#include <triqs_modest/downfolding.hpp>

Detailed Description

The projector that downfolds the energy bands onto a set of localized atomic-like orbitals.

A downfolding projector contains the projector, the kind of spin used in the projection, and the number of bands per k-point for cases when a band goes outside of the projection window.

The projectors \(P_{m\nu}^{\sigma}(\mathbf{k})\) connect the Bloch space \({\cal B}\) to \({\cal C}\). The projectors are obtained from DFT codes or Wannier90. They are defined by

\[ P_{(a,m_{a})\nu}^{\sigma}(\mathbf{k})\equiv e^{-i \mathbf{k} R_a} \langle \chi_{m_{a}}^{R_a \sigma} | \psi_{\nu}^{\sigma}(\mathbf{k}) \rangle, \]

where \( | \chi_{m_{a}}^{R_a \sigma} \rangle \) is a Wannier function localized at atom \( a \) with index \(m_a\) at position \(R_a\) and \( | \psi_{\nu}^{\sigma}(\mathbf{k}) \rangle \) is the Kohn-Sham wavefunction. The relation between the Wannier and Bloch function is therefore

\[ | \chi_{m_{a}}^{R_a \sigma} \rangle = \sum_{\mathbf{k} \nu} e^{-i \mathbf{k} R_a} \bigl(P^\sigma_{(a,m_{a})\nu} (\mathbf{k})\bigr)^* | \psi_{\nu}^{\sigma}(\mathbf{k}) \rangle. \]

Some properties:

Basis change in \(\cal C\) space: They are given by a unitary matrix \(U\), the projector transforms as \( P^{'\sigma}_{m\nu}(\mathbf{k}) = U^{\dagger}_{m, m'} P^{\sigma}_{m'\nu}(\mathbf{k}).\)
Partial unitarity property: In general \(P\) is not unitary as \(N_\nu^{\mathbf{k}} > M\). However, if the Wannier functions are reorthonormalized with respect to the truncated band basis, we have \( \sum_{ \nu} P^{\sigma}_{m\nu}(\mathbf{k}) P^{\dagger\sigma}_{\nu m'}(\mathbf{k}) = \delta_{mm'} \).

Definition at line 111 of file downfolding.hpp.

Public Member Functions
nda::matrix_const_view< dcomplex >	P (long sigma, long k_idx) const
	Get \( P_{m\nu}^{\sigma}(\mathbf{k}) \) for a given \( \mathbf{k} \) and \( \sigma \).

downfolding_projector	rotate_local_basis (nda::array< nda::matrix< dcomplex >, 2 > const &U) const
	Rotates the local basis of the downfolding projector.

Public Attributes
nda::array< long, 2 >	n_bands_per_k
	Number of bands for each k-point and \( \sigma \).

nda::array< dcomplex, 4 >	P_k
	Projector \( P_{m\nu}^{\sigma}(\mathbf{k}) \).

spin_kind_e	spin_kind
	Spin kind of the one-body data.

Friends
std::ostream &	operator<< (std::ostream &out, downfolding_projector const &proj)
	Print information about a downfolding_projector object.

Member Function Documentation

◆ P()

nda::matrix_const_view< dcomplex > triqs::modest::downfolding_projector::P	(	long	sigma,
		long	k_idx
	)		const

inline

Get \( P_{m\nu}^{\sigma}(\mathbf{k}) \) for a given \( \mathbf{k} \) and \( \sigma \).

Parameters

sigma	Spin index \( \sigma \).
k_idx	Index of the k-point in the grid.

Returns: Matrix view of \( P_{m\nu}^{\sigma}(\mathbf{k}) \) in \( (m, \nu) \) for the given \( \mathbf{k} \) and \( \sigma \).

Definition at line 132 of file downfolding.hpp.

◆ rotate_local_basis()

downfolding_projector triqs::modest::downfolding_projector::rotate_local_basis ( nda::array< nda::matrix< dcomplex >, 2 > const & U ) const

Rotates the local basis of the downfolding projector.

Definition at line 23 of file downfolding.cpp.

Friends And Related Symbol Documentation

◆ operator<<

std::ostream & operator<<	(	std::ostream &	out,
		downfolding_projector const &	proj
	)

friend

Print information about a downfolding_projector object.

Definition at line 26 of file printing.cpp.

Member Data Documentation

◆ n_bands_per_k

nda::array<long, 2> triqs::modest::downfolding_projector::n_bands_per_k

Number of bands for each k-point and \( \sigma \).

Definition at line 114 of file downfolding.hpp.

◆ P_k

nda::array<dcomplex, 4> triqs::modest::downfolding_projector::P_k

Projector \( P_{m\nu}^{\sigma}(\mathbf{k}) \).

Definition at line 113 of file downfolding.hpp.

◆ spin_kind

spin_kind_e triqs::modest::downfolding_projector::spin_kind

Spin kind of the one-body data.

Definition at line 112 of file downfolding.hpp.

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

triqs_modest/downfolding.hpp
triqs_modest/downfolding.cpp