#include <triqs/lattice/brillouin_zone.hpp>

Detailed Description

A Brillouin zone class.

Bring triqs::lattice::brillouin_zone into the triqs::mesh namespace.

A Brillouin zone is the primitive unit cell in reciprocal space. It is defined by the set of reciprocal lattice basis vectors \( \{ \mathbf{b}_1, \dots, \mathbf{b}_d \} \), which satisfy

\[ \mathbf{b}_i \cdot \mathbf{a}_j = 2 \pi \delta_{ij} \; , \]

where \( \{ \mathbf{a}_1, \dots, \mathbf{a}_d \} \) are the basis vectors of the Bravais lattice in real space.

In matrix notation, we can write this as a system of linear equations \( A^T B = 2 \pi I \), where \( A = \big( \mathbf{a}_1 \cdots \mathbf{a}_d \big) \) and \( B = \big( \mathbf{b}_1 \cdots \mathbf{b}_d \big) \) are the matrices containing the basis vectors as their columns and \( I \) is the identity matrix.

Definition at line 62 of file brillouin_zone.hpp.

Public Types
using	value_t = k_t
	Value type of a Brillouin Zone.

Public Member Functions
	brillouin_zone ()
	Construct a Brillouin zone for a simple cubic lattice with lattice constant \( a = 1 \).
	brillouin_zone (bravais_lattice bl)
	Construct a Brillouin zone for a given Bravais lattice.
bool	contains (k_t const &) const
	Check if a given vector \( \mathbf{k} \) is part of the domain.
void	deserialize (auto &ar)
	Deserialize the Brillouin from a generic archive.
auto const &	lattice () const
	Get the underlying Bravais lattice.
template<typename K>
k_t	lattice_to_real_coordinates (K const &v) const
	Transform a vector \( \mathbf{v} \) from the reciprocal lattice basis \( \{ \mathbf{b}_1, \dots, \mathbf{b}_d \} \) to the standard basis \( \{ \mathbf{e}_1, \dots, \mathbf{e}_d \} \).
auto	ndim () const
	Get the number of dimensions of the underlying Bravais lattice.
	operator bravais_lattice () const
	Conversion to the underlying triqs::lattice::bravais_lattice object.
bool	operator== (brillouin_zone const &bz) const
	Equal-to comparison operator.
template<typename K>
k_t	real_to_lattice_coordinates (K const &v_tilde) const
	Transform a vector \( \tilde{\mathbf{v}} \) from the standard basis \( \{ \mathbf{e}_1, \dots, \mathbf{e}_d \} \) to the reciprocal lattice basis \( \{ \mathbf{b}_1, \dots, \mathbf{b}_d \} \).
matrix_t const &	reciprocal_matrix () const
	Get the matrix \( B^T \) containing the reciprocal basis vectors as its rows.
matrix_t const &	reciprocal_matrix_inv () const
	Get the inverse matrix \( \left( B^T \right)^{-1} \).
void	serialize (auto &ar) const
	Serialize the Brillouin zone to a generic archive.
matrix_t const &	units () const
	Get the matrix \( B^T \) containing the reciprocal basis vectors as its rows.

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

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

Constructor & Destructor Documentation

◆ brillouin_zone() [1/2]

triqs::lattice::brillouin_zone::brillouin_zone ( )

inline

Construct a Brillouin zone for a simple cubic lattice with lattice constant \( a = 1 \).

The reciprocal basis vectors defining the BZ are given by \( B = 2 \pi I \), where \( I \) is the \( 3 \times 3 \) identity matrix.

Definition at line 72 of file brillouin_zone.hpp.

◆ brillouin_zone() [2/2]

triqs::lattice::brillouin_zone::brillouin_zone ( bravais_lattice bl )

explicit

Construct a Brillouin zone for a given Bravais lattice.

The reciprocal basis vectors defining the BZ are given by \( B = 2 \pi \left( A^T \right)^{-1} \), where \( A \) is the matrix containing the basis vectors of the given Bravais lattice as its columns.

Parameters

bl	Bravais lattice.

Definition at line 37 of file brillouin_zone.cpp.

Member Function Documentation

◆ deserialize()

void triqs::lattice::brillouin_zone::deserialize ( auto & ar )

inline

Deserialize the Brillouin from a generic archive.

Parameters

ar	Archive to deserialize from.

Definition at line 166 of file brillouin_zone.hpp.

◆ lattice_to_real_coordinates()

template<typename K>

k_t triqs::lattice::brillouin_zone::lattice_to_real_coordinates ( K const & v ) const

inlinenodiscard

Transform a vector \( \mathbf{v} \) from the reciprocal lattice basis \( \{ \mathbf{b}_1, \dots, \mathbf{b}_d \} \) to the standard basis \( \{ \mathbf{e}_1, \dots, \mathbf{e}_d \} \).

The transformed vector is given by

\[ \tilde{\mathbf{v}} = \sum_{i=1}^{d} \mathbf{b}_i v_i = B \mathbf{v} \; , \]

where \( B \) is the matrix with the reciprocal lattice basis vectors as its columns.

Parameters

v	Vector \( \mathbf{v} \) to be transformed.

Returns: Vector \( \tilde{\mathbf{v}} \) in the standard basis.

Definition at line 118 of file brillouin_zone.hpp.

◆ operator==()

bool triqs::lattice::brillouin_zone::operator== ( brillouin_zone const & bz ) const

inline

Equal-to comparison operator.

Returns: True, if the underlying Bravais lattice and the reciprocal basis vectors are equal.

Definition at line 140 of file brillouin_zone.hpp.

◆ real_to_lattice_coordinates()

template<typename K>

k_t triqs::lattice::brillouin_zone::real_to_lattice_coordinates ( K const & v_tilde ) const

inlinenodiscard

Transform a vector \( \tilde{\mathbf{v}} \) from the standard basis \( \{ \mathbf{e}_1, \dots, \mathbf{e}_d \} \) to the reciprocal lattice basis \( \{ \mathbf{b}_1, \dots, \mathbf{b}_d \} \).

This is the inverse transformation of lattice_to_real_coordinates(). The transformed vector is therefore given by \( \mathbf{v} = B^{-1} \tilde{\mathbf{v}} \).

Parameters

v_tilde Vector \( \tilde{\mathbf{v}} \) in the standard basis.

Returns: Vector \( \mathbf{v} \) in the lattice basis.

Definition at line 132 of file brillouin_zone.hpp.

◆ serialize()

void triqs::lattice::brillouin_zone::serialize ( auto & ar ) const

inline

Serialize the Brillouin zone to a generic archive.

Parameters

ar	Archive to serialize to.

Definition at line 160 of file brillouin_zone.hpp.

◆ h5_read

void h5_read	(	h5::group	g,
		std::string const &	name,
		brillouin_zone &	bz )

friend

Read a triqs::lattice::brillouin_zone from HDF5.

Parameters

g	h5::group to be read from.
name	Name of the subgroup.
bz	Brillouin zone object to be read into.

Definition at line 49 of file brillouin_zone.cpp.

◆ h5_write

void h5_write	(	h5::group	g,
		std::string const &	name,
		brillouin_zone const &	bz )

friend

Write a triqs::lattice::brillouin_zone to HDF5.

Parameters

g	h5::group to be written to.
name	Name of the subgroup.
bz	Brillouin zone object to be written.

Definition at line 43 of file brillouin_zone.cpp.

◆ operator<<

std::ostream & operator<<	(	std::ostream &	sout,
		brillouin_zone const &	bz )

friend

Write a triqs::lattice::brillouin_zone to a std::ostream.

Parameters

sout	std::ostream object.
bz	Brillouin zone to be written.

Returns: Reference to std::ostream object.

Definition at line 149 of file brillouin_zone.hpp.

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

triqs/lattice/brillouin_zone.hpp
triqs/lattice/brillouin_zone.cpp

Detailed Description

Public Types

Public Member Functions

Static Public Member Functions

Friends

Constructor & Destructor Documentation

◆ brillouin_zone() [1/2]

◆ brillouin_zone() [2/2]

Member Function Documentation

◆ deserialize()

◆ lattice_to_real_coordinates()

◆ operator==()

◆ real_to_lattice_coordinates()

◆ serialize()

Friends And Related Symbol Documentation

◆ h5_read

◆ h5_write

◆ operator<<