Local Green's function

Detailed Description

Central quantity of the DMFT self-consistency condition - the local Green's function.

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.

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 local Green's function on a \( M \times M \) mesh.
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.

Local Green's function using a TB Hamiltonian
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.
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.

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 other overloads (gloc) for more details.

Template Parameters

Mesh	The mesh type.

Parameters

mesh	(DLR) imaginary frequency mesh.
obe	One-body elements on a fixed grid.
mu	Chemical potential \( \mu \).

Returns

\( G_{\mathrm{loc}}^{\sigma} \), the local Green's function.

Definition at line 161 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 other overloads (gloc) for more details.

Template Parameters

Mesh	The mesh type.

Parameters

mesh	The mesh on which \( G_{\mathrm{loc}} \) will be computed.
obe	One-body elements containing the TB Hamiltonian.
mu	Chemical potential \( \mu \).
opt	Container for options related to integration of the BZ.

Returns

\( G_{\mathrm{loc}}^{\sigma} \), the local Green's function.

Definition at line 165 of file obe_tb.hpp.

gloc() [3/4]

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 local Green's function on a \( M \times M \) mesh.

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 \(N_{M}\).

Template Parameters

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

Parameters

obe	One-body elements on a fixed grid.
mu	Chemical potential \( \mu \).
Sigma_dynamic	The dynamic part of the embedded self-energy in the embedded view, \( \Sigma_{\text{dynamic}}[\alpha, \sigma] \).
Sigma_static	The static part of the embedded self-energy in the embedded view, \( \Sigma_{\text{static}}[\alpha,\sigma] \).

Returns

\( G_{\mathrm{loc}}^{\sigma} \), the local Green's function in the full \( \mathcal{C} \) space.

Definition at line 107 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 other overloads (gloc) for more details.

Template Parameters

Mesh	The mesh type.

Parameters

obe	One-body elements containing the TB Hamiltonian.
mu	Chemical potential \( \mu \).
Sigma_dynamic	The dynamic part of the embedded self-energy in the embedded view, \( \Sigma_{\text{dynamic}}[\alpha, \sigma] \).
Sigma_static	The static part of the embedded self-energy in the embedded view, \( \Sigma_{\text{static}}[\alpha,\sigma] \).
opt	Container for options related to integration of the BZ.

Returns

\( G_{\mathrm{loc}}^{\sigma} \), the local Green's function.

Definition at line 123 of file obe_tb.hpp.