data index \( d(n) = n - n_{\text{min}} \) and same \( \beta \), particle statistics and hash value as the current mesh.

Definition at line 308 of file imfreq.hpp.

operator[]() [1/2]

mesh_point_t triqs::mesh::imfreq::operator[] ( closest_mesh_point_t< value_t > const & cmp ) const

inlinenodiscard

Subscript operator to access a mesh point by a Matsubara frequency \( i\omega_n \) contained in a triqs::mesh::closest_mesh_point_t.

Parameters

cmp	triqs::mesh::closest_mesh_point_t containing \( i\omega_n \).

Returns

mesh_point_t with the Matsubara index \( n(i\omega_n) = n \), data index \( d(i\omega_n) = n - n_{\text{min}} \) and same \( \beta \), particle statistics and hash value as the current mesh.

Definition at line 298 of file imfreq.hpp.

operator[]() [2/2]

mesh_point_t triqs::mesh::imfreq::operator[] ( long d ) const

inlinenodiscard

Subscript operator to access a mesh point by its data index \( d \in \{0, 1, \ldots, N-1\} \).

Parameters

d	Data index \( d \) of the mesh point.

Returns

mesh_point_t with the Matsubara index \( n(d) = d + n_{\text{min}} \), data index \( d \) and same \( \beta \), particle statistics and hash value as the current mesh.

Definition at line 288 of file imfreq.hpp.

serialize()

void triqs::mesh::imfreq::serialize ( auto & ar ) const

inline

Serialize the mesh to a generic archive.

Parameters

ar	Archive to serialize to.

Definition at line 384 of file imfreq.hpp.

to_data_index() [1/3]

data_index_t triqs::mesh::imfreq::to_data_index ( closest_mesh_point_t< value_t > const & cmp ) const

inlinenodiscard

Map a Matsubara frequency \( i\omega_n \) contained in a triqs::mesh::closest_mesh_point_t to its data index \( d(i\omega_n) \).

Parameters

cmp	triqs::mesh::closest_mesh_point_t containing \( i\omega_n \).

Returns

Data index \( d(i\omega_n) = n - n_{\text{min}} \).

Definition at line 253 of file imfreq.hpp.

to_data_index() [2/3]

data_index_t triqs::mesh::imfreq::to_data_index ( index_t n ) const

inlinenodiscardnoexcept

Map a Matsubara index \( n \in \{ n_{\text{min}}, \dots, n_{\text{max}} \} \) to its corresponding data index \( d(n) \).

Parameters

n	Matsubara index \( n \) to map.

Returns

Data index \( d(n) = n - n_{\text{min}} \).

Definition at line 230 of file imfreq.hpp.

to_data_index() [3/3]

data_index_t triqs::mesh::imfreq::to_data_index ( matsubara_freq const & iw ) const

inlinenodiscardnoexcept

Map a Matsubara frequency \( i\omega_n \) to its data index \( d(i\omega_n) \).

Parameters

iw	Matsubara frequency to map.

Returns

Data index \( d(i\omega_n) = n - n_{\text{min}} \).

Definition at line 241 of file imfreq.hpp.

to_index() [1/2]

index_t triqs::mesh::imfreq::to_index ( closest_mesh_point_t< value_t > const & cmp ) const

inlinenodiscard

Map a Matsubara frequency \( i\omega_n \) contained in a triqs::mesh::closest_mesh_point_t to its Matsubara index \( n(i\omega_n) \).

Parameters

cmp	triqs::mesh::closest_mesh_point_t containing \( i\omega_n \).

Returns

Matsubara index \( n(i\omega_n) = n \).

Definition at line 276 of file imfreq.hpp.

to_index() [2/2]

index_t triqs::mesh::imfreq::to_index ( data_index_t d ) const

inlinenodiscard

Map a data index \( d \in \{0, 1, \ldots, N-1\} \) to the corresponding Matsubara index \( n(d) \).

Parameters

d	Data index \( d \) to map.

Returns

Matsubara index \( n(d) = d + n_{\text{min}} \).

Definition at line 264 of file imfreq.hpp.

to_value()

matsubara_freq triqs::mesh::imfreq::to_value ( index_t n ) const

inlinenodiscard

Map a Matsubara index \( n \in \{ n_{\text{min}}, \dots, n_{\text{max}} \} \) to its corresponding Matsubara frequency \( i\omega_n \).

Parameters

n	Matsubara index \( n \) to map.

Returns

Matsubara frequency \( i\omega_n \).

Definition at line 317 of file imfreq.hpp.

Friends And Related Symbol Documentation

h5_read

void h5_read ( h5::group g, std::string name, imfreq & m )

friend

Read a triqs::mesh::imfreq mesh from HDF5.

Parameters

g	h5::group to be read from.
name	Name of the subgroup.
m	Mesh object to be read into.

Definition at line 418 of file imfreq.hpp.

h5_write

void h5_write ( h5::group g, std::string name, imfreq const & m )

friend

Write a triqs::mesh::imfreq mesh to HDF5.

Parameters

g	h5::group to be written to.
name	Name of the subgroup.
m	Mesh object to be written.

Definition at line 402 of file imfreq.hpp.

operator<<

std::ostream & operator<< ( std::ostream & sout, imfreq const & m )

friend

Write a triqs::mesh::imfreq mesh to a std::ostream.

Parameters

sout	std::ostream object.
m	Mesh to be written.

Returns

Reference to std::ostream object.

Definition at line 374 of file imfreq.hpp.

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The documentation for this class was generated from the following file:

• triqs/mesh/imfreq.hpp