Tight-binding Hamiltonian \( H(\mathbf{k}) \) on a 3D lattice.
A tight-binding Hamiltonian is defined by a set of lattice vectors \( \mathbf{R} \) and the associated hopping (overlap) matrices \( t(\mathbf{R}) \) between orbitals. It represents
\[ H(\mathbf{k}) = \sum_\mathbf{R} t(\mathbf{R}) \, e^{2 \pi i \, \mathbf{k} \cdot \mathbf{R}} \; ,
\]
and provides access to the hopping matrices, band energies and eigenvectors at a list of k-points, comparison, stream output and HDF5 serialization.
Definition at line 40 of file tb_hk.hpp.
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| tb_hk () |
| | Default constructor: a single zero hopping matrix for the lattice vector at the origin.
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| | tb_hk (fourier_polynomial< 2, 3 > fp) |
| | Construct a tight-binding Hamiltonian from an existing matrix-valued Fourier polynomial.
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| | tb_hk (std::vector< std::array< long, 3 > > Rs, std::vector< nda::array< dcomplex, 2 > > hoppings) |
| | Construct a tight-binding Hamiltonian from lattice vectors and their hopping matrices.
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| nda::array< double, 2 > | eigenvalues (nda::array_view< double, 2 > k) const |
| | Compute the band-basis energies of \( H(\mathbf{k}) \) for a list of k-points.
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| eigenvectors_t | eigenvectors (nda::array_view< double, 2 > k) const |
| | Compute the band-basis energies and eigenvectors of \( H(\mathbf{k}) \) for a list of k-points.
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| auto | elements () |
| | Get a lazy range over the (R-vector, hopping-matrix) pairs.
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| auto | elements () const |
| | Get a lazy range over the (R-vector, hopping-matrix) pairs (const overload).
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| std::vector< std::array< long, 3 > > | get_R_list () const |
| | Get the list of real-space lattice vectors.
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| auto | hoppings () |
| | Get a lazy range over the hopping matrices, one per R-vector.
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| auto | hoppings () const |
| | Get a lazy range over the hopping matrices, one per R-vector (const overload).
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| long | n_orbitals () const |
| | Get the number of orbitals, i.e. the dimension of the Hamiltonian matrices.
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| bool | operator== (tb_hk const &tb) const |
| | Equal-to comparison operator.
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| fourier_polynomial ()=default |
| | Default constructor: construct an empty Fourier polynomial.
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| auto | get_coeff_arr () |
| | Get the packed coefficient array.
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| long | get_R_idx (std::array< long, kdim > R) const |
| | Get the storage index of a given R-vector.
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| auto const & | get_R_list () const |
| | Get the list of real-space lattice vectors.
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| auto const & | get_R_mat () const |
| | Get the packed matrix of lattice vectors.
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| long | n_R () const |
| | Get the number of R-vectors.
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| auto | operator() (T... ks) const |
| | Evaluate the Fourier polynomial, optionally as a lazy CLEF expression.
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fourier_polynomial & | operator= (fourier_polynomial const &)=default |
| | Copy assignment operator.
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| auto | operator[] (std::array< long, kdim > R) |
| | Access the coefficient associated with a given R-vector.
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void | h5_read_impl (h5::group g, std::string const &name, char const *exp_format) |
| | HDF5 read helper for derived classes: reads R_list and coeff_arr in place; rebuilds R_mat.
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void | h5_write_impl (h5::group g, std::string const &name, char const *format) const |
| | HDF5 write helper for derived classes: writes R_list and coeff_arr under the given format tag.
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nda::array< dcomplex, coeff_dim+1 > | coeff_arr |
| | Packed Fourier coefficients, of shape [nR, coeff_shape...].
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std::vector< std::array< long, kdim > > | R_list |
| | Lattice vectors \( \mathbf{R} \) as integer arrays.
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nda::matrix< double > | R_mat |
| | Lattice vectors as doubles, of shape [nR, kdim], for BLAS.
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| nda::array< double, 2 > triqs::experimental::lattice::tb_hk::eigenvalues |
( |
nda::array_view< double, 2 > | k | ) |
const |
Compute the band-basis energies of \( H(\mathbf{k}) \) for a list of k-points.
For each k-point the Hamiltonian is diagonalized and the resulting energies are returned in the band basis:
\[ H_{k,mn} = U_{\mathbf{k},ma}^\dagger \left( \sum_j t(\mathbf{R}_j)_{ab} e^{2 \pi i \mathbf{k} \cdot
\mathbf{R}_j} \right) U_{\mathbf{k},nb} \; ,
\]
with lattice vectors \( \{\mathbf{R}_j\} \) and associated overlap (hopping) matrices \( \{t(\mathbf{R}_j)_{ab}\} \).
The k-points must be given in units of the reciprocal lattice vectors (from 0 to 1).
- Parameters
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| k | k-points as an array of shape [nk, 3] in units of the reciprocal lattice vectors. |
- Returns
- Band-basis energies as a real matrix of shape [nk, nbands].
Definition at line 36 of file tb_hk.cpp.
| std::pair< nda::array< double, 2 >, nda::array< dcomplex, 3 > > triqs::experimental::lattice::tb_hk::eigenvectors |
( |
nda::array_view< double, 2 > | k | ) |
const |
Compute the band-basis energies and eigenvectors of \( H(\mathbf{k}) \) for a list of k-points.
For each k-point the Hamiltonian is diagonalized and both the band energies and the eigenvectors are returned:
\[ H_{k,mn} = U_{\mathbf{k},ma}^\dagger \left( \sum_j t(\mathbf{R}_j)_{ab} e^{2 \pi i \mathbf{k} \cdot
\mathbf{R}_j} \right) U_{\mathbf{k},nb} \; ,
\]
with lattice displacements \( \{\mathbf{R}_j\} \) and associated overlap (hopping) matrices \( \{t(\mathbf{R}_j)_{ab}\} \).
The k-points must be given in units of the reciprocal lattice vectors (from 0 to 1).
- Parameters
-
| k | k-points as an array of shape [nk, 3] in units of the reciprocal lattice vectors. |
- Returns
- Tuple containing the band-basis energies as a real matrix of shape [nk, nbands] and the complex eigenvectors as an array of shape [nk, nbands, nOrbitals].
Definition at line 46 of file tb_hk.cpp.
| std::ostream & operator<< |
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std::ostream & | out, |
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tb_hk const & | tb ) |
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friend |
Write a tight-binding Hamiltonian to a std::ostream.
Prints the number of lattice vectors and the orbital dimension, and, for small Hamiltonians (fewer than 12 lattice vectors), the hopping matrix associated with each lattice vector.
- Parameters
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| out | std::ostream object. |
| tb | Tight-binding Hamiltonian to be written. |
- Returns
- Reference to the std::ostream object.
Definition at line 185 of file tb_hk.hpp.
