Local Green's function

Detailed Description

Compute the local Green's function defined as: $$ [ G_{\mathrm{loc}}^{\sigma} ]_{m m'} = \sum_{\mathbf{k}} P_{m\nu}^{\sigma}(\mathbf{k}) \Big [ (\omega + \mu)\delta_{\nu\nu'} - H^{\sigma}_{\nu\nu'}(\mathbf{k}) - [P_{m\nu}^{\sigma}]^{\dagger}\Sigma_{\mathrm{embed}}P_{m'\nu'}^{\sigma}(\mathbf{k}) \Big ]^{-1} [P_{m'\nu'}^{\sigma}]^{\dagger},$$ where \(\omega\) is a frequency (either real- or Matsubra), \(\mu\) is the chemical potential, \(H(\mathbf{k})\) is the one-body Hamiltonian, \(P(\mathbf{k})\) are the projectors from the band to the orbital basis, and \(\Sigma_{\mathrm{embed}}\) is the embedded self-energy.

Functions
template<typename Mesh >
block2_gf< Mesh, matrix_valued >	triqs::modest::gloc (Mesh const &mesh, one_body_elements_tb const &obe, double mu, triqs::lattice::bz_int_options const &opt)
	Compute the local Green's function without a self-energy.
template<typename Mesh >
block2_gf< Mesh, matrix_valued >	triqs::modest::gloc (one_body_elements_tb const &obe, double mu, block2_gf< Mesh, matrix_valued > const &Sigma_dynamic, nda::array< nda::matrix< dcomplex >, 2 > const &Sigma_static, triqs::lattice::bz_int_options const &opt)
	Compute the local Green's function without a self-energy.

Local Green's function using a fixed k-grid
template<typename Mesh >
block2_gf< Mesh, matrix_valued >	triqs::modest::gloc (one_body_elements_on_grid const &obe, double mu, block2_gf< Mesh, matrix_valued > const &Sigma_dynamic, nda::array< nda::matrix< dcomplex >, 2 > const &Sigma_static)
	compute G𝓒 local Green's function on Mesh(MxM)
template<typename Mesh >
block2_gf< Mesh, matrix_valued >	triqs::modest::gloc (Mesh const &mesh, one_body_elements_on_grid const &obe, double mu)
	Compute the local Green's function without a self-energy.

Function Documentation

gloc() [1/4]

template<typename Mesh >

block2_gf< Mesh, matrix_valued > triqs::modest::gloc	(	Mesh const &	mesh,
		one_body_elements_on_grid const &	obe,
		double	mu
	)

#include <triqs_modest/gloc_fixed_grid.hpp>

Compute the local Green's function without a self-energy.

See gloc for more details.

Template Parameters

Mesh	The mesh type

Parameters

mesh	mesh triqs::meshes::{imfreq, dlr_imfreq}
obe	one-body elements
mu	chemical potential

Returns

The local Green's function.

Definition at line 146 of file gloc_fixed_grid.hpp.

gloc() [2/4]

template<typename Mesh >

block2_gf< Mesh, matrix_valued > triqs::modest::gloc	(	Mesh const &	mesh,
		one_body_elements_tb const &	obe,
		double	mu,
		triqs::lattice::bz_int_options const &	opt
	)

#include <triqs_modest/obe_tb.hpp>

Compute the local Green's function without a self-energy.

See gloc for more details.

Template Parameters

Mesh

Parameters

mesh	The mesh on which Gloc will be computed
obe	A one_body_elements object containing the tb_hamiltonian
mu	Chemical potential
opt	Container for options related integration of the BZ

Returns

gloc[alpha][sigma], the local Green's function

Definition at line 147 of file obe_tb.hpp.

gloc() [3/4]

template<typename Mesh >

template< typename Mesh > block2_gf< Mesh, matrix_valued > triqs::modest::gloc	(	one_body_elements_on_grid const &	obe,
		double	mu,
		block2_gf< Mesh, matrix_valued > const &	Sigma_dynamic,
		nda::array< nda::matrix< dcomplex >, 2 > const &	Sigma_static
	)

#include <triqs_modest/gloc_fixed_grid.hpp>

compute G𝓒 local Green's function on Mesh(MxM)

Template Parameters

Mesh	The mesh type (triqs::mesh::{dlr_imfreq,imfreq})

Parameters

obe	one_body_elements_on_grid
mu	chemical potential
Sigma_dynamic	The dynamic part of the embedded self-energy in the embedded view, Sigma_dynamic[alpha, sigma].
Sigma_static	The static part of the embedded self-energy in the embedded view, Sigma_static[alpha,sigma]

Returns

gloc[0, sigma], the local Green's function in the full C space.

When the one-body dispersion is defined as fixed k-grid, which is the case when working with DFT codes (e.g., VASP, Wien2k, Elk) or performing charge self-consistent calculations with any DFT code, \(H(\mathbf{k})\) is diagonal in the band basis and reduces to \(\varepsilon_{\nu}^{\sigma}(\mathbf{k})\). The local Green's function becomes: $$ [ G_{\mathrm{loc}}^{\sigma} ]_{m m'} = \sum_{\mathbf{k}} P_{m\nu}^{\sigma}(\mathbf{k}) \Big [ (\omega + \mu - \varepsilon_{\nu}^{\sigma}(\mathbf{k}))\delta_{\nu\nu'} - [P_{m\nu}^{\sigma}]^{\dagger}\Sigma_{\mathrm{embed}}P_{m'\nu'}^{\sigma}(\mathbf{k}) \Big ]^{-1} [P_{m'\nu'}^{\sigma}]^{\dagger}.$$ For performance reasons, we can avoid performing the matrix inverstion in the larger band basis ( \(N_{\nu}\)) using the Woodbury formula which allows us to perform the matrix inversion in the smaller orbital basis \(N_{M}\).

Definition at line 92 of file gloc_fixed_grid.hpp.

gloc() [4/4]

template<typename Mesh >

block2_gf< Mesh, matrix_valued > triqs::modest::gloc	(	one_body_elements_tb const &	obe,
		double	mu,
		block2_gf< Mesh, matrix_valued > const &	Sigma_dynamic,
		nda::array< nda::matrix< dcomplex >, 2 > const &	Sigma_static,
		triqs::lattice::bz_int_options const &	opt
	)

#include <triqs_modest/obe_tb.hpp>

Compute the local Green's function without a self-energy.

See gloc for more details.

Template Parameters

Mesh

Parameters

obe	A one_body_elements object containing the tb_hamiltonian
mu	Chemical potential
Sigma_dynamic	The dynamic part of the embedded self-energy.
Sigma_static	The static part of the embedded self-energy.
opt	Container for options related integration of the BZ

Returns

gloc[alpha][sigma], the local Green's function

Definition at line 105 of file obe_tb.hpp.