#include <triqs/lattice/tight_binding.hpp>

Detailed Description

Tight-binding Hamiltonian on a Bravais lattice with fully localised orbitals.

The Hamiltonian is parametrised by a set of lattice displacements \( \{ \mathbf{R}_j \} \) (given in units of the lattice basis vectors) and the associated overlap (hopping) matrices \( \{ t_{\mathbf{R}_j} \} \) between orbitals in the unit cell. The Bloch Hamiltonian in reciprocal space is obtained by the discrete Fourier transform

\[ h_{\mathbf{k}} = \sum_j t_{\mathbf{R}_j} \, e^{2 \pi i \, \mathbf{k} \cdot \mathbf{R}_j} \; , \]

where the momentum \( \mathbf{k} \) is expressed in units of the reciprocal lattice basis vectors.

The orbital overlap within a unit cell (the on-site block at \( \mathbf{R} = 0 \)) is the identity matrix unless explicitly overridden by the user-provided hoppings.

Definition at line 83 of file tight_binding.hpp.

Public Member Functions
	tight_binding (bravais_lattice bl, hopping_dict hoppings)
	Construct a tight-binding Hamiltonian on a given Bravais lattice from a hopping dictionary.
	tight_binding (bravais_lattice bl, std::vector< nda::vector< long > > displ_vec, std::vector< nda::matrix< dcomplex > > overlap_mat_vec)
	Construct a tight-binding Hamiltonian on a given Bravais lattice from explicit displacement and overlap lists.
auto	dispersion (int n_l) const
	Compute the dispersion on a regular Brillouin zone mesh with n_l points per dimension.
template<typename K> requires (nda::ArrayOfRank<K, 1> or nda::ArrayOfRank<K, 2>)
auto	dispersion (K const &k) const
	Compute the dispersion, i.e. the eigenvalue spectrum of \( h_{\mathbf{k}} \), for a given momentum vector (or array of momentum vectors).
auto	dispersion (mesh::brzone const &k_mesh) const
	Compute the dispersion on a given Brillouin zone mesh.
auto const &	displ_vec () const
	Get the list of displacement vectors, in units of the lattice basis vectors.
auto	fourier (int n_l) const
	Compute the Fourier transform on a regular Brillouin zone mesh with n_l points per dimension.
template<typename K> requires (nda::ArrayOfRank<K, 1> or nda::ArrayOfRank<K, 2>)
auto	fourier (K const &k) const
	Compute the Fourier transform for a given momentum vector (or array of momentum vectors).
auto	fourier (mesh::brzone const &k_mesh) const
	Compute the Fourier transform on a given Brillouin zone mesh.
bravais_lattice const &	lattice () const
	Get the underlying Bravais lattice.
template<typename R>
r_t	lattice_to_real_coordinates (R const &x) const
	Transform a vector from the lattice basis to the standard basis.
long	n_orbitals () const
	Number of orbitals (also the size of the Bloch Hamiltonian matrix \( h_{\mathbf{k}} \)).
bool	operator!= (tight_binding const &tb) const
	Not-equal-to comparison operator (negation of operator==()).
bool	operator== (tight_binding const &tb) const
	Equal-to comparison operator. True if the underlying lattice, displacements and overlap matrices all match.
auto const &	overlap_mat_vec () const
	Get the list of overlap (hopping) matrices, aligned with the displacement vectors.

Static Public Member Functions
static tight_binding	h5_read_construct (h5::group g, std::string subgroup_name)
	Construct a tight-binding Hamiltonian by reading it from HDF5.
static std::string	hdf5_format ()
	Get the HDF5 format tag.

Friends
template<typename F>
void	foreach (tight_binding const &tb, F f)
	Apply a callable to every (displacement, overlap matrix) pair stored in the tight-binding Hamiltonian.
void	h5_write (h5::group fg, std::string subgroup_name, tight_binding const &tb)
	Write a tight-binding Hamiltonian to HDF5.
std::ostream &	operator<< (std::ostream &sout, tight_binding const &tb)
	Write a tight-binding Hamiltonian to a std::ostream.

Constructor & Destructor Documentation

◆ tight_binding() [1/2]

triqs::lattice::tight_binding::tight_binding	(	bravais_lattice	bl,
		std::vector< nda::vector< long > >	displ_vec,
		std::vector< nda::matrix< dcomplex > >	overlap_mat_vec )

Construct a tight-binding Hamiltonian on a given Bravais lattice from explicit displacement and overlap lists.

The matrix structure of each overlap matrix is with respect to the orbitals in the unit cell. The displacement and overlap lists must have the same length, and every overlap matrix must be square with size equal to the number of orbitals in the unit cell.

Parameters

bl	Underlying Bravais lattice.
displ_vec	List of displacement vectors, in units of the lattice basis vectors.
overlap_mat_vec	List of overlap (hopping) matrices, one per displacement.

Definition at line 69 of file tight_binding.cpp.

◆ tight_binding() [2/2]

triqs::lattice::tight_binding::tight_binding	(	bravais_lattice	bl,
		hopping_dict	hoppings )

Construct a tight-binding Hamiltonian on a given Bravais lattice from a hopping dictionary.

Parameters

bl	Underlying Bravais lattice.
hoppings	Hopping dictionary mapping displacement vectors to their overlap matrices.

Definition at line 76 of file tight_binding.cpp.

Member Function Documentation

◆ dispersion() [1/3]

auto triqs::lattice::tight_binding::dispersion ( int n_l ) const

inline

Compute the dispersion on a regular Brillouin zone mesh with n_l points per dimension.

Parameters

n_l	Number of grid-points along each reciprocal direction.

Returns: Tensor-valued Green's function defined on the regular Brillouin zone mesh, with its data initialised with the band energies at every mesh point.

Definition at line 254 of file tight_binding.hpp.

◆ dispersion() [2/3]

template<typename K>
requires (nda::ArrayOfRank<K, 1> or nda::ArrayOfRank<K, 2>)

auto triqs::lattice::tight_binding::dispersion ( K const & k ) const

inline

Compute the dispersion, i.e. the eigenvalue spectrum of \( h_{\mathbf{k}} \), for a given momentum vector (or array of momentum vectors).

Parameters

k	Momentum vector (or an array of momentum vectors) in units of the reciprocal lattice basis vectors.

Returns: Real-valued array of length n_orbitals containing the band energies at \( \mathbf{k} \), or an array of such band-energy arrays when an array of momenta is passed.

Definition at line 219 of file tight_binding.hpp.

◆ dispersion() [3/3]

auto triqs::lattice::tight_binding::dispersion ( mesh::brzone const & k_mesh ) const

inline

Compute the dispersion on a given Brillouin zone mesh.

Parameters

k_mesh Brillouin zone mesh on which to evaluate the band energies.

Returns: Tensor-valued Green's function defined on k_mesh, with its data initialised with the band energies at every mesh point (one real value per orbital).

Definition at line 240 of file tight_binding.hpp.

◆ fourier() [1/3]

auto triqs::lattice::tight_binding::fourier ( int n_l ) const

inline

Compute the Fourier transform on a regular Brillouin zone mesh with n_l points per dimension.

Parameters

n_l	Number of grid-points along each reciprocal direction.

Returns: Matrix-valued Green's function defined on the regular Brillouin zone mesh, with its data initialised with the Fourier transform \( h_{\mathbf{k}} \) at every mesh point.

Definition at line 205 of file tight_binding.hpp.

◆ fourier() [2/3]

template<typename K>
requires (nda::ArrayOfRank<K, 1> or nda::ArrayOfRank<K, 2>)

auto triqs::lattice::tight_binding::fourier ( K const & k ) const

inline

Compute the Fourier transform for a given momentum vector (or array of momentum vectors).

The Bloch Hamiltonian is given by

\[ h_{\mathbf{k}} = \sum_j t_{\mathbf{R}_j} \, e^{2 \pi i \, \mathbf{k} \cdot \mathbf{R}_j} \; , \]

with lattice displacements \( \{ \mathbf{R}_j \} \) and associated overlap (hopping) matrices \( \{ t_{\mathbf{R}_j} \} \). The momentum \( \mathbf{k} \) is expressed in units of the reciprocal lattice basis vectors.

Parameters

k	Momentum vector (or an array of momentum vectors) in units of the reciprocal lattice basis vectors.

Returns: Complex matrix \( h_{\mathbf{k}} \) (or an array of such matrices, one per input momentum).

Definition at line 163 of file tight_binding.hpp.

◆ fourier() [3/3]

auto triqs::lattice::tight_binding::fourier ( mesh::brzone const & k_mesh ) const

inline

Compute the Fourier transform on a given Brillouin zone mesh.

Parameters

k_mesh Brillouin zone mesh on which to evaluate the Bloch Hamiltonian.

Returns: Matrix-valued Green's function defined on k_mesh, with its data initialised with the Fourier transform \( h_{\mathbf{k}} \) at every mesh point.

Definition at line 189 of file tight_binding.hpp.

◆ h5_read_construct()

tight_binding triqs::lattice::tight_binding::h5_read_construct	(	h5::group	g,
		std::string	subgroup_name )

inlinestatic

Construct a tight-binding Hamiltonian by reading it from HDF5.

Parameters

g	h5::group to be read from.
subgroup_name	Name of the subgroup.

Returns: The reconstructed tight-binding Hamiltonian.

Definition at line 310 of file tight_binding.hpp.

◆ lattice_to_real_coordinates()

template<typename R>

r_t triqs::lattice::tight_binding::lattice_to_real_coordinates ( R const & x ) const

inline

Transform a vector from the lattice basis to the standard basis.

Equivalent to calling lattice_to_real_coordinates() on the underlying Bravais lattice.

Parameters

x	Vector in the lattice basis.

Returns: Vector in the standard basis.

Definition at line 131 of file tight_binding.hpp.

◆ foreach

template<typename F>

void foreach	(	tight_binding const &	tb,
		F	f )

friend

Apply a callable to every (displacement, overlap matrix) pair stored in the tight-binding Hamiltonian.

Parameters

tb	Tight-binding Hamiltonian to iterate over.
f	Callable invoked as f(R, t_R) for each stored displacement R and its overlap matrix t_R.

Definition at line 142 of file tight_binding.hpp.

◆ h5_write

void h5_write	(	h5::group	fg,
		std::string	subgroup_name,
		tight_binding const &	tb )

friend

Write a tight-binding Hamiltonian to HDF5.

Parameters

fg	h5::group to be written to.
subgroup_name	Name of the subgroup.
tb	Tight-binding Hamiltonian to be written.

Definition at line 294 of file tight_binding.hpp.

◆ operator<<

std::ostream & operator<<	(	std::ostream &	sout,
		tight_binding const &	tb )

friend

Write a tight-binding Hamiltonian to a std::ostream.

Parameters

sout	std::ostream object.
tb	Tight-binding Hamiltonian to be written.

Returns: Reference to std::ostream object.

Definition at line 276 of file tight_binding.hpp.

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

triqs/lattice/tight_binding.hpp
triqs/lattice/tight_binding.cpp

Detailed Description

Public Member Functions

Static Public Member Functions

Friends

Constructor & Destructor Documentation

◆ tight_binding() [1/2]

◆ tight_binding() [2/2]

Member Function Documentation

◆ dispersion() [1/3]

◆ dispersion() [2/3]

◆ dispersion() [3/3]

◆ fourier() [1/3]

◆ fourier() [2/3]

◆ fourier() [3/3]

◆ h5_read_construct()

◆ lattice_to_real_coordinates()

Friends And Related Symbol Documentation

◆ foreach

◆ h5_write

◆ operator<<