Embedding Theory

Self-energy embedding theory (SEET) is an embedding framework for systems where only part of the orbital space needs a non-perturbative treatment. The formulation emphasized in the 2017 finite-temperature quantum embedding paper is functional: SEET is built as an approximation to the Luttinger-Ward functional $\Phi[G]$, so a self-consistent SEET solution inherits the conservation and thermodynamic consistency properties of a $\Phi$-derivable approximation.¹

The motivation is practical. GF2 and GW can treat large orbital spaces, but they are weak-coupling approximations. Exact diagonalization, configuration interaction, coupled cluster impurity solvers, or quantum Monte Carlo can treat strong correlation, but only in a small active space. SEET combines these two layers at the level of the self-energy: the whole system is first described by a weak method, and selected correlated subspaces are then corrected by a stronger solver.

Functional Construction

Assume the orbital space is separated into correlated subspaces $A_i$ and a remainder $R$. The subspaces $A_i$ are intended to contain orbitals whose correlations are not well described by a weak method. In the simplest non-overlapping case, the SEET approximation to the Luttinger-Ward functional is

Here $\Phi^\mathrm{tot}_\mathrm{weak}$ is the weak-coupling functional for the full system. The term $\Phi^{A_i}_\mathrm{weak}$ is the part of that weak functional whose interaction vertices all lie inside subspace $A_i$. The term $\Phi^{A_i}_\mathrm{strong}$ replaces that subspace contribution with a higher-level non-perturbative result.¹

This equation encodes the double-counting correction. The weak method already includes an approximate contribution from diagrams internal to $A_i$. SEET subtracts that weak internal contribution and adds the strong one. Everything outside the correlated subspace, and all weak correlations coupling the subspace to its environment, remains in $\Phi^\mathrm{tot}_\mathrm{weak}$.

Taking the functional derivative with respect to $G$ gives the corresponding subspace self-energy,

For matrix elements entirely in the remainder $R$, and for inter-subspace blocks, the self-energy is kept at the weak level. This is why the method is called self-energy embedding: the strongly correlated subspace self-energy is embedded into an environment self-energy generated by the full weak-coupling calculation.

Exact Limits

The functional form makes several limits transparent.¹

If the interaction vanishes, the self-energy vanishes and the method is exact. If the correlated subspace is the entire orbital space and the strong solver is exact, SEET reduces to the exact solution. If the strong solver is replaced by the same weak method used for the full system, the correction cancels and the full weak-coupling solution is recovered.

These limits matter because they identify the control parameter of SEET. Increasing the correlated subspace, or improving the solver used inside it, moves the result toward the exact solution. The method is most useful when the difficult correlations are concentrated in a small number of orbitals.

Choosing Correlated Orbitals

The 2017 analysis stresses that SEET does not require the correlated orbitals to be chosen a priori as local atomic-like orbitals. They can be selected after a weak-coupling calculation. A common criterion is based on the one-particle density matrix: orbitals whose occupations are far from 0 or 2 are candidates for the correlated subspace.¹

This is a useful diagnostic for molecular and realistic-basis calculations. In a weakly correlated closed-shell system, natural-orbital occupations are close to 2 for occupied orbitals and close to 0 for virtual orbitals. Occupations in between indicate multi-reference character or strong static correlation, and therefore mark orbitals that may need a non-perturbative impurity treatment.

Local-orbital choices are also possible. They are natural for materials where the correlated physics is known to live on a set of local $d$ or $f$ orbitals, or on a cluster of neighboring sites. The important point is that SEET treats the choice of subspace as a controllable approximation rather than as a fixed definition of the method.

Hybridization Function

For a selected subspace $A$, the rest of the system acts as a dynamical environment. This environment is summarized by a hybridization function $\Delta$. The cleanest way to see it is to write the inverse Green’s-function problem in block form. Define

and

The hybridization is the Schur complement contribution from eliminating the remainder space,

The Green’s function projected into the correlated subspace is then

The hybridization function is therefore not an adjustable potential. It is the frequency-dependent object that makes the open subspace reproduce the effect of coupling to the rest of the system.¹

Impurity Problem

SEET maps each correlated subspace to a quantum impurity problem. The impurity contains the orbitals in $A$, the bare two-electron interactions restricted to $A$, and a bath chosen to reproduce $\Delta$. In the basic functional construction, the noninteracting impurity Green’s function is

An impurity solver produces an interacting impurity Green’s function $G^\mathrm{imp}$. The strong subspace self-energy is obtained from the impurity Dyson equation,

This self-energy then replaces the weak internal self-energy in the SEET expression. Configuration-interaction solvers and their restricted-active-space variants were used in the 2017 SEET/GW paper because they can handle general hybridization functions and general four-index interactions at low temperature.²

In the SEET formulation based on $\Phi[G]$, the impurity interactions remain instantaneous bare Coulomb interactions transformed into the correlated subspace. This differs from $GW$+DMFT-style formulations based on screened interactions, where the impurity problem may require frequency-dependent interactions.¹

Double Counting

The double-counting term is not empirical in SEET. It is the weak self-energy evaluated inside the same subspace and with the same restricted interactions that the strong solver will replace. For a generic weak method this is the term $\Sigma^A_\mathrm{weak}$ in

The 2017 SEET+GW paper emphasizes an important practical point for $GW$ double counting. One must recompute the subspace polarization and screened interaction using the truncated Green’s functions and transformed interactions inside $A$. One should not simply truncate the full-system polarization or the full-system screened interaction to the subspace.²

With decomposed subspace interactions $\bar v^\mu$, the subspace polarization used for the $GW$ double-counting term has the form

The subspace screened interaction is then built from this polarization, and the $GW$ double-counting self-energy is evaluated only within $A$:

This is the term subtracted from the full weak $GW$ self-energy before the strong impurity self-energy is embedded.

Self-Consistency

A practical SEET calculation follows two nested ideas.¹, ²

1. Solve the full system with a weak, self-consistent method such as GF2 or GW.
2. Select correlated orbitals, often by natural occupations or by a local-orbital criterion.
3. Transform the weak self-energy, Green’s function, one-body terms, and the needed two-electron integrals into the chosen subspace basis.
4. Construct the double-counting self-energy for each correlated subspace.
5. Build and solve the impurity problem for each subspace.
6. Replace the weak internal subspace self-energy by the strong impurity self-energy.
7. Solve the Dyson equation for the full system with the embedded self-energy.
8. Update the hybridization and repeat until the Green’s function, self-energy, and energy are converged.

The outer weak-method loop can also be repeated after the embedded solution is obtained. In many natural-orbital calculations this outer update has a smaller effect than the inner impurity self-consistency, while in local bases it can be more important.²

Relation to DMFT and GW+DMFT

The 2017 finite-temperature embedding paper also uses the functional viewpoint to relate SEET to DMFT and $GW$+DMFT. DMFT can be understood as a special case where the correlated subspaces are local orbitals, nonlocal self-energy contributions are neglected, and the impurity problem supplies the local self-energy. SEET is more general in two ways: the correlated subspace can be chosen adaptively, and nonlocal weak-coupling self-energy contributions remain present through $\Sigma^\mathrm{tot}_\mathrm{weak}-\Sigma^A_\mathrm{weak}$.¹

Compared with $GW$+DMFT, SEET formulated with $\Phi[G]$ avoids a frequency-dependent impurity interaction. The effects of the environment enter through the embedding self-energy and hybridization, while the strong solver sees bare interactions restricted to the selected subspace. This makes SEET a systematic route for combining a weak method such as GF2 or GW with accurate impurity solvers for selected orbitals.