#include <triqs_modest/embedding.hpp>

Detailed Description

The embedding class.

The embedding class encodes the mapping between impurities and the correlated space \( \mathcal{C} \) (cf. ModEST implementation notes). In particular, it maps the individual "blocks" within an impurity self-energy to a block and spin of the embedded self-energy function, which can then be upfolded (downfolded) to obtain the lattice self-energy and the local Green's function.

Definition at line 32 of file embedding.hpp.

Public Member Functions
	embedding (std::vector< long > sigma_embed_decomposition, std::vector< std::vector< long > > imp_decompositions, nda::array< imp_block_t, 2 > psi, std::vector< std::string > sigma_names)
	Construct an embedding object.
template<typename Mesh>
std::vector< std::pair< block_gf< Mesh, matrix_valued >, block_matrix_t > >	make_zero_imp_self_energies (Mesh const &mesh)
	Create zero-initialized impurity self-energies.
Accessors
imp_block_t	operator[] (long alpha, long sigma) const
	Bracket accessor: \( \psi[\alpha,\sigma] \).
nda::array< imp_block_t, 2 >	psi_map () const
	The mapping table \( \psi \).
long	n_alpha () const
	Number of blocks in \( \alpha \) for the \( \Sigma_{\text{embed}} \).
long	n_sigma () const
	Number of blocks in \( \sigma \) for the \( \Sigma_{\text{embed}} \).
long	n_gamma (long imp_idx) const
	Number of blocks in \( \gamma \) for the \( \Sigma_{\text{imp}} \) [imp_idx].
long	n_impurities () const
	Number of impurities.
std::vector< std::string >	sigma_names () const
	The names of the sigma indices.
std::vector< long >	imp_decomposition (long imp_idx) const
	The impurity decomposition.
C2PY_IGNORE gf_struct2_t	embed_block_structure () const
	Block structure (names and dimensions) for the embedded self-energy \( \Sigma_{\text{embed}} \).
std::vector< gf_struct_t >	imp_block_structure () const
	Block structure (names and dimensions) for each impurity solver.
std::string	description (bool verbosity=false) const
	Summarize the embedding object.
Embedding transformations
Methods that return a new embedding with a modified \( \psi \) map.
bool	operator== (embedding const &other) const =default
embedding	drop_imp (long imp_idx) const
	Remove an impurity from the embedding table \( \psi \).
embedding	replace_imp (long imp_idx_old, long imp_idx_new) const
	Redirect all \( \psi \) entries from one impurity to another.
embedding	swap_sigma (long alpha) const
	Swap the \( \sigma \) (spin) assignment for \( \alpha \) block alpha.
embedding	swap_sigma (std::vector< long > alphas) const
	Swap the \( \sigma \) (spin) assignment for all \( \alpha \) blocks in alphas.
embedding	slice_sigma () const
	Slice the embedding to a single \( \sigma \) channel ("ud").
embedding	split_imp (long imp_idx, std::function< bool(long)> p) const
	Split impurity imp_idx using a predicate.
embedding	split_imp (long imp_idx, std::vector< long > const &block_list) const
	Split impurity imp_idx based on block_list.
embedding	split_imp (long imp_idx, std::initializer_list< const char * > x)=delete
embedding	split_imp_block (long imp_idx, long gamma, std::vector< long > const &new_dims) const
	Split a single \( \gamma \) block of an impurity into multiple blocks.
embedding	merge_embed_block_by_imp () const
	Merge consecutive \( \alpha \) blocks belonging to the same impurity into a single block.
Embed and extract operations
Methods that map data between impurity and embedded representations.
template<int Rank>
std::vector< std::vector< nda::array< dcomplex, Rank > > >	extract (std::vector< nda::array< dcomplex, Rank > > const &X) const
	Extract impurity data from embedded arrays.
template<int Rank>
std::vector< nda::array< dcomplex, Rank > >	embed (std::vector< std::vector< nda::array< dcomplex, Rank > > > const &imps_blocks) const
	Embed impurity data into the full correlated space.
template<typename Mesh>
block2_gf< Mesh, matrix_valued >	embed (std::vector< block_gf< Mesh, matrix_valued > > const &Sigma_imp_vec) const
	Embed Green's function containers.
template<typename Mesh>
std::vector< block_gf< Mesh, matrix_valued > >	extract (block2_gf< Mesh, matrix_valued > const &g_loc) const
	Extract Green's function containers.
template<typename Mesh>
std::pair< block2_gf< Mesh, matrix_valued >, block2_matrix_t >	embed (std::vector< block_gf< Mesh, matrix_valued > > const &Sigma_imp_vec, std::vector< block_matrix_t > const &Sigma_imp_static_vec) const
	Embed pairs Green's function containers and matrices.
block2_matrix_t	embed (std::vector< block_matrix_t > const &Sigma_imp_static_vec) const
	Embed block matrices into the full correlated space.
std::vector< block_matrix_t >	extract (block2_matrix_t const &matrix_C) const
	Extract block matrices from the embedded representation.

Friends
void	h5_read (h5::group g, std::string const &name, embedding &x)
void	h5_write (h5::group g, std::string const &name, embedding const &x)
	h5 read/write
std::ostream &	operator<< (std::ostream &out, embedding const &E)

Constructor & Destructor Documentation

◆ embedding()

triqs::modest::embedding::embedding	(	std::vector< long >	sigma_embed_decomposition,
		std::vector< std::vector< long > >	imp_decompositions,
		nda::array< imp_block_t, 2 >	psi,
		std::vector< std::string >	sigma_names )

Construct an embedding object.

Parameters

sigma_embed_decomposition	Decomposition (list of block sizes) for \( \Sigma_{\text{embed}} \): \([ \alpha ] = \) block size. Note that the decompositions are given for \( \alpha \), \( \gamma \) index, and are independent of \( \sigma \).
imp_decompositions	List of decomposition of solvers: [n_imp][ \( \gamma \)] = block size.
psi	The mapping \( \psi[\alpha,\sigma]\) -> n_imp, \( \gamma \), \( \tau \).
sigma_names	Names for the values of the \( \sigma \) index (e.g., "up", "down").

Definition at line 17 of file embedding.cpp.

Member Function Documentation

◆ description()

std::string triqs::modest::embedding::description ( bool verbosity = false ) const

Summarize the embedding object.

Definition at line 101 of file printing.cpp.

◆ drop_imp()

embedding triqs::modest::embedding::drop_imp ( long imp_idx ) const

Remove an impurity from the embedding table \( \psi \).

Remove the impurity imp_idx, map the corresponding \( \alpha \) blocks to -1 (i.e. no longer connected to an impurity) and relabel the other impurities with a number > imp_idx.

Parameters

imp_idx Index of the impurity to remove from the table \( \psi \).

Returns: New embedding with the updated \( \psi \) map.

Definition at line 274 of file embedding.cpp.

◆ embed() [1/4]

template<typename Mesh>

block2_gf< Mesh, matrix_valued > triqs::modest::embedding::embed ( std::vector< block_gf< Mesh, matrix_valued > > const & Sigma_imp_vec ) const

inline

Embed Green's function containers.

Embed impurity solver self-energies into an embedded self-energy.

\( \Sigma_{\text{embed}}^{ασ} = \Sigma_{\text{imp}}^{ψ(α,σ)} \),
\( \psi(\alpha,\sigma) \to \) (n_imp, \( \gamma \), \( \tau \)).

Parameters

Sigma_imp_vec A list of impurity self-energies.

Returns: \( \Sigma_{\text{embed}}[\alpha, \sigma] \) in \( \mathcal{C} \) space using the embed decomposition.

Definition at line 476 of file embedding.hpp.

◆ embed() [2/4]

template<typename Mesh>

std::pair< block2_gf< Mesh, matrix_valued >, block2_matrix_t > triqs::modest::embedding::embed	(	std::vector< block_gf< Mesh, matrix_valued > > const &	Sigma_imp_vec,
		std::vector< block_matrix_t > const &	Sigma_imp_static_vec ) const

inline

Embed pairs Green's function containers and matrices.

Embed the impurity self-energy decomposed as a list of dynamic and a list of static parts.

Template Parameters

Mesh	The type of mesh.

Parameters

Sigma_imp_vec	A list of dynamic impurity self-energies.
Sigma_imp_static_vec	A list of static impurity self-energies.

Returns: The self-energy in the \( \mathcal{C} \) space as a pair (dynamic, static).

Definition at line 535 of file embedding.hpp.

◆ embed() [3/4]

block2_matrix_t triqs::modest::embedding::embed ( std::vector< block_matrix_t > const & Sigma_imp_static_vec ) const

Embed block matrices into the full correlated space.

Embeds static (frequency-independent) impurity data stored as block matrices into the \( (\alpha, \sigma) \)-indexed embedded representation. This is the matrix analogue of the frequency-dependent embed operation.

Parameters

Sigma_imp_static_vec List of block matrices, one per impurity.

Returns: Embedded block2 matrix in \( \mathcal{C} \) space.

Definition at line 418 of file embedding.cpp.

◆ embed() [4/4]

template<int Rank>

std::vector< nda::array< dcomplex, Rank > > triqs::modest::embedding::embed ( std::vector< std::vector< nda::array< dcomplex, Rank > > > const & imps_blocks ) const

inline

Embed impurity data into the full correlated space.

This method maps impurity solver data into the embedded representation using the \( \psi \) mapping table. Given data from each impurity solver organized by blocks \( (\gamma, \tau) \), the method constructs the full embedded arrays indexed by \( (\alpha, \sigma) \).

The algorithm proceeds as follows:

Initialize zero arrays for each \( (\alpha, \sigma) \) block with dimensions from sigma_embed_decomp.
For each \( (\alpha, \sigma) \), look up \( \psi(\alpha, \sigma) = (n\_imp, \gamma, \tau) \) and copy the corresponding impurity block data.
Assemble the blocks into full diagonal tensors for each spin channel \( \sigma \).

Blocks with imp_idx == -1 (disconnected from any impurity) remain zero.

The template parameter Rank determines the dimensionality of the data arrays. For frequency-dependent quantities (Rank ∈ {3,5}), the first dimension is the frequency index.

Template Parameters

Rank	The rank of the underlying nda arrays (e.g., 2 for static, 3 for frequency-dependent).

Parameters

imps_blocks Nested vector [n_imp][block] of impurity data arrays.

Returns: Vector of embedded arrays, one per spin channel \( \sigma \).

Definition at line 408 of file embedding.hpp.

◆ embed_block_structure()

gf_struct2_t triqs::modest::embedding::embed_block_structure ( ) const

Block structure (names and dimensions) for the embedded self-energy \( \Sigma_{\text{embed}} \).

Definition at line 153 of file embedding.cpp.

◆ extract() [1/3]

template<typename Mesh>

std::vector< block_gf< Mesh, matrix_valued > > triqs::modest::embedding::extract ( block2_gf< Mesh, matrix_valued > const & g_loc ) const

inline

Extract Green's function containers.

Template Parameters

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

Parameters

g_loc Block2Gf of gloc in \( M \times M \) space.

Returns: Local impurity Green's function.

Definition at line 502 of file embedding.hpp.

◆ extract() [2/3]

std::vector< block_matrix_t > triqs::modest::embedding::extract ( block2_matrix_t const & matrix_C ) const

Extract block matrices from the embedded representation.

Extracts static (frequency-independent) impurity data from the embedded \( (\alpha, \sigma) \)-indexed representation back into per-impurity block matrices. This is the inverse of the block matrix embed operation.

Parameters

matrix_C Embedded block2 matrix in \( \mathcal{C} \) space.

Returns: List of block matrices, one per impurity.

Definition at line 426 of file embedding.cpp.

◆ extract() [3/3]

template<int Rank>

std::vector< std::vector< nda::array< dcomplex, Rank > > > triqs::modest::embedding::extract ( std::vector< nda::array< dcomplex, Rank > > const & X ) const

inline

Extract impurity data from embedded arrays.

This method performs the inverse of the embed operation: it takes data in the full correlated space and extracts the individual impurity blocks using the reverse mapping \( \psi^{-1} \).

The algorithm proceeds in two steps:

For each \( (\alpha, \sigma) \) pair, extract the diagonal block from the input array X using the sigma_embed_decomp decomposition.
For each impurity, use the reverse_psi mapping to gather the appropriate blocks into the impurity's block structure, indexed by \( (\gamma, \tau) \).

The template parameter Rank determines the dimensionality of the underlying data arrays. For frequency-dependent quantities (Rank ∈ {3,5}), the first dimension is interpreted as the frequency index.

Template Parameters

Rank	The rank of the underlying nda arrays (e.g., 2, 4 for static, 3, 5 for frequency-dependent).

Parameters

X	Vector of arrays in the embedded space, one per spin channel \( \sigma \).

Returns: Nested vector [n_imp][block] of extracted impurity data arrays.

Definition at line 338 of file embedding.hpp.

◆ imp_block_structure()

std::vector< gf_struct_t > triqs::modest::embedding::imp_block_structure ( ) const

Block structure (names and dimensions) for each impurity solver.

Definition at line 161 of file embedding.cpp.

◆ imp_decomposition()

std::vector< long > triqs::modest::embedding::imp_decomposition ( long imp_idx ) const

inlinenodiscard

The impurity decomposition.

Definition at line 131 of file embedding.hpp.

◆ make_zero_imp_self_energies()

template<typename Mesh>

std::vector< std::pair< block_gf< Mesh, matrix_valued >, block_matrix_t > > triqs::modest::embedding::make_zero_imp_self_energies ( Mesh const & mesh )

inline

Create zero-initialized impurity self-energies.

Constructs a list of zero-initialized self-energy containers for each impurity, with both dynamic (frequency-dependent) and static components. The block structure of each impurity's self-energy is determined by imp_block_structure().

Each impurity receives a pair consisting of:

A block_gf initialized to zero on the provided mesh (dynamic part).
A block_matrix_t of zero matrices matching the block dimensions (static part).

Template Parameters

Mesh	The mesh type (e.g., imfreq, refreq, dlr_imfreq).

Parameters

mesh	The frequency mesh for the dynamic self-energy.

Returns: List of (dynamic, static) self-energy pairs, one per impurity.

Definition at line 286 of file embedding.hpp.

◆ merge_embed_block_by_imp()

embedding triqs::modest::embedding::merge_embed_block_by_imp ( ) const

Merge consecutive \( \alpha \) blocks belonging to the same impurity into a single block.

Merges consecutive \( \alpha \) blocks that belong to the same atom into a single block. Atom boundaries are detected by monitoring the \( \gamma \) index: within one atom the \( \gamma \) values are strictly increasing; when \( \gamma \) resets or the impurity index changes a new group is started. Disconnected blocks (imp_idx == -1) are never merged and pass through unchanged.

Returns: New embedding with a coarser block structure suitable for two-particle quantities.

Definition at line 200 of file embedding.cpp.

◆ n_alpha()

long triqs::modest::embedding::n_alpha ( ) const

inlinenodiscard

Number of blocks in \( \alpha \) for the \( \Sigma_{\text{embed}} \).

Definition at line 116 of file embedding.hpp.

◆ n_gamma()

long triqs::modest::embedding::n_gamma ( long imp_idx ) const

inlinenodiscard

Number of blocks in \( \gamma \) for the \( \Sigma_{\text{imp}} \) [imp_idx].

Definition at line 122 of file embedding.hpp.

◆ n_impurities()

long triqs::modest::embedding::n_impurities ( ) const

inlinenodiscard

Number of impurities.

Definition at line 125 of file embedding.hpp.

◆ n_sigma()

long triqs::modest::embedding::n_sigma ( ) const

inlinenodiscard

Number of blocks in \( \sigma \) for the \( \Sigma_{\text{embed}} \).

Definition at line 119 of file embedding.hpp.

◆ operator==()

bool triqs::modest::embedding::operator== ( embedding const & other ) const

default

◆ operator[]()

imp_block_t triqs::modest::embedding::operator[]	(	long	alpha,
		long	sigma ) const

inline

Bracket accessor: \( \psi[\alpha,\sigma] \).

Returns: The corresponding impurity block (n_imp, \( \gamma \), \( \tau \)).

Definition at line 110 of file embedding.hpp.

◆ psi_map()

nda::array< imp_block_t, 2 > triqs::modest::embedding::psi_map ( ) const

inlinenodiscard

The mapping table \( \psi \).

Definition at line 113 of file embedding.hpp.

◆ replace_imp()

embedding triqs::modest::embedding::replace_imp	(	long	imp_idx_old,
		long	imp_idx_new ) const

Redirect all \( \psi \) entries from one impurity to another.

Rewires every \( (\alpha, \sigma) \) that currently points to imp_idx_old so that it points to imp_idx_new instead. The impurity at imp_idx_old remains in the embedding but becomes dangling — no \( \alpha \) block maps to it any longer. Call drop_imp(imp_idx_old) afterwards to remove the dangling entry and compact the indices.

Parameters

imp_idx_old	The impurity whose \( \alpha \) connections are redirected.
imp_idx_new	The impurity that will receive those connections.

Returns: New embedding with the updated \( \psi \) map (dangling impurity retained).

Definition at line 287 of file embedding.cpp.

◆ sigma_names()

std::vector< std::string > triqs::modest::embedding::sigma_names ( ) const

inlinenodiscard

The names of the sigma indices.

Definition at line 128 of file embedding.hpp.

◆ slice_sigma()

embedding triqs::modest::embedding::slice_sigma ( ) const

Slice the embedding to a single \( \sigma \) channel ("ud").

Converts a spinful embedding (with two spin channels, e.g., "up" and "down") into a single-channel embedding labelled "ud". This is useful when working with systems with spin-orbit coupling or spin-averaged quantities, like the density-density susceptibility.

Returns: New embedding with a single spin channel.

Definition at line 190 of file embedding.cpp.

◆ split_imp() [1/3]

embedding triqs::modest::embedding::split_imp	(	long	imp_idx,
		std::function< bool(long)>	p ) const

Split impurity imp_idx using a predicate.

Partitions the blocks of impurity imp_idx into two new impurities based on predicate p. Blocks for which p(block_idx) returns true stay with imp_idx; the rest are assigned to a new impurity inserted at imp_idx + 1.

Parameters

imp_idx	Impurity number.
p	Predicate taking a block index and returning true/false.

Returns: New embedding with the updated \( \psi \) map.

Definition at line 301 of file embedding.cpp.

◆ split_imp() [2/3]

embedding triqs::modest::embedding::split_imp	(	long	imp_idx,
		std::initializer_list< const char * >	x )

delete

◆ split_imp() [3/3]

embedding triqs::modest::embedding::split_imp	(	long	imp_idx,
		std::vector< long > const &	block_list ) const

Split impurity imp_idx based on block_list.

For impurity imp_idx, split part of its blocks into a new impurity. The blocks that were split off remain mapped to their original alpha blocks with the updated imp_idx.

Parameters

imp_idx	Impurity number.
block_list	A list of blocks of imp_idx to split off.

Returns: New embedding with the updated \( \psi \) map.

Definition at line 336 of file embedding.cpp.

◆ split_imp_block()

embedding triqs::modest::embedding::split_imp_block	(	long	imp_idx,
		long	gamma,
		std::vector< long > const &	new_dims ) const

Split a single \( \gamma \) block of an impurity into multiple blocks.

Splits block gamma of impurity imp_idx into multiple blocks whose sizes are given by new_dims. The sum of new_dims must equal the original dimension of the block. For example, a block of dimension 5 could be split into blocks of dimensions 2 (e.g.) and 3 (t2g).

Parameters

imp_idx	The index of the impurity to operate on.
gamma	The block index of the impurity to split.
new_dims	New dimensions of the split blocks; must sum to the original block dimension.

Returns: New embedding with the updated impurity structure and \( \psi \) mapping.

Definition at line 345 of file embedding.cpp.

◆ swap_sigma() [1/2]

embedding triqs::modest::embedding::swap_sigma ( long alpha ) const

Swap the \( \sigma \) (spin) assignment for \( \alpha \) block alpha.

For block \( \alpha \) transforms \( (\sigma,\sigma') \to (\tau, \tau') \) to \( (\sigma, \sigma') \to (\tau', \tau)\).

Parameters

alpha The index of the alpha block whose spin assignment is swapped.

Returns: New embedding with the updated \( \psi \) map.

Definition at line 173 of file embedding.cpp.

◆ swap_sigma() [2/2]

embedding triqs::modest::embedding::swap_sigma ( std::vector< long > alphas ) const

Swap the \( \sigma \) (spin) assignment for all \( \alpha \) blocks in alphas.

For each block \( \alpha \) in alphas, transforms \( (\sigma,\sigma') \to (\tau, \tau') \) to \( (\sigma, \sigma') \to (\tau', \tau)\).

Parameters

alphas A list of alpha blocks whose spin assignments are swapped.

Returns: New embedding with the updated \( \psi \) map.

Definition at line 182 of file embedding.cpp.

◆ h5_read

void h5_read	(	h5::group	g,
		std::string const &	name,
		embedding &	x )

friend

Definition at line 96 of file h5.cpp.

◆ h5_write

void h5_write	(	h5::group	g,
		std::string const &	name,
		embedding const &	x )

friend

h5 read/write

Definition at line 104 of file h5.cpp.

◆ operator<<

std::ostream & operator<<	(	std::ostream &	out,
		embedding const &	E )

friend

Definition at line 167 of file printing.cpp.

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

triqs_modest/embedding.hpp
triqs_modest/embedding.cpp
triqs_modest/printing.cpp

Detailed Description

Public Member Functions

Friends

Constructor & Destructor Documentation

◆ embedding()

Member Function Documentation

◆ description()

◆ drop_imp()

◆ embed() [1/4]

◆ embed() [2/4]

◆ embed() [3/4]

◆ embed() [4/4]

◆ embed_block_structure()

◆ extract() [1/3]

◆ extract() [2/3]

◆ extract() [3/3]

◆ imp_block_structure()

◆ imp_decomposition()

◆ make_zero_imp_self_energies()

◆ merge_embed_block_by_imp()

◆ n_alpha()

◆ n_gamma()

◆ n_impurities()

◆ n_sigma()

◆ operator==()

◆ operator[]()

◆ psi_map()

◆ replace_imp()

◆ sigma_names()

◆ slice_sigma()

◆ split_imp() [1/3]

◆ split_imp() [2/3]

◆ split_imp() [3/3]

◆ split_imp_block()

◆ swap_sigma() [1/2]

◆ swap_sigma() [2/2]

Friends And Related Symbol Documentation

◆ h5_read

◆ h5_write

◆ operator<<