# ComDMFT v.2.0: Fully Self-Consistent ab initio GW+EDMFT for the Electronic
Structure of Correlated Quantum Materials

Byungkyun Kang  Patrick Semon  Corey Melnick  Gabriel Kotliar  Sangkook Choi
Condensed Matter Physics and Materials Science Department, Brookhaven National
Laboratory, Upton, NY 11973, USA  College of Arts and Sciences, University of
Delaware, Newark, DE 19716, USA  Department of Physics and Astronomy, Rutgers
University, NJ 08854, USA  School of Computational Sciences, Korea Institute
for Advanced Study, Seoul 02455, Republic of Korea

###### Abstract

ComDMFT is a parallel computational package designed to study the electronic
structure of correlated quantum materials from first principles. Our approach
is based on the combination of first-principles methods and dynamical mean
field theories. In version 2.0, we implemented fully-diagrammatic GW+EDMFT
from first-principles. In this approach, correlated electrons are treated
within full GW+EDMFT and the rest are treated within full-GW, seamlessly. This
implementation enables the electronic structure calculation of quantum
materials with weak, intermediate, and strong electron correlation without
prior knowledge of the degree of electron correlation.

###### keywords:

correlated quantum materials; first-principles; dynamical mean-field theory;
GW approximation; electronic structure

††journal: Computer Physics Communications

PROGRAM SUMMARY/NEW VERSION PROGRAM SUMMARY

Program Title: ComDMFT
Licensing provisions(please choose one): GPLv3
Programming language: fortran2008, C++, and Python3

Nature of problem:
In order to comprehend the electronic structure of correlated quantum
materials, a non-perturbative and parameter-free ab initio method is required.
One of the promising approaches is fully-diagrammatic GW+EDMFT; however, there
is no available open-source package which implements the ab initio full
GW+EDMFT methodology.

Solution method:
We implemented ab initio all electron full GW+EDMFT methodology for the
electronic structure calculation of correlated quantum materials and compare
the results with those of LQSGW+DMFT.

Additional comments:
ComDMFTv2 is built on top open source codes, ComDMFTv1 [1], Wannier90 [2],
ComCTQMC[3] and FlapwMBPT [4] codes.

##  1 Introduction

In materials with a large number of electrons, the quantum mechanical
interactions between these electrons often dictate the materials properties.
These so-called correlated quantum materials can display a range of
exceptional emergent phenomena, including metal-insulator transition, colossal
magnetoresistance, and high-temperature superconductivity. Recently,
correlated quantum materials received renewed attentions due to their
potential for quantum information science applications. An excellent example
is the Fe-based superconductors, which have demonstrated potential as qubit
hosts which are topologically-protected. [5, 6, 7, 8].

One of the grand challenges in condensed matter physics as well as materials
science is to predict correlated quantum materials properties from first
principles. Treating strongly correlated quantum materials require new
concepts, algorithms, as well as computer codes. To this end, various ideas
have been realized and applied for the quantitative understanding of
correlated quantum materials properties [9]. Among them, one of the most
successful methodologies is dynamical mean-field theory (DMFT) [10, 11, 12,
13, 14, 15, 16, 17, 18, 19]. DMFT establishes a mapping from a quantum many-
body problem onto a quantum impurity problem. Quantities obtained within
quantum impurity problems are embedded into the Green’s function of the
quantum many-body problems. This idea has been applied extensively to model
Hamiltonians and have been described correlated electron phenomena such as
Mott physics, [16, 20] Hund metal physics [21] and heavy fermion physics [22].

The success of model-based DMFT approaches has led to the integration of DMFT
with ab initio methodologies. This combination is based on the concept of
quantum embedding, which entails separating electrons into two groups [23, 24,
25]: correlated electrons and itinerant electrons. Correlated electron
phenomena are addressed by DMFT, while first-principles methods are used to
describe itinerant electrons and to generate materials-specific quantum
impurity problems. An example of this successful union is the DFT+DMFT
methodology [26, 27, 28]. This approach can explain various phenomena specific
to correlated quantum materials such as paramagnetic Mott transitions[29] and
volume collapse transitions[30]. The development of the continuous-time Monte-
Carlo approaches to quantum impurity problems [31] has enabled the analysis of
broad classes of quantum impurity problems, making them the popular choice for
DMFT-based ab initio calculations. Recent important advances in the DFT+DMFT
methodology were introduced to compute total energies[32], free energies[33],
and forces[34], as well as more accurate double-counting scheme
corrections[35].

The DFT+DMFT methodology has attracted a lot of attention and is supported by
the development of various software programs. Reusable libraries, such as
TRIQS[36] and ALPS[37], have been created to aid DFT+DMFT implementation.
Standalone impurity solvers, iQIST[38], W2dynamics[39] and ComCTQMC[3], are
available. DFT+DMFT packages have been developed: ComDMFT[1]; EDMFTF[40]
integrated with Wien2K[41]; TRIQS/DFTTools [42] built on top of TRIQS[36] and
Wien2k[41]; DMFTwDFT[43, 32] bulit on top of EDMFTF package impurity
solver[44]; D-Core [45] built on top of TRIQS[42, 36], ALPS[37], Quantum
Espresso[46], and OpenMX[47]; AMULET[48] integrated with Quantum Espresso[46]
and Elk[49]; and Wien2k+W2dynamics. Furthermore, DFT+DMFT functionality has
been added to ab initio codes such as Abinit[50] and RSPt[51].

Density Functional Theory combined with Dynamical Mean Field Theory (DFT+DMFT)
has been very successful in predicting the properties of static equilibrium
properties of correlated quantum materials. Moreover, it has been proven that
the predicted dynamics of electronic excitations agrees well with experimental
data from many spectroscopic tools. It requires, however, the introduction of
parameters in constructing quantum impurity problem. For examples, Coulomb
interactions of the correlated subshell are required.

The success of DMFT has motivate the search for its justification and
extension via diagrammatic Green’s function approaches based on functionals
[52, 53, 54]. These approaches can be understood as Feynmann-diagrammatic
approaches merging many-body perturbation theory (MBPT) and DMFT. By defining
exchange-correlational functionals using Green’s function ($G$) and the
screened Coulomb interaction ($W$)[55, 56], these approaches provide a
systematic way to calculate the properties of correlated quantum materials. An
example of this functional approach is the GW+EDMFT approach [57, 58, 59]; by
accounting for exchange and correlation effects within the GW approximation
and extended DMFT[60, 61, 62], this approach merges these two theories. This
approach is advantageous as it is free of adjustable parameters; once a
projector is chosen to describe the correlated orbitals, the method proceeds
without any further input from the user and defines the quantum impurity
problem for the correlated orbitals self-consistently. Related approaches with
the same aim are the ab initio density matrix embedded theory [63] and full
Cell DMFT. The latter is formulated in terms of the Green’s function ($G$) and
the bare Coulomb interaction ($V$) [64]. The Self-Energy Embedding Theory [65]
is another example, patterned after the periodized cluster dynamical mean-
field theory (PCDMFT)[66].

Various earlier schemes of GW+EDMFT have been proposed. Two examples are the
partial self-consistent scheme [67] and the multitier GW+EDMFT approach[59].
These combine GW and DMFT, but not in a fully self consistent manner. The
former integrates DMFT with quasiparticle GW approaches such as one-shot
GW[68, 69, 70], QSGW[71, 72], LQSGW[73, 1], and screened exchange[74]. Coulomb
interactions of the correlated subshells are obtained within constrained
random phase approximation[75]. In this approaches, DMFT is treated as a one-
shot correction to the corresponding quasiparticle Hamiltonian. On the other
hand, the latter entails self-consistency within a subspace of the full
Hilbert space, while the remaining Hilbert space is treated with a one-shot
GW. These approaches have been applied to various correlated quantum materials
[76, 77, 78, 79], exhibiting the capability of full implementation. Within the
field of correlated quantum materials, there is a pressing requirement for a
package that implements the ab initio full GW+EDMFT methodology. Currently,
few such packages exist and none of them supports a fully self-consistent
GW+DMFT scheme. ComDMFT[1] supports the ab initio LQSGW+DMFT methodology [73]
and Questaal[72] implements the QSGW+DMFT methodology.

This paper describes the ComDMFT v2.0 package, developed at Brookhaven
National Lab. It is the first implementation of full GW+EDMFT. As an open
source code for the community, it builds on earlier open source packages
ComDMFT v.1.0 [1], the FlapwMBPT code[4] for ab initio GW calculations, the
Wannier90 code[2] for the construction of correlated orbitals, and the
ComCTQMC code[3] for the solution of quantum impurity problems. The package is
composed of five modules, written in FORTRAN2008 and Python3, namely: ComWann,
Comlocal, ComWeiss, ComDC, and ComEmbed.

In this paper, we introduce a full GW+EDMFT methodology and to provide
benchmark results on SrVO3 and NiO. Section 2 outlines the general theoretical
framework, while Section 3 provides a detailed description of our
implementation. Benchmark results are presented in Section 4 together with
calculations with LQSGW+DMFT for the same materials for comparison.

##  2 Theoretical Background

A very useful approach in many body theory is provided by the functional
formalism of Luttinger and Ward [53] and Baym and Kadanoff [52]. Here, the
grand potential ($\Omega$) is viewed as a functional of the electron Green’s
function ($G$) and electron self-energy ($\Sigma$). $\Omega[G,\Sigma]$ is
stationary at the physical $G$ and the corresponding $\Sigma$. Approximations
to this exact functional result in approximate many body methods. This
approach has been later extended to construct a new functional, $\Gamma$: [55,
56]

$\begin{split}\Gamma[G,\Sigma,W,\Pi]=&-\operatorname{Tr}[log(-G_{H}^{-1})]-\operatorname{Tr}[log(1-G_{H}\Sigma)]-\operatorname{Tr}[G\Sigma]\\\
&+\frac{1}{2}\operatorname{Tr}[log(1-V\Pi)]+\frac{1}{2}\operatorname{Tr}[\Pi
W]+\Psi[G,W].\\\ \end{split}$ (1)

Here, $\operatorname{Tr}$ is used to denote the sum over all relevant indices
(crystal momentum, state indices, and spin indices) in addition to Matsubara
frequency summation, with a prefactor of $1/\beta$. $G_{H}$ and $V$ denote the
Hartree Green’s function and bare Coulomb interaction, respectively. $\Psi$ is
a functional expressed as a summation of all two-particle irreducible diagrams
constructed from $G$ and $W$. Approximations to $\Psi$ result in approximate
many body methods. Within this construction, $\Gamma$ is regarded as the
functional of Green’s function ($G$), electron self-energy ($\Sigma$),
screened Coulomb interaction($W$), and polarizability ($\Pi$). These can be
shown to be stationary at the physical $G$ and $W$, along with the
corresponding $\Sigma$ and $\Pi$. These conditions lead to the following
equations:

$\begin{split}\frac{\delta\Gamma}{\delta
G}=0\to&\Sigma=\frac{\delta\Psi}{\delta G}\\\ \end{split}$ (2)
$\begin{split}\frac{\delta\Gamma}{\delta\Sigma}=0\to&G=\left(G_{H}^{-1}-\Sigma\right)^{-1}\\\
\end{split}$ (3) $\begin{split}\frac{\delta\Gamma}{\delta
W}=0\to&\Pi=-2\frac{\delta\Psi}{\delta W}\\\ \end{split}$ (4)
$\begin{split}\frac{\delta\Gamma}{\delta\Pi}=0\to&W=\left(V^{-1}-\Pi\right)^{-1}\\\
\end{split}$ (5)

This approach can be utilized to comprehend various diagrammatic
approximations. For instance, the first-order diagram in $W$
($\Psi_{GW}(G,W)=-\frac{1}{2}\operatorname{Tr}GWG$) corresponds to the
renowned GW approximation[80, 55]. $\Psi_{GW}(G,W)$ leads to the following
self-energy and polarizability.

$\begin{split}\Sigma_{GW}(1,2)&=\frac{\delta\Psi_{GW}}{\delta
G(2,1)}=-G(1,2)W(1,2)\\\ \Pi_{GW}(1,2)&=-2\frac{\delta\Psi_{GW}}{\delta
W(2,1)}=G(1,2)G(2,1)\\\ \end{split}$ (6)

Here, the notation $1$ denotes
$(\mathbf{r}_{1},\sigma_{1},\tau_{1})=(\mathbf{x}_{1},\tau_{1})=x_{1}$, where
$\mathbf{r}$, $\sigma$ and $\tau$ are the position vector, spin index, and
imaginary time, respectively.

In order to incorporate strong correlation effects, the DMFT idea can be
applied. This amounts to constraining the $G$ and $W$ in the $\Psi$ functional
to the correlated orbital space, and is referred to as extended DMFT
(EDMFT)[60, 61, 62]. Rigorously, the correlated orbitals
$\varphi_{\alpha}^{f}(\mathbf{r})$ are introduced, where $\alpha$ is a
composite index for the correlated orbital and spin, and $f$ denotes
“fermion”. The projection ($\hat{P}^{f}$) and embedding ($\hat{E}^{f}$) of the
fermionic quantities can then be defined accordingly.

$\begin{split}\widetilde{G}_{\alpha\beta}(\tau_{1},\tau_{2})=&\hat{P}_{\alpha\beta}^{f}(G)(\tau_{1},\tau_{2})=\int
d\mathbf{x}_{1}\mathbf{x}_{2}{\varphi_{\alpha}^{f}}^{*}(\mathbf{x}_{1})G(\mathbf{x}_{1},\tau_{1},\mathbf{x}_{2},\tau_{2})\varphi_{\beta}^{f}(\mathbf{x}_{2})\\\
\widetilde{G}(1,2)=&\hat{E}^{f}(\widetilde{G})(1,2)=\sum_{\alpha\beta}\varphi_{\alpha}^{f}(\mathbf{x}_{1})\widetilde{G}_{\alpha\beta}(\tau_{1},\tau_{2}){\varphi_{\beta}^{f}}^{*}(\mathbf{x}_{2})\end{split}$
(7)

Here, $\ \widetilde{}\ $ denotes a physical quantity projected onto the
correlated orbital space and $\int d\mathbf{x}_{1}=\sum_{\sigma_{1}}\int
d\mathbf{r}_{1}$.

For the projection ($\hat{P}^{b}$) and embedding ($\hat{E}^{b}$) of bosonic
quantities, it is convenient to define orthonormalized product basis
functions. To develop these functions, we initially construct
unorthonormalized product basis functions, $\vartheta$.

$\begin{split}\vartheta_{\alpha\beta}(\mathbf{x})={\varphi_{\alpha}^{f}}^{*}(\mathbf{x}){\varphi_{\beta}^{f}}(\mathbf{x})\end{split}$
(8)

The orthonormalized product basis function, $\varphi^{b}$ can be represented
as a linear combination of $\vartheta$:

$\begin{split}\varphi_{\alpha}^{b}(\mathbf{x})=\sum_{\beta\gamma}\vartheta_{\beta\gamma}(\mathbf{x})X_{\beta\gamma,\alpha}\end{split}$
(9)

For the construction of $\varphi^{b}$, we employed canonical
orthonormalization. The overlap matrix ($S_{\alpha\beta;\gamma\delta}=\int
d\mathbf{x}\vartheta_{\alpha\beta}^{*}(\mathbf{x})\vartheta_{\gamma\delta}(\mathbf{x})$)
was diagonalized by a unitary matrix $U$.

$\begin{split}U^{\dagger}SU=D\end{split}$ (10)

where $D$ is the diagonal matrix of the eigenvalues of $S$. Then $X$ is given
by

$\begin{split}X_{\beta\gamma,\alpha}=U_{\beta\gamma;\alpha}D_{\alpha}^{-1/2}\end{split}$
(11)

By using the orthonormalized product basis, we can define the projection and
embedding of the bosonic quantities in the following way.

$\begin{split}\widetilde{W}_{\alpha\beta}(\tau_{1},\tau_{2})=&\hat{P}_{\alpha\beta}^{b}(W)(\tau_{1},\tau_{2})=\int
d\mathbf{x}_{1}d\mathbf{x}_{2}{\varphi_{\alpha}^{b}}^{*}(\mathbf{x}_{1})W(\mathbf{x}_{1},\tau_{1},\mathbf{x}_{2},\tau_{2}){\varphi_{\beta}^{b}}(\mathbf{x}_{2})\\\
\widetilde{W}(1,2)=&\hat{E}^{b}(\widetilde{W})(1,2)=\sum_{\alpha\beta}\varphi_{\alpha}^{b}(\mathbf{x}_{1})\widetilde{W}_{\alpha\beta}(\tau_{1},\tau_{2}){\varphi_{\beta}^{b}}^{*}(\mathbf{x}_{2})\end{split}$
(12)

Alternatively, the projected bosonic quantities can be defined by using
unorthonormalized product basis.

$\begin{split}\widetilde{W}_{\alpha\beta\gamma\delta}=&\hat{P}^{b}_{\alpha\beta\gamma\delta}(W)(\tau_{1},\tau_{2})=\int
d\mathbf{x}_{1}d\mathbf{x}_{2}{\vartheta_{\alpha\delta}}^{*}(\mathbf{x}_{1})W(\mathbf{x}_{1},\tau_{1},\mathbf{x}_{2}\tau_{2})\vartheta_{\gamma\beta}(\mathbf{x}_{2})\\\
\end{split}$ (13)

Figure 1: Diagrammatic representations of the $\Psi$ functional within full
GW+EDMFT approach. (a) GW $\Psi$ functional (b) EDMFT $\Psi$ functional and
(c) local-GW $\Psi$ functional. Full and wiggly lines represent Green’s
function ($G$) and screened Coulomb interaction ($W$), respectively. Black and
gray colors represent quantities in the full Hilbert space and in the
correlated subspace, respectively.

The EDMFT $\Psi$ functional ($\Psi_{EDMFT}(\widetilde{G},\widetilde{W})$) is
given by all two-particle irreducible diagrams constructed from
$\widetilde{G}$ and $\widetilde{W}$. The electron self-energy and
polarizability within EDMFT is given by the following equations.

$\begin{split}\Sigma_{EDMFT}(1,2)=\frac{\delta\Psi_{EDMFT}}{\delta
G(2,1)}=\frac{\delta\Psi_{EDMFT}}{\delta\widetilde{G}(2,1)}\\\
\Pi_{EDMFT}(1,2)=-2\frac{\delta\Psi_{EDMFT}}{\delta
W(2,1)}=-2\frac{\delta\Psi_{EDMFT}}{\delta\widetilde{W}(2,1)}\\\ \end{split}$
(14)

The functional approach provides a natural way to merge these two diagrammatic
approaches [57, 58, 59]. When we merge $\Psi_{GW}(G,W)$ and
$\Psi_{EDMFT}(\widetilde{G},\widetilde{W})$, the double-counted diagrams
should be subtracted. In GW+EDMFT, the GW $\Psi$ functional in the correlated
space
($\Psi_{DC}(\widetilde{G},\widetilde{W})=-\frac{1}{2}\operatorname{Tr}\widetilde{G}\widetilde{W}\widetilde{G}$)
provides the double counted diagrams. In particular, the double-counting self-
energy and polarizability are given by

$\begin{split}\Sigma_{DC}(1,2)&=\frac{\delta\Psi_{DC}}{\delta
G(2,1)}=\frac{\delta\Psi_{DC}}{\delta\widetilde{G}(2,1)}=-\widetilde{G}(1,2)\widetilde{W}(1,2)\\\
\Pi_{DC}(1,2)&=-2\frac{\delta\Psi_{DC}}{\delta
W(2,1)}=-2\frac{\delta\Psi_{DC}}{\delta\widetilde{W}(2,1)}=\widetilde{G}(1,2)\widetilde{G}(2,1)\\\
\end{split}$ (15)

Finally, we can derive the $\Psi$ functional of GW+EDMFT
($\Psi_{GW+EDMFT}(G,W)$) illustrated in Fig. 1.

$\begin{split}\Psi_{GW+EDMFT}(G,W)=-\frac{1}{2}\operatorname{Tr}GWG+\Psi_{EDMFT}(\widetilde{G},\widetilde{W})+\frac{1}{2}\operatorname{Tr}\widetilde{G}\widetilde{W}\widetilde{G}.\end{split}$
(16)

Then, the electron self-energy within GW+EDMFT is given by the following
equations. For the correlated states,

$\Sigma(1,2)=-G(1,2)W(1,2)+\frac{\delta\Psi_{EDMFT}}{\delta\widetilde{G}(2,1)}+\widetilde{G}(1,2)\widetilde{W}(1,2).$
(17)

For the itinerant states,

$\Sigma_{GW}(1,2)=-G(1,2)W(1,2).$ (18)

The polarizability within GW+EDMFT is given by the following equations. For
the correlated states,

$\Pi(1,2)=G(1,2)G(2,1)-2\frac{\delta\Psi_{EDMFT}}{\delta\widetilde{W}(2,1)}-\widetilde{G}(1,2)\widetilde{G}(2,1).$
(19)

For the itinerant states,

$\Pi(1,2)=G(1,2)G(1,2).$ (20)

Figure 2:  The GW+EDMFT approach involves three self-consistency loops: the GW
loop (blue), the EDMFT loop (red), and the double counting loop (green). This
diagram seeks to emphasize the need for self-consistent determination of
three-different terms in both fermionic and bosonic self-energies. For more
information on the practical implementation of the self-consistent loop,
please refer to Fig. 3.

The GW+EDMFT approximation provides a set of self-consistent equations, which
can be solved to obtain $G$, $\Sigma$, $W$, and $\Pi$ at their stationary
conditions. Central equations are the fermionic and bosonic Dyson’s equations,
as shown in eqs. (LABEL:eq:f_dyson) and (LABEL:eq:b_dyson), respectively. In
addition, three distinct self-consistency loops must be considered: the GW
loop, EDMFT loop, and double counting loop. A summary of these equations and
loops is provided in Fig. 2. Here we note that Dyson’s equations for Green’s
function can be written in terms of $G_{GW}$

$G_{GW}=\left(G_{H}^{-1}-\Sigma_{GW}\right)^{-1}$ (21)
$G=\\{G_{GW}^{-1}-\hat{E}^{f}(\widetilde{\Sigma}_{EDMFT}-\widetilde{\Sigma}_{DC})\\}^{-1}.$
(22)

In the same way, Dyson’s equation for screened Coulomb interaction can be
written in terms of $W_{GW}$

$W_{GW}=\left(V^{-1}-\Pi_{GW}\right)^{-1}$ (23)
$W=\\{W_{GW}^{-1}-\hat{E}^{b}(\widetilde{\Pi}_{EDMFT}-\widetilde{\Pi}_{DC})\\}^{-1}$
(24)

##  3  Ab initio fully self-consistent GW+EDMFT implementation in ComDMFT v2.0

ComDMFT v2.0 comprises eight components and a Python script: FlapwMBPT[4],
Wannier90[2], ComBasis, ComWeiss, ComLocal, ComCTQMC[3], ComDC, and ComEmbed.
These components are executed in sequence by the Python script “gwdmft.py”.

All the other components are developed using object-oriented programming
features in python3 and fortran 2008. The implemented resources (classes) are
reused by several components and can easily be reused by future developers.
All the generated data by individual components are saved in a single hdf5
file and can be accessed by other components based on a data-map upon inquiry
with keys (file name, group name, data name).

###  3.1 Hilbert space and interpolation

For the GW+EDMFT implementation, we will consider three Hilbert spaces: the
full space (F), the correlated space (C), and the low-energy space (L). The
full space is the entire Hilbert space. To label the basis functions in the
full space, we will use an upper-case Latin alphabet
$|\varphi_{I}^{f/b}\rangle$ or a position vector together with a spin index
$|\mathbf{x}\rangle=|\mathbf{r}\sigma\rangle$. Here $f$ and $b$ in the
superscript denote “Fermion” and “Boson” respectively. The correlated space is
a subspace of the full space. They are spanned by correlated orbitals for
fermionic quantities and orthonormalized product basis sets of the correlated
orbitals for bosonic quantities. We will use a lower-case Greek alphabet to
label the basis functions in the correlated space, e.g.
$|\varphi_{\alpha}^{f/b}\rangle$.

The low-energy space is a subspace of the full space, and a parent vector
space of the correlated space. We construct this Hilbert space (L) for the
crystal momentum ($\mathbf{k}$) interpolation of $G^{GW}$ and $W^{GW}$. These
are two quantities obtained from ab initio GW calculations. Due to the heavy
numerical cost, it is not feasible to acquire them at an arbitrary crystal
momentum resolution. Therefore, we take advantage of Wannier interpolation to
interpolate these two quantities under the constraint of a restricted Hilbert
space. Maximally-localized Wannier functions are used as the localized basis
sets for fermionic quantities. Typically, they are constructed to span in the
low-energy window (typically chosen in $E_{F}\pm 10eV$ as a frozen energy
window). The basis functions for the bosonic quantities are constructed by
calculating orthonormalized product basis functions of the low-energy
fermionic basis functions. We will use lower-case Latin alphabet to label the
basis functions in the low-energy space, e.g. $|\varphi_{i}^{f/b}\rangle$.
After the basis function construction, we project $G^{GW}$ and $W^{GW}$ into
the low-energy space and then interpolate them in the momentum space.

We also interpolate the Green’s function, $G^{GW}$, and the screened Coulomb
interaction, $W^{GW}$, onto a finer Matsubara frequency grid. The simulation
temperature determines the frequency grid for both $G^{GW}$ and $W^{GW}$. For
the ab initio GW calculation, we employ the FlapwMPBT package; however, GW
calculations at temperatures lower than 1000K can be computationally costly.
As an alternative, we approximate $G^{GW}$ and $W^{GW}$ at temperatures lower
than 1000K via interpolation of the quantities obtained at 1000K.

For a function (A) in the low-energy space and correlated space, we utilize
$\overline{A}$ and $\widetilde{A}$ notations, respectively. To exemplify, $A$
represents the function $A$ in the full space, which is calculated on a coarse
crystal momentum and frequency grid. $\overline{A}$ denotes the same function
in the low-energy space, calculated on a fine crystal momentum and frequency
grid using interpolation techniques. Lastly, $\widetilde{A}$ is denoting the
function in the correlated space, calculated on a fine frequency grid.

For the projection of a parent space onto a subspace, we employ the notation
$\hat{P}$, while $\hat{E}$ is used for the embedding of a subspace quantity.
Subscripts on the $\hat{E}$ and $\hat{P}$ denote the initial and destination
vector spaces, and $F$, $L$, and $C$ are used to represent full space, low-
energy space, and correlated space, respectively. Additionally, superscripts
“f” and “b” signify “Fermionic” and “Bosonic”. To illustrate, $\hat{P}_{F\to
L}^{f}(A)$ denotes the fermionic projection of $A$ from the full space to the
low-energy space.

###  3.2 Full GW+EDMFT implementation

Figure 3: ComDMFT v2.0 components and the flowchart for full GW+EDMFT
calculation. Inputs, outputs and important equations each component solves are
also shown. Colored equations show GW (blue), EDMFT (red), and double counting
(green) calculations shown in Fig. 2

Fig. 3 provides the complete GW+EDMFT flowchart. To begin, FlapwMBPT computes
the GW Green’s function ($G_{GW}$) and GW screened Coulomb interaction
($W_{GW}$). Subsequently, ComWann constructs localized orbitals
$\varphi_{i}^{f}(\mathbf{r})$ and $\varphi_{\alpha}^{f}(\mathbf{r})$ as well
as bosonic basis functions $\varphi_{i}^{b}(\mathbf{r})$ and
$\varphi_{\alpha}^{b}(\mathbf{r})$. These basis functions in the low-energy
space and correlated space facilitate fermionic projection ($\hat{P}^{f}$),
fermionic embedding ($\hat{E}^{f}$), bosonic projection ($\hat{P}^{b}$) and
bosonic embedding ($\hat{E}^{b}$). ComLocal then calculates the local Green’s
function ($\widetilde{G}_{loc}$) and local screened Coulomb interaction
$\widetilde{W}_{loc}$ under the GW+EDMFT approximation. Following this, ComDC
obtains the double-counted self-energy ($\Sigma_{DC}$) and double-counting
polarizability ($\Pi_{DC}$) within the local GW approximation. ComCTQMC and
ComWeiss together evaluate the fermionic and bosonic Weiss fields, and compute
the EDMFT self-energy ($\Sigma_{EDMFT}$) and EDMFT polarizability
($\Pi_{EDMFT}$). Finally, ComEmbed embeds the EDMFT and double-counting
objects into $G_{GW}$ and $W_{GW}$ in the full space. This loop is iteratively
solved until a converged solution is achieved.

###  3.3 FlapwMBPT and full GW calculation

We conducted a first-principles study to obtain the GW Green’s Function,
$G_{GW}$, and the GW screened Coulomb Interaction, $W_{GW}$, using FlapwMBPT
[4]. To this end, we modified FlapwMBPT to calculate the GW self-energy,
$\Sigma_{GW}$ and GW polarizability, $\Pi_{GW}$, from $G$ and $W$ within the
framework of GW+EDMFT.

$\begin{split}\Sigma_{GW}(\mathbf{k}^{c},i\omega_{n}^{c})=&-\sum_{\mathbf{R}^{c}}\int_{0}^{\beta^{c}}d\tau
G(\mathbf{R}^{c},\tau)\circ
W(-\mathbf{R}^{c},-\tau)e^{-i(\mathbf{k}^{c}\cdot\mathbf{R}^{c}-\omega_{n}^{c}\tau)}\\\
P_{GW}(\mathbf{k}^{c},i\nu_{n}^{c})=&\sum_{\mathbf{R}^{c}}\int_{0}^{\beta^{c}}d\tau
G(\mathbf{R}^{c},\tau)\circ
G(-\mathbf{R}^{c},-\tau)e^{-i(\mathbf{k}^{c}\cdot\mathbf{R}^{c}-\nu_{n}^{c}\tau)}\\\
\end{split}$ (25)

Here, superscript “c” in $\mathbf{k}^{c}$, $\mathbf{R}^{c}$,
$i\omega_{n}^{c}$, $i\nu_{n}^{c}$ and $\beta^{c}$ denotes “coarse” frequency
and momentum grids, and $\circ$ denotes element-wise product in the position
and spin space. Then FlapwMBPT calculates $G_{GW}$ and $W_{GW}$ in the
following way.

$\begin{split}G_{GW}(\mathbf{k}^{c},i\omega_{n}^{c})=&\left(G_{H}^{-1}(\mathbf{k}^{c},i\omega_{n}^{c})-\Sigma_{GW}(\mathbf{k}^{c},i\omega_{n}^{c})\right)^{-1}\\\
W_{GW}(\mathbf{k}^{c},i\omega_{n}^{c})=&\left(V^{-1}(\mathbf{k}^{c})-P_{GW}(\mathbf{k}^{c},i\omega_{n}^{c})\right)^{-1}\\\
\end{split}$ (26)

Hartree energy matrix element in the position representation is calculated by
using $G$ within GW+EDMFT

$\begin{split}\Sigma_{H}(\mathbf{x},\mathbf{x}^{\prime},\mathbf{k}^{c})=&\delta(\mathbf{x}-\mathbf{x}^{\prime})\frac{1}{N_{\mathbf{k}}}\sum_{\mathbf{k_{1}^{c}}}\int
d\mathbf{x}^{\prime\prime}G(\mathbf{x},\mathbf{x},\mathbf{k}_{1}^{c},\tau=0^{-})V(\mathbf{x},\mathbf{x}^{\prime\prime},\mathbf{k}^{c}=0)\\\
\end{split}$ (27)

###  3.4 FlapwMBPT and the quasiparticle Hamiltonian construction

For the purpose of Wannier interpolation of $G^{GW}$ and $W^{GW}$ in the low-
energy space, localized basis sets are required. We construct Wannier
functions spanning the low-energy space (Hilbert space in the energy window of
$-10\ eV<E-E_{F}<10\ eV$) by using the Wannier90 package. The Wannier90
package requires the best mean-field Hamiltonian to reproduce the fully
interacting GW Green’s function, $G^{GW}$. By linearizing the GW self-energy
$\Sigma^{GW}$ in the frequency, we construct a GW quasiparticle Hamiltonian at
each iteration in the following manner.

$\begin{split}H_{GW}^{QP}(\mathbf{k}^{c})=&\sqrt{Z_{GW}(\mathbf{k}^{c})}\left(H_{H}(\mathbf{k}^{c})+\Sigma_{GW}(\mathbf{k}^{c},\omega=0)\right)\sqrt{Z_{GW}(\mathbf{k}^{c})}\\\
Z_{GW}(\mathbf{k}^{c})=&\left(1-\frac{\partial\Sigma_{GW}(\mathbf{k}^{c},\omega)}{\partial\omega}\right)^{-1}\bigg{|}_{\omega=0}\\\
\end{split}$ (28)

Here, $H_{H}$ denotes Hartree Hamiltonian. By diagonalizing
$H_{GW}^{QP}(\mathbf{k}^{c})$, we obtain energy eigenvectors
$|n\mathbf{k}^{c}\rangle$, where $n$ is the band index. These eigenvectors are
then utilized in order to calculate the inputs for the Wannier90 package. It
is important to note that $H_{GW}^{QP}$ is solely constructed in order to
build Wannier functions.

###  3.5 Wannier90 and ComWann: localized basis set construction for low-
energy space and correlated space

Calculating local quantities such as $\widetilde{G}_{loc}$ and
$\widetilde{W}_{loc}$ presents a challenge due to the need to calculate
$G_{GW}$ and $W_{GW}$ on a fine $\mathbf{k}$-point grid. The accuracy of
$\widetilde{G}_{loc}$ and $\widetilde{W}_{loc}$ is dependent on the momentum
resolution of the aforementioned quantities; however, obtaining them from
first principles is too computationally expensive. Wannier interpolation can
be employed to interpolate $\overline{G}_{GW}(\mathbf{k}^{c},i\omega_{n})$ and
$\overline{W}_{GW}(\mathbf{k}^{c},i\omega_{n})$ into those on a fine
$\mathbf{k}$ grid instead.

For the Wannier interpolation of fermionic quantities in low-energy space,
Maximally Localized Wannier Functions (MLWFs) spanning the low-energy space
are constructed. To find orbitals with maximum localizations (or minimum of
the total orbital spreads), quasiparticle wavefunction information is needed
for Wannier90. For this purpose, we utilized the eigenfunctions
($|\psi_{n\mathbf{k}}\rangle$) of $H_{GW}^{QP}(\mathbf{k}^{c})$ as stated in
Eq. 28.

The other input to the Wannier90 package is trial orbitals
$\langle\mathbf{r}|T_{i\mathbf{R}}\rangle$, which provide the initial
condition for the total spread minimization. We employed muffin-tin orbitals
of the LAPW basis set for our trial orbitals. The radial function of an
initial trial orbital can be any linear combination of any muffin-tin orbitals
with the same angular momentum character and that of the centered atom.
ComWann takes the linear combination which maximizes
$\sum_{n,\mathbf{k}}^{E_{froz}^{min}<E_{n,\mathbf{k}}<E_{froz}^{max}}|\langle\psi_{n\mathbf{k}}|T_{i\mathbf{k}}\rangle|^{2}$.
Here, $E_{froz}^{min/max}$ is the lower/upper bound of the frozen energy
window, and $|T_{i\mathbf{k}}\rangle$ is the Bloch sum of
$|T_{i\mathbf{R}}\rangle$. In ComWann, the angular momentum part of the trial
orbitals is approximately defined as the cubic-harmonics $Y_{lm}$ by default
when there is no spin-orbit coupling present.

$\begin{split}Y_{lm}=\begin{cases}\frac{i}{\sqrt{2}}\left(Y_{l}^{-|m|}-(-1)^{m}Y_{l}^{|m|}\right),&m<0\\\
Y_{l}^{0},&m=0\\\
\frac{1}{\sqrt{2}}\left(Y_{l}^{-|m|}+(-1)^{m}Y_{l}^{|m|}\right),&m>0\\\
\end{cases}\end{split},$ (29)

where $Y_{l}^{m}$ are the spherical harmonics. That is, a subscript $m$
corresponds to the cubic harmonics, and a superscript $m$ denotes spherical
harmonics.

For a system with spin-orbit coupling, the angular part is approximately the
spin-angular function ($\Omega_{l,a,m}$):

$\begin{split}\Omega_{l,a=\pm\frac{1}{2},m}=\sum_{s\pm
1/2}C_{a,s}^{l,m}Y_{l}^{m-s}(\hat{r})u_{s}.\end{split}$ (30)

Here, $u_{s}$ is a spinor and $C_{a,s}^{l,m}=\langle
l,m-s,\frac{1}{2},s|l+a,m\rangle$. Then, following the procedure outlined in
Sec. 2, we constructed orthonormalized product basis from the MLWFs
($|\psi_{i}^{f}\rangle$) spanning low-energy space and obtained a basis set
for the bosonic quantities in that space.

We select a subset of orbitals with desired center positions and angular
momentum characters from MLWFs and label them as correlated orbitals, which
constitute the basis set of fermionic quantities in the correlated space.
Subsequently, we construct an orthonormalized product basis of these
correlated orbitals, thereby obtaining a basis set for the bosonic quantities
in the correlated space, following the procedure detailed in Sec. 2.

Now, we possess all the necessary components for the projection and embedding
between three distinct spaces. The projection and embedding between Full space
and low-energy space can be accomplished via the following approach.

$\begin{split}\hat{P}_{F\to L,ij}^{f/b}(A)=\sum_{MN\in
F}\langle\varphi_{i}^{f/b}|\varphi_{M}^{f/b}\rangle
A_{MN}\langle\varphi_{N}^{f/b}|\varphi_{j}^{f/b}\rangle\\\ \hat{E}_{L\to
F,MN}^{f/b}(\overline{A})=\sum_{ij\in
L}\langle\varphi_{M}^{f/b}|\varphi_{i}^{f/b}\rangle\overline{A}_{ij}\langle\varphi_{j}^{f/b}|\varphi_{N}^{f/b}\rangle\\\
\end{split}$ (31)

Projection and embedding between low-energy space and correlated space can be
achieved in the following way.

$\begin{split}\hat{P}_{L\to C,\alpha\beta}^{f/b}(\overline{A})=\sum_{ij\in
L}\langle\varphi_{\alpha}^{f/b}|\varphi_{i}^{f/b}\rangle\overline{A}_{ij}\langle\varphi_{j}^{f/b}|\varphi_{\beta}^{f/b}\rangle\\\
\hat{E}_{C\to L,ij}^{f/b}(\widetilde{A})=\sum_{\alpha\beta\in
C}\langle\varphi_{i}^{f/b}|\varphi_{\alpha}^{f/b}\rangle\widetilde{A}_{\alpha\beta}\langle\varphi_{\beta}^{f/b}|\varphi_{j}^{f/b}\rangle\\\
\end{split}$ (32)

Finally, a mapping between the full space and the correlated-energy space can
be constructed through the following projection and embedding.

$\begin{split}\hat{P}_{F\to C,\alpha\beta}^{f/b}(A)=\sum_{MN\in
F}\langle\varphi_{\alpha}^{f/b}|\varphi_{M}^{f/b}\rangle
A_{MN}\langle\varphi_{N}^{f/b}|\varphi_{\beta}^{f/b}\rangle\\\ \hat{E}_{C\to
F,MN}^{f/b}(\widetilde{A})=\sum_{\alpha\beta\in
C}\langle\varphi_{M}^{f/b}|\varphi_{\alpha}^{f/b}\rangle\widetilde{A}_{\alpha\beta}\langle\varphi_{\beta}^{f/b}|\varphi_{N}^{f/b}\rangle\\\
\end{split}$ (33)

###  3.6 ComLocal

ComLocal calculates the local Green’s function ($\widetilde{G}_{loc}$) and
local screened Coulomb interaction ($\widetilde{W}_{loc}$) in the following
way. First, with the input of $G_{GW}(\mathbf{k}^{c},i\omega_{n}^{c})$ and
$W_{GW}(\mathbf{k}^{c},i\omega_{n}^{c})$ of FlapwMBPT, ComLocal projects them
into the low-energy space.

$\overline{G}_{GW}(\mathbf{k}^{c},i\omega_{n}^{c})=\hat{P}_{F\to
L}^{f}\left(G_{GW}(\mathbf{k}^{c},i\omega_{n}^{c})\right)$ (34)
$\overline{W}_{GW}(\mathbf{k}^{c},i\nu_{n}^{c})=\hat{P}_{F\to
L}^{b}\left(W_{GW}(\mathbf{k}^{c},i\nu_{n}^{c})\right)$ (35)

Next, ComLocal Wannier-interpolates them from the coarse k-grid to the fine
k-grid, to obtain $\overline{G}_{GW}(\mathbf{k}^{f},i\omega_{n}^{c})$ and
$\overline{W}_{GW}(\mathbf{k}^{f},i\omega_{n}^{c})$. Subsequently, the
interpolation from the coarse Matsubara grid to fine-Matsubara grid is carried
out to acquire $\overline{G}_{GW}(\mathbf{k}^{f},i\omega_{n}^{f})$ and
$\overline{W}_{GW}(\mathbf{k}^{f},i\omega_{n}^{f})$. Then, EDMFT and double-
counting quantities are embedded into
$\overline{G}_{GW}(\mathbf{k}^{f},i\omega_{n}^{f})$ and
$\overline{W}_{GW}(\mathbf{k}^{f},i\omega_{n}^{f})$.

$\overline{G}(\mathbf{k}^{f},i\omega_{n}^{f})=\\{\overline{G}_{GW}^{-1}(\mathbf{k}^{f},i\omega_{n}^{f})-\hat{E}_{C\to
L}^{f}(\widetilde{\Sigma}_{EDMFT}(i\omega_{n}^{f})-\widetilde{\Sigma}_{DC}(i\omega_{n}^{f}))\\}^{-1}$
(36)
$\overline{W}(\mathbf{k}^{f},i\omega_{n}^{f})=\\{\overline{W}_{GW}^{-1}(\mathbf{k}^{f},i\omega_{n}^{f})-\hat{E}_{C\to
L}^{b}(\widetilde{\Pi}_{EDMFT}(i\omega_{n}^{f})-\widetilde{\Pi}_{DC}(i\omega_{n}^{f}))\\}^{-1}$
(37)

Finally, we project $\overline{G}$ and $\overline{W}$ into the correlated
subspace and obtain local quantities.

$\widetilde{G}_{loc}(i\omega_{n}^{f})=\hat{P}_{L\to
C}^{f}\left(\overline{G}(\mathbf{k}^{f},i\omega_{n}^{f})\right)$ (38)
$\widetilde{W}_{loc}(i\omega_{n}^{f})=\hat{P}_{L\to
C}^{b}\left(\overline{W}(\mathbf{k}^{f},i\omega_{n}^{f})\right)$ (39)

###  3.7 ComDC

ComDC is the module to calculate the double-counted electron self-energy and
polarizability. Local GW self-energy has been calculated in the following way.

$\widetilde{\Sigma}_{DC,\alpha,\beta}(i\omega_{n})=-\sum_{\gamma\delta}\int
d\tau\widetilde{G}_{loc,\delta\gamma}(\tau)\widetilde{W}_{loc,\alpha\gamma\beta\delta}(\tau)e^{i\omega_{n}\tau}.$
(40)

where $\alpha$,$\beta$,$\gamma$ and $\delta$ are composite indices for both
orbital and spin (spin-angular function index) in the system without (with)
spin-orbit coupling. Double-counting polarizability is calculated in the
following way.

$\widetilde{P}_{\alpha\beta\gamma\delta}(i\nu_{n})\allowbreak=\int
d\tau\widetilde{G}_{\delta\beta}(\tau)\widetilde{G}_{\gamma\alpha}(-\tau)\allowbreak
e^{i\nu_{n}\tau}.$ (41)

###  3.8 ComWeiss

ComWeiss calculates bosonic as well as fermionic Weiss field in the following
way.

$\widetilde{\mathcal{G}}={\left(\widetilde{G}_{loc}^{-1}+\widetilde{\Sigma}_{EDMFT}\right)^{-1}}$
(42)
$\widetilde{\mathcal{U}}={\left(\widetilde{W}_{loc}^{-1}+\widetilde{\Pi}_{EDMFT}\right)^{-1}}$

###  3.9 ComCTQMC

In order to solve the impurity model action as required by ComDMFT,
hybridization-expansion continuous time quantum Monte Carlo (CTQMC) is
employed. CTQMC is an established stochastic approach for obtaining exact
numerical solutions of an impurity model, namely a small interacting system
embedded in an electron bath. The EDMFT action of the impurity model, required
for GW+DMFT, is

$\begin{split}S=&-\sum_{\alpha\chi}\iint_{0}^{\beta}d\tau
d\tau^{\prime}c^{\dagger}_{\alpha}(\tau)\widetilde{\mathcal{G}}_{\alpha\chi}^{-1}(\tau-\tau^{\prime})c_{\chi}(\tau^{\prime})\\\
&\quad\quad+\frac{1}{2}\sum_{\alpha\chi\gamma\delta}\iint_{0}^{\beta}d\tau
d\tau^{\prime}c^{\dagger}_{\alpha}(\tau)c^{\dagger}_{\chi}(\tau^{\prime})\widetilde{\mathcal{U}}_{\alpha\chi\gamma\delta}(\tau-\tau^{\prime})c_{\gamma}(\tau^{\prime})c_{\delta}(\tau),\end{split}$
(43)

where $c^{\dagger}_{\alpha}$ creates an electron in the generalized orbital
$\alpha$ (which includes both spin and orbital degrees of freedom), $\beta$ is
the inverse temperature.

The present study postulates that the frequency-dependent interaction is
assumed to take the following form.

$\widetilde{\mathcal{U}}_{\alpha\chi\gamma\delta}(i\nu_{n})=\widetilde{U}_{\alpha\chi\gamma\delta}+F^{0}(i\nu_{n})\delta_{\alpha\delta}\delta_{\chi\gamma},$
(44)

We take into account only the dynamical screening of the Slater-Condon
parameter $F^{0}$ for the sake of convenience in the numerical algorithm based
on hybridization-expansion CTQMC. $\widetilde{U}_{\alpha\chi\gamma\delta}$ are
frequency-independent and are approximated by their values at
$i\nu_{n}=\infty$. The fermionic Weiss field is divided into two parts:

$\widetilde{\mathcal{G}}_{\alpha\chi}^{-1}(i\omega_{n})=(i\omega_{n}+\mu)\delta_{\alpha\chi}-\widetilde{t}_{\alpha\chi}-\widetilde{\Delta}_{\alpha\chi}(i\omega_{n}),$
(45)

Here, $\widetilde{t}$ is the impurity-level matrix and $\widetilde{\Delta}$ is
the hybridization function matrix. By solving the quantum impurity action, we
obtain the self-energy ($\widetilde{\Sigma}_{EDMFT}$) and the
polarizability($\widetilde{\Pi}_{EDMFT}$).

###  3.10 ComEmbed

In the final step, EDMFT and double-counting quantities were incorporated into
the $G_{GW}(\mathbf{k}^{c},i\omega_{n}^{c})$ and
$W_{GW}(\mathbf{k}^{c},i\nu_{n}^{c})$ of FlapwMBPT.

$\begin{split}G=\\{G_{GW}^{-1}-\hat{E}_{C\to
F}^{f}(\widetilde{\Sigma}_{EDMFT}-\widetilde{\Sigma}_{DC})\\}^{-1}\\\
W=\\{W_{GW}^{-1}-\hat{E}_{C\to
F}(\widetilde{\Pi}_{EDMFT}-\widetilde{\Pi}_{DC})\\}^{-1}\\\ \end{split}$ (46)

The obtained Green’s function, $G(\mathbf{k}^{c},i\omega_{n}^{c})$, and the
screened Coulomb interaction, $W(\mathbf{k}^{c},i\nu_{n}^{c})$, will be
utilized to calculate the self-energy,
$\Sigma_{GW}(\mathbf{k}^{c},i\omega_{n}^{c})$, and the polarization function,
$P_{GW}(\mathbf{k}^{c},i\nu_{n}^{c})$ in eq. LABEL:gw_self_energy.

###  3.11 Causal Optimization

Truncations in the $\Psi$ functional of the Luttinger-Ward (or Baym-Kadanoff)
functional[52, 53, 54] provides a roadmap to build GW+EDMFT theory. Recently,
it has been shown that the free energy functional can be multivalued [81, 82,
83, 84, 85]. Moreover, it has been shown that this multivaluedness can bring
about noncausality of the Greens functions and self energies[81]. Noncausal
Green’s functions can also arise from non-local extension of dynamical mean
field theory[86, 66, 87]. To address this problem, we used the causal
optimization method[88] which replaces non-causal objects with best-possible
causal objects. In this implementation, we used the method to find the closest
causal $\Pi_{EDMFT}$, $\mathcal{U}$, and $W_{loc}$, if these quantities are
non-causal.

###  3.12 Database architecture

Figure 4:  Grouped data structure within GW+EDMFT.

In ComDMFT v2.0, data exchange between components is facilitated by a
database. This version supports writing and reading data with the parallel
HDF5 IO library. Each component stores data in a singular HDF5 file called
“global.dat”. Keys are provided to ensure the data is easily retrieved when
needed. There are four HDF5 groups included in the file: ”Full Space”, ”Low
energy space”, ”Correlated Space”, and ”Correlated Problem” which are denoted
as different Hilbert spaces in Sec. 3.1. Data in each space is stored in its
corresponding HDF5 group. Additionally, the “Correlated Problem” group is
designed to store local quantities of each inequivalent impurity problem. Fig.
4 provides an overview of the groups and important data stored in the
“global.dat”. For example, $G$ and $W$ obtained from the FlapwMBPT calculation
are stored in the “full space group”, while $\hat{P}_{F\to L}^{f/b}$ and
$\hat{E}_{L\to F}^{f/b}$ are stored in the “Low-energy space” group.

##  4 Results

To evaluate the performance of fully self-consistent GW+EDMFT, we calculated
the electronic structure of several quantum materials, with different degrees
of correlation, NiO and SrVO3 at a temperature of 1000K. We compared the full
GW+EDMFT results with experiments, as well as with those of LQSGW+DMFT. We
find that the intermediate quantities such as the Weiss fields are quite
different in the two methods, but the final physical results are in rather
good agreement.

###  4.1 NiO

NiO is a pardigmatic Mott insulator and it has been intensively studied [89,
90, 91, 92, 93, 94, 95, 96]. Experimentally, NiO is a paramagnetic insulator;
however, traditional ab initio electronic structure methodologies based on
one-particle picture predict it to be metallic when it is not ordered
magnetically. We use NiO to illustrate the GW+EDMFT method and compare the
results to LQSGW+DMFT. Both calculations treat the Ni-3$d$ orbitals as
correlated orbitals, and the simulation temperature is $1000$ K.

####  4.1.1 GW self-consistency

Figure 5:  (a) and (b) show the Wannier-interpolated GW bandstructures
($H_{GW}^{QP}$ in eq. (28)) of NiO at the first and last iterations of the
self-consistent loop in Fig. 3, respectively.

Figure 5 (a) and (b) show the Wannier-interpolated GW bandstructures
($H_{GW}^{QP}$ in eq. (28)), obtained at the first and last iterations of the
GW+EDMFT self-consistent loop in Figure 3, respectively. The GW self-energy,
as expressed in equation LABEL:gw_self_energy, does not exhibit any divergence
near the chemical potential. Consequently, $\Sigma_{GW}$ can be linearized
safely even in this charge transfer insulator. This implies that the EDMFT
self-energy, described in eq. 14, must have a divergence near the chemical
potential to open a Mott insulating gap. Furthermore, the importance of GW
self-consistency is evident when comparing figures 5 (a) and (b). The band
width of two low-lying bands (largely originated from Ni-$e_{g}$ and O-$p$)
increased by 50%. A strong downshift of the occupied Ni-$t_{2g}$ bands is also
observed. Note that this is not the GW+EDMFT bandstructure, and that the
spectral function within GW+EDMFT can be obtained by embedding EDMFT and
double counting self-energy to the GW Green’s Function.

####  4.1.2 Hybridization functions

Figure 6:  Imaginary parts of hybridization functions associated with (a)
Ni-$t_{2g}$ and (b) Ni-$e_{g}$ of NiO. (c) and (d) are zoom in view of (a) and
(b), respectively. Orange and blue colors show the results within full
GW+EDMFT and LQSGW+DMFT, respectively

Fig. 6 illustrates the imaginary part of the hybridization functions
associated with Ni-$d$ orbitals on the imaginary frequency axis. Results from
both LQSGW+DMFT and GW+EDMFT are displayed in blue and orange, respectively.
It is observed that Ni-$e_{g}$ orbitals have a stronger hybridization near the
chemical potential compared to Ni-$t_{2g}$ orbitals in both methodologies.
Additionally, GW+EDMFT reveals a substantially enhanced hybridization function
when compared to the LQSGW+DMFT result. Within LQSGW+DMFT, it is found to be
near-zero at $\sim$ 200 eV. However, a peak is present even at an energy of
$\sim$ 400 eV in the GW+EDMFT hybridization function. Unlike conventional
implementations of DFT+DMFT or LQSGW+EDMFT, where the hybridizing states are
limited by the energy window (e.g. $-10\>\mathrm{eV}<E-E_{F}<10$ eV) or the
basis functions (e.g., the constructed Wannier orbitals), GW+DMFT allows for
hopping of electrons from or into all the states in the full Hilbert space. We
note, however, that the high energy features which show up in GW+EDMFT (but no
other existing method), and therefore look shocking, have little effect on the
local self-energy and polarizability. Indeed, CTQMC finds identical self-
energies when we dramatically adjusted these hybridization functions in order
to test the consequences of these high energy features.111We forced the
hybridization to smoothly transition from the low-energy to asymptotic tail
regions in these tests using a spline interpolation, and then we re-ran CTQMC,
to check that the Green’s functions and self-energies remained within the
error bars of the solution.

####  4.1.3 Bosonic Weiss field

Figure 7:  (a) The monopole part ($F^{0}$ of Slater integrals) of the bosonic
Weiss field $F^{0}$ associated with Ni-$d$ orbitals in NiO. (b) zoom-in view
of (a). Orange and blue colors show the results within full GW+EDMFT and
LQSGW+DMFT, respectively

Fig. 7 (a) presents the monopole part ($F^{0}$) of the bosonic Weiss field
associated with Ni-$d$ orbitals. Fig. 7 (b) displays a zoomed-in view of the
same figure. The static value of $F^{0}$ is 10 $eV$ within full GW+EDMFT,
which is 40$\%$ greater than that acquired from LQSGW+DMFT. Moreover, this
value further increases to 15 $eV$ at a frequency of 10 $eV$, indicating a
strong frequency dependence in the low energy region. The dynamical screening
captured by the Lang-Firsov approximation is suggested to be a pathway for
renormalization of the hybridization function[97]. Discussion on
renormalization of the hybridization function due to the dynamical screening
is in the following section.

####  4.1.4 EDMFT self-energies

Figure 8:  Imaginary parts of EDMFT self-energies associated with (a)
Ni-$t_{2g}$ and (b) Ni-$e_{g}$ in NiO. Orange and blue colors show the results
within full GW+EDMFT and LQSGW+DMFT, respectively

Fig. 8 presents the imaginary parts of the EDMFT self-energies for the
Ni-$t_{2g}$ and Ni-$e_{g}$ orbitals on the imaginary frequency axis. The blue
and orange lines refer to LQSGW+DMFT and GW+EDMFT, respectively. The
Ni-$t_{2g}$ orbital self-energy displays a linear dependence on the imaginary
frequency near the chemical potential; this is a characteristic of Fermi-
liquid behavior. In contrast, the imaginary part of the Ni-$e_{g}$ self-energy
reveals the onset of the divergence near the chemical potential, indicating a
Mott-insulating phase. This is supported by the impurity Green’s function in
the static limit; on the Matsubara axis, Ni-$e_{g}$ impurity Green’s function
approches to zero values. Interestingly, the Ni-$t_{2g}$ orbital quasi-
particle weights within the two different theories are found to be comparable,
with values of 0.71 and 0.47 within full GW+EDMFT and LQSGW+DMFT,
respectively. This is surprisingly similar, given the significant differences
between the fermionic and bosonic Weiss fields in the two theories. This
implies a compensation of the enhanced fermionic Weiss field by the enhanced
bosonic Weiss fields.

####  4.1.5 Double-counting self-energies

Figure 9:  Imaginary parts of the double-counting self-energies associated
with (a) Ni-$t_{2g}$ and (b) Ni-$e_{g}$ in NiO. Orange and blue colors show
the results within full GW+EDMFT and LQSGW+DMFT, respectively

Fig. 9 displays the double-counting self-energies associated with Ni-$d$
orbitals on the imaginary frequency axis within the local-GW approximation.
The results obtained from the LQSGW+DMFT and GW+EDMFT methodologies are shown
in blue and orange lines, respectively. Similarly, it is observed that neither
the Ni-$e_{g}$ nor the Ni-$t_{2g}$ orbitals exhibit any divergence in their
double-counting self-energies near zero frequency. However, there are
quantitative differences in the quasiparticle-weights between the two
methodologies; the Ni-$t_{2g}$ orbitals possess quasiparticle-weights of 0.88
and 0.60 in GW+EDMFT and LQSGW+DMFT, respectively, whereas for the Ni-$e_{g}$
orbitals, the quasiparticle-weights are 0.88 and 0.47.

####  4.1.6 Spectral functions

Figure 10:  NiO electronic structures within (a) non spin-polarized LDA and
(b) non spin-polarized LQSGW, (c) paramagnetic LQSGW+DMFT and (d) paramagnetic
full GW+EDMFT, respectively

Fig. 10 presents the electronic structures of NiO without magnetic order with
various approaches: LDA, LQSGW, LQSGW+DMFT and GW+EDMFT. GW+EDMFT spectral
function are obtained by the the analytical continuation of the momentum-
resolved Greens function on the imaginary frequency axis. Results from the non
spin-polarized LDA and non spin-polarized LQSGW approaches indicate that NiO
is a metal when it is not magnetically ordered, as evidenced by the presence
of electronic states near the chemical potential. In contrast, GW+EDMFT and
LQSGW+DMFT predict that NiO is an insulator.

####  4.1.7 Projected density of states

Figure 11:  Projected density of states of NiO within (a) LQSGW+DMFT and (b)
GW+EDMFT.

Fig. 11 demonstrates projected density of states (DOS) from LQSGW+DMFT and
GW+EDMFT. GW+EDMFT data is obtained through analytical continuation of the
orbital-resolved Greens function on the imaginary frequency axis. This data
reveals that for Ni-$e_{g}$ orbitals, both methodologies open up a well-
defined Mott insulating gap, as indicated by the strong divergence of the
EDMFT self-energy. The Ni-$t_{2g}$ orbitals are fully-filled as expected. The
separations between the upper and lower Hubbard bands as well as between the
upper Hubbard band and Ni-$t_{2g}$ orbitals are both $\sim 6$ eV within
GW+EDMFT. This value is consistent with the measured XPS-BIS peak separation
of $\sim 6$ eV[98].

###  4.2 SrVO3

SrVO3 has been recognized to be one of the simplest correlated transition
metal oxides. It is a cubic perovskite with a three V-d bands crossing the
Fermi level. Photoemission and inverse photoemission spectra both exhibit
satellite peaks[99], making it a topical subject for DMFT methodologies[68,
100, 101, 102, 59, 69, 103, 104, 78, 43, 105, 106]. In this paper, we apply
GW+EDMFT to this metallic phase of SrVO3 and compare the results with other
methodologies. For our calculations, the V-3$d$ orbitals were treated as
correlated orbitals, and the simulation temperature was set to T=1000K.

####  4.2.1 GW self-consistency

Figure 12:  (a) and (b) show the Wannier-interpolated GW bandstructures
($H_{GW}^{QP}$ in eq. (28)) of SrVO3 at the first and last iterations of the
self-consistent loop in Fig. 3, respectively.

Fig. 12 (a) and (b) display the Wannier-interpolated GW bandstructures
($H_{GW}^{QP}$) in eq. (28), obtained at the first and last iterations of the
self-consistent loop in Fig. 3 respectively. As the GW self-energy in eq.
LABEL:gw_self_energy does not exhibit any divergence near the chemical
potential, a linearization of the self-energy can be safely applied to
construct $H_{GW}^{QP}$ in this correlated metallic phase. Notably, the GW
bandstructure at the first and last iteration is almost identical, indicating
that the EDMFT and double counting quantities compensate each other.

####  4.2.2 Hybridization function

Figure 13:  Imaginary parts of hybridization functions associated with (a)
V-$t_{2g}$ and (b) Ni-$e_{g}$ of SrVO3. (c) and (d) are zoom-in view of (a)
and (b), respectively. Orange and blue colors show the results within full
GW+EDMFT and LQSGW+DMFT, respectively Figure 14:  (a) The monopole part
($F^{0}$ of Slater integrals) of the bosonic Weiss field $F^{0}$ associated
with V-$d$ orbitals in SrVO3. (b) zoom-in view of (a). Orange and blue colors
show the results within full GW+EDMFT and LQSGW+DMFT, respectively

Fig. 13 displays the imaginary part of the hybridization functions for the
V-$t_{2g}$ and V-$e_{g}$ orbitals on the imaginary frequency axis. Blue and
orange colors in the figure correspond to the results obtained from LQSGW+DMFT
and GW+EDMFT, respectively. As in NiO, the magnitude of this hybridization is
very different for the same reasons discussed in Sec. 4.1.

####  4.2.3 Bosonic Weiss field

Fig. 7 (a) presents the monopole part of the bosonic Weiss field associated
with V-$d$ orbitals ($F^{0}$ of Slater integrals). Fig. 14 (b) is a zoomed-in
view. The displayed blue and orange lines represent the results obtained by
means of LQSGW+DMFT and GW+EDMFT, respectively. The static value of the
monopole terms is 4 eV in both theories. As is the case for NiO, GW+EDMFT
exhibits a stronger frequency dependence in the low-energy region, with the
monopole term reaching 15 $eV$ at a frequency of 10 eV. As discussed in 4.1,
this renormalizes the hybridization.

####  4.2.4 EDMFT self-energies

Figure 15:  Imaginary parts of EDMFT self-energies associated with (a)
V-$t_{2g}$ and (b) V-$e_{g}$ in SrVO3. Orange and blue colors show the results
within full GW+EDMFT and LQSGW+DMFT, respectively Figure 16:  Imaginary parts
of the double-counting self-energies associated with (a) V-$t_{2g}$ and (b)
V-$e_{g}$ in SrVO3. Orange and blue colors show the results within full
GW+EDMFT and LQSGW+DMFT, respectively

Fig. 15 illustrates the imaginary parts of the EDMFT self-energies for both
V-$t_{2g}$ and V-$e_{g}$ orbitals as represented by the blue and orange lines,
respectively, on the imaginary frequency axis. These results confirm the
presence of sizable electron correlations for both orbitals. The quasiparticle
weights of V-$t_{2g}$ were determined to be 0.58 and 0.53 within LQSGW+DMFT
and GW+EDMFT, respectively, while the quasiparticle weights of V-$e_{g}$ were
0.73 and 0.70 within LQSGW+DMFT and GW+EDMFT, respectively. These results
suggest the cancellation between the enhanced hybridization and Weiss field,
as indicated by the nearly identical quasiparticle weights in both approaches.

####  4.2.5 Double-counting self-energy

Figure 17:  SrVO3 electronic structures within (a) non spin-polarized LDA and
(b) non spin-polarized LQSGW, (c) paramagnetic LQSGW+DMFT and (d) paramagnetic
full GW+EDMFT, respectively

Fig. 16 depicts the double-counted self-energies of V-$t_{2g}$ and V-$e_{g}$
orbitals in the local-GW approximation on the imaginary frequency axis. Blue
and orange lines indicate the results of LQSGW+DMFT and GW+EDMFT,
respectively. It is observed that both of these approaches exhibit
qualitatively similar outcomes, indicating that the V-$t_{2g}$ orbital is more
correlated than V-$d_{z^{2}}$. Additionally, while the self-energies provided
by LQSGW+DMFT and GW+EDMFT are almost identical, the quasiparticle weight in
the local GW is slightly greater than that of EDMFT in GW+EDMFT.

####  4.2.6 Spectral functions

Fig. 17 displays the electronic structures of SrVO3 computed with various ab
initio approaches: non spin-polarized Local Density Approximation (LDA), non
spin-polarized LQSGW method, LQSGW+DMFT, and full GW+EDMFT. Momentum-resolved
spectral function data on the real frequency axis is derived from the
analytical continuation of the momentum-resolved Greens function on the
imaginary frequency axis. All approaches lead to qualitatively similar band
structures in the vicinity of the Fermi level. Three bands cross the Fermi
level and are distinct from the other bands. The hole excitation energies at
the $\Gamma$ point obtained from LDA, LQSGW, LQSGW+DMFT and GW+EDMFT are 1.0,
0.5, 0.7, and 0.7 eV, respectively, compared to the experimental value of
approximately 0.5 eV[107, 108, 109, 108].

####  4.2.7 Projected density of states

Figure 18:  Projected density of states of SrVO3 within (a) LQSGW+DMFT and (b)
GW+EDMFT.

Fig. 18 illustrates the projected density of states of both LQSGW+DMFT and
GW+EDMFT. Results from both theories present similar features, with three peak
structures located at -4 eV, 0 eV, and 2 eV matching well with each other.
With both theories, we did not observe strong satellite peaks on either side
of the Fermi level, which agrees with the recent claim that defect states are
the source of the observed satellite peak at -1.5 eV[107]. Additionally, only
a V-$e_{g}$ peak located at 2 eV was observed above the Fermi level.

##  5 Conclusions

We presented the implementation of fully self-consistent ab initio GW+EDMFT
for the electronic structure of correlated quantum materials in ComDMFT. This
implementation enables the electronic structure calculation of quantum
materials within weak, intermediate, and strong electron correlation without
prior knowledge of the degree of electron correlation. We tested our
implementation by calculating the electronic structures of NiO and SrVO3 and
obtained qualitatively similar results to LQSGW+DMFT.

##  6 Acknowledgments

This paper was supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences as a part of the Computational Materials Science
Program. S.C. was supported by a KIAS individual Grant (No. CG090601) at Korea
Institute for Advanced Study. This research used resources of the National
Energy Research Scientific Computing Cen- ter (NERSC), a U.S. Department of
Energy Office of Science User Facility operated under Contract No. DE-
AC02-05CH11231.

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