#include <triqs_modest/downfolding.hpp>

Detailed Description

The projector that downfolds the one-body dispersion (ν) onto local orbitals (m).

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

A downfoldin 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

$$ P_{(a,m_{a})\nu}^{\sigma}(\mathbf{k})\equiv e^{-i \mathbf{k} R_a} \bra{\chi_{m_{a}}^{R_a \sigma}}\ket{\psi_{\nu}^{\sigma}(\mathbf{k})},$$

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

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

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 72 of file downfolding.hpp.

Public Member Functions
nda::matrix_const_view< dcomplex >	P (long sigma, long k_idx) const
	P^σ(k)_mν, returned as a matrix in (m ν)

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
	n_bands_per_k [k_idx, σ'] = # of nu

nda::array< dcomplex, 4 >	P_k
	Pk[alpha][k_idx, σ', m_alpha, nu].

spin_kind_e	spin_kind

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

Member Function Documentation

◆ P()

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

inline

P^σ(k)_mν, returned as a matrix in (m ν)

Definition at line 85 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

printing

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

n_bands_per_k [k_idx, σ'] = # of nu

Definition at line 75 of file downfolding.hpp.

◆ P_k

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

Pk[alpha][k_idx, σ', m_alpha, nu].

Definition at line 74 of file downfolding.hpp.

◆ spin_kind

spin_kind_e triqs::modest::downfolding_projector::spin_kind

Definition at line 73 of file downfolding.hpp.

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

triqs_modest/downfolding.hpp
triqs_modest/downfolding.cpp