#include <triqs/lattice/bravais_lattice.hpp>

Detailed Description

A Bravais lattice class.

A Bravais lattice in \( d \) dimensions is defined by

a set of linear independent d-dimensional basis vectors \( \{ \mathbf{a}_1, \dots, \mathbf{a}_d \} \) and
the positions, \( \{ \mathbf{r}_1, \dots, \mathbf{r}_m \} \), of atomic orbitals within each unit cell.

Optionally, the atomic orbitals can be named.

The infinite lattice points \( \mathbf{R}^{\mathbf{n}} \) (see bravais_lattice::point_t) are generated by

\[ \mathbf{R}^{\mathbf{n}} = \sum_{i=1}^{d} \mathbf{a}_i n_i = \mathbf{A} \mathbf{n} \; , \]

where \( \mathbf{n} = (n_1, \dots, n_d) \in \mathbb{Z}^d \) is an index vector (or more formally the vector \(\mathbf{R}^{\mathbf{n}} \) in the lattice basis) and \( \mathbf{A} = \big( \mathbf{a}_1 \cdots \mathbf{a}_d \big) \) is the matrix with the basis vectors as its columns.

Note: Although the supported dimensions are 1, 2 and 3, the index vectors are always 3-dimensional, i.e. \(\mathbf{n} = (n_1, n_2, n_3) \). Indices \( n_j \) with \( j > d \) are simply ignored when computing the corresponding lattice vector.

Definition at line 75 of file bravais_lattice.hpp.

Classes
class	point_t
	Lattice point of a Bravais lattice. More...

Public Types
using	index_t = std::array<long, 3>
	Index type for lattice points on the Bravais lattice.

Public Member Functions
	bravais_lattice ()
	Construct a simple cubic lattice with lattice constant \( a = 1 \).
	bravais_lattice (matrix_t const &units, std::vector< r_t > orbital_positions=std::vector< r_t >{{0, 0, 0}}, std::vector< std::string > atom_orb_name={})
	Construct a Bravais Lattice with given basis vectors and positions of atomic orbitals with optional names.
bool	contains (r_t const &) const
	Check if a given vector \( \mathbf{r} \) is part of the domain.
void	deserialize (auto &ar)
	Deserialize the Bravais lattice from a generic archive.
template<typename R>
r_t	lattice_to_real_coordinates (R const &v) const
	Transform a vector \( \mathbf{v} \) from the lattice basis \( \{ \mathbf{a}_1, \dots, \mathbf{a}_d \} \) to the standard basis \( \{ \mathbf{e}_1, \dots, \mathbf{e}_d \} \).
long	n_orbitals () const
	Get the number of atomic orbitals in the unit cell.
int	ndim () const
	Get the number of dimensions of the Bravais lattice.
bool	operator== (bravais_lattice const &bl) const
	Equal-to comparison operator.
auto const &	orbital_names () const
	Get the list of orbital names.
auto const &	orbital_positions () const
	Get the list of atomic orbital positions \( \{\mathbf{r}_1, \dots, \mathbf{r}_m\} \).
template<typename R>
r_t	real_to_lattice_coordinates (R const &v_tilde) const
	Transform a vector \( \tilde{\mathbf{v}} \) from the standard basis \( \{ \mathbf{e}_1, \dots, \mathbf{e}_d \} \) to the lattice basis \( \{ \mathbf{a}_1, \dots, \mathbf{a}_d \} \).
void	serialize (auto &ar) const
	Serialize the Bravais lattice to a generic archive.
point_t	to_point (index_t const &n) const
	Convert an index vector to a lattice point.
matrix_t const &	units () const
	Get the matrix \( \mathbf{A}^T \) containing 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, bravais_lattice &bl)
	Read a triqs::lattice::bravais_lattice from HDF5.
void	h5_write (h5::group g, std::string const &name, bravais_lattice const &bl)
	Write a triqs::lattice::bravais_lattice to HDF5.
std::ostream &	operator<< (std::ostream &sout, bravais_lattice const &bl)
	Write a triqs::lattice::bravais_lattice to a std::ostream.

Constructor & Destructor Documentation

◆ bravais_lattice() [1/2]

triqs::lattice::bravais_lattice::bravais_lattice ( )

inline

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

The only atomic orbital is placed at the origin with no name.

Definition at line 193 of file bravais_lattice.hpp.

◆ bravais_lattice() [2/2]

triqs::lattice::bravais_lattice::bravais_lattice	(	matrix_t const &	units,
		std::vector< r_t >	orbital_positions = std::vector<r_t>{{0, 0, 0}},
		std::vector< std::string >	atom_orb_name = {} )

Construct a Bravais Lattice with given basis vectors and positions of atomic orbitals with optional names.

The matrix \( \mathbf{A}^T \) containing the basis vectors as its rows is required to be square. The number of dimensions of the Bravais lattices is determined by the size of the matrix.

Parameters

units	Matrix with the basis vectors \( \{ \mathbf{a}_1, \dots, \mathbf{a}_d \} \) as its rows.
orbital_positions	Atomic orbital positions \( \{ \mathbf{r}_1, \dots, \mathbf{r}_m \} \) in the unit cell.
atom_orb_name	Optional names for the atomic orbitals.

Definition at line 38 of file bravais_lattice.cpp.

Member Function Documentation

◆ deserialize()

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

inline

Deserialize the Bravais lattice from a generic archive.

Parameters

ar	Archive to deserialize from.

Definition at line 300 of file bravais_lattice.hpp.

◆ lattice_to_real_coordinates()

template<typename R>

r_t triqs::lattice::bravais_lattice::lattice_to_real_coordinates ( R const & v ) const

inlinenodiscard

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

The transformed vector is given by

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

where \( A \) is the matrix with the 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 239 of file bravais_lattice.hpp.

◆ operator==()

bool triqs::lattice::bravais_lattice::operator== ( bravais_lattice const & bl ) const

inline

Equal-to comparison operator.

Returns: True, if the number of dimensions, unit cell basis vectors and the atomic orbital positions and names are equal.

Definition at line 270 of file bravais_lattice.hpp.

◆ real_to_lattice_coordinates()

template<typename R>

r_t triqs::lattice::bravais_lattice::real_to_lattice_coordinates ( R const & v_tilde ) const

inlinenodiscard

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

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

Parameters

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

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

Definition at line 253 of file bravais_lattice.hpp.

◆ serialize()

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

inline

Serialize the Bravais lattice to a generic archive.

Parameters

ar	Archive to serialize to.

Definition at line 294 of file bravais_lattice.hpp.

◆ to_point()

point_t triqs::lattice::bravais_lattice::to_point ( index_t const & n ) const

inlinenodiscard

Convert an index vector to a lattice point.

Parameters

n	Index vector \( \mathbf{n} \).

Returns: bravais_lattice::point_t corresponding to the given index array.

Definition at line 263 of file bravais_lattice.hpp.

◆ h5_read

void h5_read	(	h5::group	g,
		std::string const &	name,
		bravais_lattice &	bl )

friend

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

Parameters

g	h5::group to be read from.
name	Name of the subgroup.
bl	Bravais lattice object to be read into.

Definition at line 74 of file bravais_lattice.cpp.

◆ h5_write

void h5_write	(	h5::group	g,
		std::string const &	name,
		bravais_lattice const &	bl )

friend

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

Parameters

g	h5::group to be written to.
name	Name of the subgroup.
bl	Bravais lattice object to be written.

Definition at line 65 of file bravais_lattice.cpp.

◆ operator<<

std::ostream & operator<<	(	std::ostream &	sout,
		bravais_lattice const &	bl )

friend

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

Parameters

sout	std::ostream object.
bl	Bravais lattice to be written.

Returns: Reference to std::ostream object.

Definition at line 281 of file bravais_lattice.hpp.

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

triqs/lattice/bravais_lattice.hpp
triqs/lattice/bravais_lattice.cpp

Detailed Description

Classes

Public Types

Public Member Functions

Static Public Member Functions

Friends

Constructor & Destructor Documentation

◆ bravais_lattice() [1/2]

◆ bravais_lattice() [2/2]

Member Function Documentation

◆ deserialize()

◆ lattice_to_real_coordinates()

◆ operator==()

◆ real_to_lattice_coordinates()

◆ serialize()

◆ to_point()

Friends And Related Symbol Documentation

◆ h5_read

◆ h5_write

◆ operator<<