# Fully ab-initio electronic structure of Ca2RuO4

Francesco Petocchi  Department of Physics, University of Fribourg, 1700
Fribourg, Switzerland  Viktor Christiansson  Department of Physics, University
of Fribourg, 1700 Fribourg, Switzerland  Philipp Werner  Department of
Physics, University of Fribourg, 1700 Fribourg, Switzerland

(August 27, 2024)

###### Abstract

The reliable ab-initio description of strongly correlated materials is a long-
sought capability in condensed matter physics. The $GW$+EDMFT method is a
promising scheme, which provides a self-consistent description of correlations
and screening, and does not require user-provided parameters. In order to test
the reliability of this approach we apply it to the experimentally well
characterized perovskite compound Ca2RuO4, in which a temperature-dependent
structural deformation drives a paramagnetic metal-insulator transition. Our
results demonstrate that the nonlocal polarization and self-energy components
introduced by $GW$ are essential for setting the correct balance between
interactions and bandwidths, and that the $GW$+EDMFT scheme produces
remarkably accurate predictions of the electronic properties of this strongly
correlated material.

##  I INTRODUCTION

Density functional theory (DFT)Hohenberg and Kohn (1964); Kohn and Sham (1965)
does not provide an adequate description of the electronic structure of
strongly correlated materials, but substantial progress in the simulation of
this class of materials has been made by adding the missing correlations via
some dynamical mean-field theory (DMFT) construction.Metzner and Vollhardt
(1989); Georges and Kotliar (1992); Georges  _et al._ (1996) The widely used
DFT+DMFTKotliar  _et al._ (2006) scheme however has several drawbacks which
prevent ab-initio predictions. In particular, a true ab-initio procedure needs
to provide the interaction parameters that go into the DMFT calculation, and
the latter should be consistent with the correlated electronic structure. A
more advanced scheme is the combination of extended (E)DMFTSun and Kotliar
(2002) with the $GW$Hedin (1965) method ($GW$+EDMFT),Biermann  _et al._
(2003); Ayral  _et al._ (2013); Werner and Casula (2016) which provides a
self-consistent description of correlations and screening in a solid. Apart
from the choice of a subspace of strongly correlated orbitals, $GW$+EDMFT is
free from ad-hoc parameters and thus represents a true ab-initio method for
strongly correlated materials.

While first applications of this recently developed simulation framework have
produced promising results,Boehnke  _et al._ (2016); Nilsson  _et al._ (2017);
Petocchi  _et al._ (2020a, b) the reliability of $GW$+EDMFT still needs to be
systematically tested. A particularly relevant question is if the self-
consistently computed effective local interaction parameters, which are
obtained by several downfolding steps,Nilsson  _et al._ (2017) correctly
capture the correlation strength in the material. To clarify this point, it is
useful to investigate a compound in which a correlation-driven metal-insulator
transition (MIT) occurs without a simultaneous electronic ordering transition,
since this allows to disentangle the opening of a Mott gap from, e. g., the
effect of antiferromagnetic correlations. Here, we study the layered
perovskite Ca2RuO4, which exhibits a transition between a metal and Mott
insulator (and a simultaneous structural transition) at a temperature near 360
K,Braden  _et al._ (1998); Alexander  _et al._ (1999) which is significantly
larger than the Néel temperature of the antiferromagnetic (AFM) phase emerging
below $T_{N}\sim$110 K.Braden  _et al._ (1998) As we will show, $GW$+EDMFT
predicts solutions for the two structures which are in good agreement with the
available photo-emission data,Sutter  _et al._ (2017); Miyashita  _et al._
(2021) and no metal-insulator transition near the experimental temperature in
the absence of a structural transition. In contrast, an EDMFT treatment based
on the downfolded interactions would predict insulating solutions with a far
too large gap for both structures. This demonstrates the important role of
nonlocal screening processes in producing physically correct interaction
strengths within self-consistent $GW$+EDMFT.

The paper is organized as follows: In Sec. II we summarize the $GW$+EDMFT
method and explain its extension to materials with multiple correlated sites
within the unit cell. In Sec. III we present the results obtained for the two
studied crystal structures and provide comparisons with available
photoemission data. In Sec. IV we present our conclusions.

##  II METHOD

###  II.1 DFT and single-shot $G_{0}W_{0}$

The DFT calculations were done using the full-potential linearized augmented
plane-wave code FLEUR The FLEUR group  and a 16$\times$16$\times$16
${\mathbf{k}}$-grid. The space group of Ca2RuO4 is Pbca, and the lattice
parameters as well as the positions of the atoms are taken to be the
experimental ones from Ref. Friedt  _et al._ , 2001. The lattice parameters
are given in Table 1 for the long ($L$) and short ($S$) cell, with the latter
structure shown in Fig. 1 together with the corresponding first Brillouin
zone. The crystal structure consists of Ru atoms inside tilted and rotated O
octahedra, surrounded by a cage of Ca atoms. The long cell is characterized by
a longer $c$-axis and a slightly increased rotation angle, while the tilt
angle is reduced compared to the short cell.Friedt  _et al._ (2001) The
Wyckoff positions are for Ru: (4a)-(0,0,0), and for Ca, O(1), O(2):
(8c)-(x,y,z) with the positional parameters given in Table 1. Here, O(1) and
O(2) refers to the two inequivalent oxygen positions.

| x | y | z
---|---|---|---
S-Pbca (a=5.3945, b=5.5999, c=11.7653)
Ca | 0.0042 | 0.0559 | 0.3524
O(1) | 0.1961 | 0.3018 | 0.0264
O(2) | -􏰔0.0673 | -0.0218 | 0.1645
L-Pbca: (a=5.3606, b=5.3507, c=12.2637)
Ca | 0.0110 | 0.0269 | 0.3479
O(1) | 0.1939 | 0.3064 | 0.0147
O(2) | -0.0386 | -0.0067 | 0.1656
Table 1: Lattice parameters in Å and position parameters for the Wyckoff
positions for the long ($L$) and short ($S$) cell of Ca2RuO4. O(1) and O(2)
represent the two inequivalent oxygen positions.

We use the Wannier90 libraryMostofi  _et al._ (2008) to define a low-energy
model for the bands around the Fermi energy with three maximally localized
Wannier functionsMarzari and Vanderbilt (1997) centered on each of the four Ru
sites. For the constrained random phase approximation (cRPA)Aryasetiawan  _et
al._ (2004) and single-shot $G_{0}W_{0}$ calculations we use the SPEX
code,Friedrich  _et al._ (2010) where we employ a 7$\times$7$\times$5
${\mathbf{k}}$-grid together with 300 bands in the calculation of both the
polarization and self-energy.

Figure 1: (a) S-Pbca crystal structure drawn using the VESTA software.Momma
and Izumi (2011) The primitive unit cell consists of four formula units. The
Ru atoms (orange) are inside tilted octahedra of O (red) surrounded by a cage
of Ca (gray). (b) First Brillouin zone and high-symmetry lines.

###  II.2 Real space extension of $GW$+EDMFT

The ab-initio $GW$+EDMFT method for correlated materials has been developed
and described in Refs. Boehnke  _et al._ , 2016; Nilsson  _et al._ , 2017;
Petocchi  _et al._ , 2020a, b. In contrast to the widely-used DFT+DMFT
approach,Kotliar  _et al._ (2006) it is fully ab-initio and does not require
ad-hoc parameters such as local interactions and double-counting corrections.
The only choice which needs to be made is the definition of physically
motivated high-, intermediate-, and low-energy subspaces (or “tiers” Nilsson
_et al._ (2017)), in which correlations will be treated at the $G_{0}W_{0}$,
self-consistent $GW$, and EDMFT level, respectively. In the present case of
Ca2RuO4 we treat in the intermediate and correlated low-energy subspace the
$4d$ $t_{2g}$ orbitals of the Ru atoms, and treat the higher-energy electronic
degrees of freedom at the $G_{0}W_{0}$ level.

Spin-orbit coupling (SOC) is known to be relevant for Ruthenates,Mackenzie
_et al._ (1996) and the resulting hybridized $t_{2g}$ orbitals are usually
referred to as $\alpha$, $\beta$ and $\gamma$ multiplets. In the present
study, which is restricted to the high-temperature phases of Ca2RuO4, we
however neglect SOC, so that the orbitals retain their $t_{2g}$ character.

The $GW$+EDMFT simulation starts with a DFT calculation in the local density
approximation (LDA),Kohn and Sham (1965) as described in Sec. II.1. By means
of a single-shot $G_{0}W_{0}$ calculation, the effective “bare” interactions
$U_{\bf q}^{\text{cRPA}}$ and the effective “bare” lattice Greens functions
$G_{\bf k}^{G_{0}W_{0}}$ of the intermediate/low-energy space are computed.
With these, the self-consistent $GW$+EDMFT calculation is performed in the
intermediate/low-energy space, starting from a vanishing EDMFT polarization
and self-energy ($\Pi^{\mathrm{EDMFT}}=0$, $\Sigma^{\mathrm{EDMFT}}=0$). The
self-consistency loop fixes the EDMFT impurity problem with fermionic and
bosonic baths such that the local lattice Green’s function is identical to the
impurity Green’s function, $G_{\mathrm{loc}}=G_{\mathrm{imp}}$ (fermionic
self-consistency condition), and that the screened local lattice interaction
is identical to the screened impurity interaction,
$W_{\mathrm{loc}}=W_{\mathrm{imp}}$ (bosonic self-consistency condition).Sun
and Kotliar (2002) This is imposed for each site within the unit cell that is
treated at the EDMFT level, and hence mapped onto an impurity problem with
retarded effective interaction (here, the four Ru atoms). The local EDMFT
self-energies and polarizations replace the corresponding local $GW$
contributions, and thereby strong correlation effects can be incorporated into
the simulation without ill-defined double countings. The method is able to
treat different types of correlated atoms in a unit cell, and to couple them
at the $GW$ level, as demonstrated in Ref. Petocchi  _et al._ , 2020b.

In the case of Ca2RuO4, we have to consider correlated orbitals residing on
ions of the same species (Ru), but inequivalently arranged in the unit cell
(Fig. 1). In this situation, the local Hamiltonians of the different sites
$\mathcal{H}_{\mathrm{loc}}^{i}$ become equivalent provided that the local
problem is rotated to the diagonal or crystal-field basis via a site-dependent
rotation $\Theta^{i}$. In the crystal-field basis the local noninteracting
Hamiltonians are replaced by local energies $\varepsilon_{\mathrm{loc}}^{i}$,
which are identical for each correlated site, and hence only a single impurity
problem needs to be solved. Before the self-consistent calculation is started,
we determine the rotations $\Theta^{i}$

$\hat{\Theta}=\left(\begin{array}[]{c}\left[\Theta^{1}\right]\\\ \ldots\\\
\left[\Theta^{N_{s}}\right]\end{array}\right),$ (1)

which provide the mapping between the original and the crystal-field basis. In
the present case, $N_{s}=4$ is the number of Ru sites in the unit cell. If the
interaction is of the Kanamori form and restricted to the $t_{2g}$ or $e_{g}$
manifold, no transformation is needed in the case of a rotationally invariant
formulation ($J=J_{X}=J_{P}$, see Ref. Georges  _et al._ , 2013). This follows
from the fact that a $3\times 3$ real rotation is preserving the
SO$\left(3\right)$ invariance of the interaction.Georges  _et al._ (2013) For
the full cRPA tensor that is used as a bare interaction in $GW$+EDMFT, this is
generally not the case, especially when the orbitals are slightly splitted
already at the LDA level as in Ca2RuO4. In such systems, one needs to apply
the tensor analogue of the fermionic matrix rotation to the local effective
bosonic interactions related to each site. A brief summary of this procedure
is provided in the Appendix. As it is the case for the local energies and
hybridization functions, in the crystal field basis, the transformed
interaction tensors become equivalent among the sites, thus ensuring the
invariance of the impurity models.

In the following, we summarize the steps in the $GW$+EDMFT self-consistency
loop for locally equivalent sites:

  1. 1.

Compute the $\mathbf{k}$ dependent self-energy and polarization in the $GW$
approximation (polarization $\Pi_{\mathbf{q}}^{GG}$ and self-energy
$\Sigma_{\mathbf{k}}^{GW}$, see Ref. Nilsson  _et al._ , 2017 for details) and
replace the local components with the EDMFT results:

     * •

$\Pi_{\mathbf{q}}=\Pi_{\mathbf{q}}^{GG}-\Pi_{\mathbf{q}}^{GG}|_{\mathrm{loc}}+\Pi^{\mathrm{EDMFT}}$,

     * •

$\Sigma_{\mathbf{k}}=\Sigma_{\mathbf{k}}^{GW}-\Sigma_{\mathbf{k}}^{GW}|_{\mathrm{loc}}+\Sigma^{\mathrm{EDMFT}}$.

  2. 2.

Use the polarization and self-energy, as well as the bare propagators
$U_{\mathbf{q}}^{\text{cRPA}}$ and $G_{\mathbf{k}}^{G_{0}W_{0}}$, to extract
the local screened interaction and local lattice Green’s function:

     * •

$W_{\mathrm{loc}}=\sum_{\mathbf{q}}U_{\mathbf{q}}^{\text{cRPA}}\left[1-\Pi_{\mathbf{q}}U_{\mathbf{q}}^{\text{cRPA}}\right]^{-1}$,

     * •

$G_{\mathrm{loc}}=\sum_{\mathbf{k}}\left[\left(G_{\mathbf{k}}^{G_{0}W_{0}}\right)^{-1}-\Sigma_{\mathbf{k}}\right]^{-1}$.

  3. 3.

Impose the two self-consistency conditions (at some given site $i$):

     * •

$W_{\mathrm{loc}}^{i}=W_{\mathrm{imp}}^{i}$,

     * •

$G_{\mathrm{loc}}^{i}=G_{\mathrm{imp}}^{i}$.

  4. 4.

For that particular site, compute the bosonic and fermionic Weiss fields
$\mathcal{U}$ and $\mathcal{G}$ of the EDMFT impurity problem:

     * •

$\mathcal{U}^{i}=W_{\mathrm{loc}}^{i}\left[1+\Pi^{\mathrm{EDMFT},i}W_{\mathrm{loc}}^{i}\right]^{-1}$,

     * •

$\mathcal{G}^{i}=\left[\left(G_{\mathrm{loc}}^{i}\right)^{-1}-\Sigma^{\mathrm{EDMFT},i}\right]^{-1}$.

These two fields are then rotated to the crystal-field basis:

     * •

$\mathcal{U}^{i}\overset{\Theta^{i}}{\longrightarrow}\tilde{\mathcal{U}}^{i}$,

     * •

$\tilde{\mathcal{G}}^{i}=\left(\Theta^{i}\right)^{T}\mathcal{G}^{i}\Theta^{i}$.

  5. 5.

A continuous-time Monte Carlo impurity solverWerner  _et al._ (2006);
Hafermann  _et al._ (2013) for models with dynamically screened
interactionsWerner and Millis (2010) provides the density-density correlator
$\tilde{\chi}_{\text{imp}}$ and the impurity Green’s function
$\tilde{G}_{\mathrm{imp}}$ in the crystal field basis ($0\leq\tau\leq\beta$)

     * •

$\tilde{\chi}_{\mathrm{imp}}^{i}=\left\langle\tilde{\hat{n}}_{\alpha}\left(\tau\right)\tilde{\hat{n}}_{\beta}\left(0\right)\right\rangle$,

     * •

$\tilde{G}_{\mathrm{imp}}^{i}$,

which are used to solve two Dyson equations, and to extract the local EDMFT
polarization and self-energy:

     * •

$\tilde{\Pi}^{\mathrm{EDMFT,}i}=\tilde{\chi}_{\mathrm{imp}}^{i}\left[\tilde{\mathcal{U}}^{i}\tilde{\chi}_{\mathrm{imp}}^{i}-1\right]^{-1}$,

     * •

$\tilde{\Sigma}^{\mathrm{EDMFT,}i}=\left(\tilde{\mathcal{G}}^{i}\right)^{-1}-\left(\tilde{G}_{\mathrm{imp}}^{i}\right)^{-1}$.

  6. 6.

The updated polarization and self-energy in the crystal-field basis is then
transformed back to the orbital basis for each site using the corresponding
site-dependent rotation. This generates a set of $N_{s}$ different pairs:

     * •

$\tilde{\Pi}^{\mathrm{EDMFT,}i}\overset{\left(\Theta^{i}\right)^{T}}{\longrightarrow}\Pi^{\mathrm{EDMFT,}i}$,

     * •

$\Sigma^{\mathrm{EDMFT,}i}=\Theta^{i}\tilde{\Sigma}^{\mathrm{EDMFT,}i}\left(\Theta^{i}\right)^{T}$.

The site-diagonal $\Pi^{\mathrm{EDMFT}}$ and $\Sigma^{\mathrm{EDMFT}}$ are
then substituted back into step 1, and the loop is iterated until a converged
solution if obtained.

It is worth mentioning that, for LDA inputs with degenerate orbitals, if the
invariance conditions for $U$ and $J$ are fulfilled, the rotation of
$\mathcal{U}$ can be avoided. This is however not the case for $\Pi$, since
the charge susceptibility will reflect the occupations of the realigned
fermionic levels, thus preventing any assumption on SO$\left(3\right)$
invariance.

##  III Results

###  III.1 Interaction strengths and electronic properties of the two
structures

We now apply the above ab-initio $GW$+EDMFT scheme to investigate the
electronic structure of Ca2RuO4. This is an interesting test case for our
scheme, since Ca2RuO4 exhibits a metal-insulator transition that is not linked
to magnetic ordering. Here, we should note that spin correlations are not
expected to be accurately described by any mean-field-based theory, and in the
case of Ca2RuO4 they will be influenced by SOC. We therefore do not consider
magnetic ordering in the present study. In fact, the structural deformation
from the $L$-Pbca to the $S$-Pbca structure at 360 K, which shortens the
$\mathbf{c}$ axis, appears to be the main driving force of the transition from
a metallic high-temperature to a Mott insulating low-temperature state.Gorelov
_et al._ (2010) The experimental gap in the $S$-Pbca structure is $\sim 0.2$
eV.

To investigate the transition, we performed two separate calculations, one for
each structure, at temperatures where the corresponding electronic phases are
experimentally found. Specifically, we considered $T^{S\text{-Pbca}}=290$ K
and $T^{L\text{-Pbca}}=580$ K, using a Matsubara frequency cutoff at $150$ eV
for both the fermionic and bosonic fields. We considered all the four sites in
the lattice problem and the corresponding $GW$ contributions, but only a
single impurity problem for the calculation of the local self-energy and
polarization, using the method described in Sec. II.2. The site-dependent
rotations to the crystal-field basis yield a level alignment in agreement with
Ref. Gorelov  _et al._ , 2010, with a splitting between the lowest and the
middle levels decreasing from $0.3$ eV to $0.1$ eV as one switches from the
$S$ to the $L$ structure:

$\varepsilon_{\mathrm{loc}}^{S\text{-Pbca}}{=}\left(\begin{array}[]{c|c}\left|1\right\rangle&-0.386\\\
\left|2\right\rangle&-0.084\\\
\left|3\right\rangle&-0.072\end{array}\right),\;\varepsilon_{\mathrm{loc}}^{L\text{-Pbca}}{=}\left(\begin{array}[]{c|c}\left|1\right\rangle&-0.321\\\
\left|2\right\rangle&-0.216\\\ \left|3\right\rangle&-0.206\end{array}\right).$
(2)

Even though we perform the tensor analogue of this operation on $\mathcal{U}$
and $\Pi$ at all the Matsubara points, we report here only the effect of the
basis change on the local $U_{\mathrm{cRPA}}$ interaction at $\omega=0$:

$U_{\mathrm{cRPA}}^{S\text{-Pbca}}\left(0\right){=}\begin{pmatrix}d_{xz}&d_{yz}&d_{xy}\\\
\hline\cr 2.55&1.94&1.76\\\ \cline{1-1}\cr\lx@intercol\hfil
0.30\hfil\lx<EMAIL_ADDRESS>\cline{2-2}\cr
0.27&\lx@intercol\hfil
0.27\hfil\lx<EMAIL_ADDRESS>\hline\cr 2.34&1.74&1.75\\\ \cline{1-1}\cr\lx@intercol\hfil
0.27\hfil\lx<EMAIL_ADDRESS>\cline{2-2}\cr
0.27&\lx@intercol\hfil
0.29\hfil\lx@intercol\vrule\lx@intercol&2.54\end{pmatrix},$ (3)
$U_{\mathrm{cRPA}}^{L\text{-Pbca}}\left(0\right){=}\begin{pmatrix}d_{xz}&d_{yz}&d_{xy}\\\
\hline\cr 2.46&1.87&1.72\\\ \cline{1-1}\cr\lx@intercol\hfil
0.31\hfil\lx<EMAIL_ADDRESS>\cline{2-2}\cr
0.27&\lx@intercol\hfil
0.27\hfil\lx<EMAIL_ADDRESS>\hline\cr 2.30&1.72&1.73\\\ \cline{1-1}\cr\lx@intercol\hfil
0.28\hfil\lx<EMAIL_ADDRESS>\cline{2-2}\cr
0.28&\lx@intercol\hfil
0.30\hfil\lx@intercol\vrule\lx@intercol&2.46\end{pmatrix},$ (4)

where the upper triangular section corresponds to the density-density
interactions, while the lower one gives the Hund couplings between the
different orbitals. In the crystal-field basis the orbital with the lowest
local energy ($\left|1\right\rangle$) has the most screened value of the
interaction, while the other two have more similar values. In the $L$-Pbca
case the difference is less pronounced, because of less charge reshuffling
introduced by the rotation. These static values are in agreement with what is
usually employed in static DFT+DMFT simulations of Ca2RuO4.Gorelov  _et al._
(2010); Sutter  _et al._ (2017); Riccó  _et al._ (2018); Hao  _et al._ (2020)
In the present case, however, these interactions represent an upper bound for
the static interaction strength, since $GW$+EDMFT takes into account
additional non-local screening processes which will lead to a reduced
effective impurity interaction. On the other hand, $GW$+EDMFT treats the full
frequency-dependence of the interactions up to the considerably larger bare
values (see below).

Figure 2: Local spectral functions in the orbital basis. Panel (a) shows the
comparison between $GW$+EDMFT (red lines) and EDMFT (black lines) in the
$S$-Pbca crystal structure, while panel (b) shows the corresponding results
for the $L$-Pbca structure. Solid lines refer to the $d_{xy}$ orbital, while
dashed lines show the average of the $d_{xz}$ and $d_{yz}$ spectra.

A few tenths of iterations lead to a converged insulating solution for the
$S$-Pbca structure and a converged metallic one for the $L$-Pbca structure, in
agreement with experiment. Both are found to be stable against statistical
noise and both persist in a very large temperature window: calculations
between $T=250K$ and $T=1160K$ for both systems did not yield any metal-
insulator transition. This might seem in contrast with DFT+DMFT results
reporting a (bad) metal state for the $S$-Pbca structure at $T=580K$.Gorelov
_et al._ (2010) However, it should be noted that even if the static
interactions in $GW$+EDMFT are screened down to few eV, the correlation
strength is intrinsically higher than in a system with a corresponding
frequency-independent interaction.Werner and Casula (2016) In either case, the
absence of a temperature-dependent transition in a large temperature interval
around 360 K (for a given crystal structure) supports the scenario where the
structural transition triggers the electronic one. Similar conclusions were
drawn in a very recent study of the interplay between lattice and electronic
degrees of freedom in Ca2RuO4.Georgescu and Millis (2021)

In Fig. 2 we report the local $t_{2g}$ spectral functions in the two setups.
The charge configuration of the insulator found in the $S$-Pbca crystal
structure corresponds to a nearly fully occupied $d_{xy}$ orbital, with
$n_{xy}=0.97$, and two almost half-filled $d_{xz/yz}$ orbitals with
$n_{xz}=0.51$ and $n_{yz}=0.52$, respectively. All the bands are insulating
and the gap, measured as the distance between the two frequency points where
the spectral weight vanishes, is $\sim 0.2$ eV wide, in agreement with
experiment. The spectral weight associated with the lower Hubbard band (LHB),
of mainly $d_{xz/yz}$ character, is centered at $-1.8$ eV, while the upper
Hubbard band (UHB) is centered at $0.5$ eV. The almost filled $d_{xy}$ band
displays two prominent features separated by roughly $1.2$ eV.

The metallic state obtained for the $L$-Pbca structure has negligible orbital
polarization given by $n_{xz}=n_{yz}=0.67$ and $n_{xy}=0.66$, and all the
bands contribute to the spectral weight at the Fermi level. Comparing the
$GW$+EDMFT spectral functions with the LDA ones (see rightmost panel of Fig. 8
below) and considering the value of the effective local interaction (Eq. (6))
of $\sim 1.4$-$1.6$ eV, we deduce that the three peak structure appearing in
the $-0.5$ eV to $0.5$ eV energy window does not correspond to preformed
Hubbard bands. This is rather a consequence of a renormalization of the LDA
bands, which exhibit similar structures in the local spectral function. Also,
by comparison with the LDA spectra one can see that the weight at $\sim-1.2$
eV for the $d_{xz/yz}$ orbitals is due to interactions and, as we will discuss
later, may be interpreted as a LHB.

To assess the role of the non-local screening from the $GW$ contribution, we
also performed EDMFT calculations for both setups. This corresponds to setting
$\Pi_{\mathbf{q}}^{GG}=0$ and $\Sigma_{\mathbf{k}}^{GW}=0$ at each iteration
in the selfconsistency loop described in Sec. II.2. As can be seen from the
black lines in Fig. 2, these EDMFT results are inconsistent with experiment,
since they yield a wide-gap Mott insulating state with full orbital
polarization ($n_{xz}=n_{yz}=0.5$ and $n_{xy}=1$ configuration) for both
structures. This further illustrates the fact that in a calculation with
frequency-dependent interactions, the correlation strength is significantly
stronger than what one might expect by looking at the $\omega=0$ value. The
result also shows that for a meaningful self-consistent ab-initio description
of the compound, one needs to take into account the non-local screening
originating from the non-local interactions. In addition, the band widening
effect of the $GW$ self-energy, coming mainly from the Fock term,Ayral  _et
al._ (2017) plays an important role in setting the correct balance between the
bandwidth and correlation strength.

Information on the local states of the correlated sites can be obtained by
measuring the probability with which the Monte Carlo solver generates a given
level/spin configuration. The histogram of the occupation and spin statistics
is reported in Fig. 3 and clearly shows that the Ru atoms in the insulating
phase ($S$-Pbca structure) are in a high spin $3d^{4}$ configuration with
suppressed charge fluctuations to the $3d^{3}$ and $3d^{5}$ configurations.
This is the result of a strong Hund’s coupling in Ca2RuO4 which favors spin-1
configurations with four electrons in the $t_{2g}$ orbitals, as directly
confirmed in the right panel of the figure. Both the charge and the spin are
fluctuating more strongly in the metallic phase ($L$-Pbca structure), but
there is still a dominant spin-1 state and a dominant $3d^{4}$ configuration.
The high-temperature state of Ca2RuO4 may thus be characterized as a Hund
metal.Georges  _et al._ (2013)

Figure 3: Occupation and spin statistics for the Ru atoms. Left panel:
probability distributions for the atom to be in the indicated charge states.
Right panel: probability distributions for the absolute value of the
$z$-component of the spin.

###  III.2 Results for the $S$-Pbca structure

In this section, we will analyze in more detail the properties of the low-
temperature $S$-Pbca structure, and compare the $GW$+EDMFT results to
experimental data.

####  III.2.1 Self-energies and effective interactions

The self-consistently determined bosonic fields in $GW$+EDMFT allow us to
address the screening induced by long-range interactions and to explore how
dynamical screening affects the electronic structure. In Fig. 4(a) we
illustrate the effect of the non-local screening by comparing the imaginary
parts of the $GW$+EDMFT and EDMFT local self-energies $\Im\Sigma$ for the
insulating structure. The small imaginary self-energy associated with $d_{xy}$
reflects the fact that this orbital is in an almost band insulating state,
while the larger $\Im\Sigma$ for the $d_{xz/yz}$ orbitals is responsible for
the gapped spectra. The absence of non-local screening in EDMFT increases the
orbital polarization by almost completely filling the $d_{xy}$ orbital, so
that the corresponding self-energy is further reduced. The opposite effect
occurs in the $d_{xz/yz}$ orbitals which, being exactly half-filled, have a
larger $\Im\Sigma$ compared to the $GW$+EDMFT counterpart. A consistent
picture is obtained from the intensity of the local charge susceptibility
$-\Im\chi_{\mathrm{imp}}/\omega$ which is shown in Fig. 4(b) on the real axis.
$-\Im\chi_{\mathrm{imp}}/\omega$ has a peak at an energy comparable with the
splitting between the upper and lower bands. The overall magnitude is already
relatively small in $GW$+EDMFT and several orders of magnitude smaller in
EDMFT (not shown in the plot), where charge fluctuations are essentially
frozen.

Figure 4: Frequency-dependent local interactions and related quantities for
the $S$-Pbca crystal structure. The red (black) lines show the $GW$+EDMFT
(EDMFT) results. (a) Imaginary part of the local self-energy on the Matsubara
axis, with thick (dashed) lines showing the results for the $d_{xy}$
($d_{xz/yz}$) orbitals. (b) Strength of the local charge fluctuations in the
two channels on the real frequency axis. (c,d) Effective density-density
interaction $\mathcal{U}$ and Hund coupling $\mathcal{J}$ on the Matsubara
axis. (e) Frequency dependence of the orbitally-averaged density-density
elements of the interaction. The thin line shows the $U_{\mathrm{cRPA}}$ input
(“bare” interaction), the dashed line is the local effective interaction
$\mathcal{U}$ and the thick line is the fully screened interaction $W$. (f)
Local bosonic spectral functions showing the energies of single-particle and
collective charge excitations. In the inset the low-energy screening modes are
magnified.

One consequence of ignoring non-local charge fluctuations, reported in panel
(c), is that the EDMFT local effective interaction tensor $\mathcal{U}$ is
essentially identical to the bare $U_{\mathrm{cRPA}}$ one (compare to panel
(e)), while in $GW$+EDMFT, even in the insulating state, it is significantly
reduced for both the $d_{xz/yz}$ and $d_{xy}$ orbitals. The effective Hund
coupling $\mathcal{J}$, reported in Fig. 4(d), is almost unchanged from the
cRPA result in both methods, and it has little frequency dependence ($\sim
15\%$ difference between the bare and screened value), which shows that
screening has little effects on this quantity. The situation is different for
the effective impurity model interaction tensor, whose static value in the
orbital basis reads

$\mathcal{U}^{S\text{-Pbca}}\left(0\right){=}\begin{pmatrix}d_{xz}&d_{yz}&d_{xy}\\\
\hline\cr 2.08&1.49&1.29\\\ \cline{1-1}\cr\lx@intercol\hfil
0.29\hfil\lx<EMAIL_ADDRESS>\cline{2-2}\cr
0.27&\lx@intercol\hfil
0.27\hfil\lx<EMAIL_ADDRESS>(5)

By comparing Eq. (5) to Eq. (3) one can see that the screening affects the
three orbitals significantly.

In the last two panels of Fig. 4 we illustrate how the average of the diagonal
components of the $U_{\mathrm{cRPA}}$ tensor are modified by the screening on
the Matsubara and real frequency axis. Given the insulating state in the
$S$-Pbca structure one may not expect large effects on the effective
interaction strength, but $\mathcal{U}(\omega=0)$ is nevertheless reduced
relative to $U_{\text{cRPA}}(\omega=0)$ by nearly half an eV. The imaginary
part of the same quantities, plotted on the real-frequency axis, reveals the
characteristic energies of screening modes associated with (local and
nonlocal) charge excitations in the low-energy space, which are excluded from
the cRPA input. From Fig. 4(f) one sees that the bosonic spectral functions,
while following the cRPA behavior at high energies, exhibit two additional
modes at $0.2$ eV and $1.1$ eV in $-\Im W_{\mathrm{loc}}(\omega)$, and
similarly also for $\mathcal{U}$. We associate the peak at $0.2$ eV with
charge excitations across the gap, which has a similar magnitude (Fig. 2(a)).
The second peak is located at an energy comparable to the separation between
the main features with $d_{xy}$ and $d_{xz/yz}$ character in the occupied and
unoccupied parts of the spectra. A possible explanation for these features
involves local multiplet excitations with magnitude $\sim 3\mathcal{J}$, as
proposed in Ref. Sutter  _et al._ , 2017.

Figure 5: Momentum-resolved spectral function for Ca2RuO4 in the $S$-Pbca
crystal structure at inverse temperature $\beta=40$ eV-1 obtained by applying
the maximum entropy method to the lattice Green’s function computed along the
indicated path. The results are provided in the basis with well-defined
orbital character. Dark colors correspond to high intensities. The top panels
show the (average) result for the $d_{xz/yz}$ orbitals, while the bottom
panels show the results for the $d_{xy}$ orbital. The left panels plot the LDA
bandstructure, the middle panels the correlated electronic structure from
$GW$+EDMFT, and the right panels the local spectra (black for LDA and red for
$GW$+EDMFT).

####  III.2.2 Momentum-resolved spectra and comparison with photoemission
experiments

The momentum-resolved spectral function for the $S$-Pbca structure is shown in
the middle panel of Fig. 5 for a path along high symmetry lines in the first
Brillouin zone, as illustrated in Fig. 1(b). These results confirm the
insulating state of the system and the positions of the (weakly dispersive)
Hubbard bands deduced from the local spectral functions. They are also in good
agreement with the previous results obtained with the DFT+DMFT method,Gorelov
_et al._ (2010); Sutter  _et al._ (2017); Riccó  _et al._ (2018) although one
needs to keep in mind that the latter approach involves several adjustable
parameters. The comparison with the momentum-resolved LDA spectral functions
in the orbital basis (left panels) reveals how the dispersive $d_{xz/yz}$
bands are split into strongly broadened Hubbard bands, while the $d_{xy}$ band
is shifted to lower energies and also substantially broadened. In the latter
case, however, the broadening may be to a large extent due to the limitations
of maximum entropy analytical continuation,Jarrell and Gubernatis (1996) since
(as shown in Fig. 4(a)) the self-energy effects in the almost completely
filled $d_{xy}$ orbital are weak.

ARPES experiments on the $S$-Pbca structureSutter  _et al._ (2017) report the
presence of two main peaks in the occupied part of the spectrum. These can be
identified with the LHB and associated with the $d_{xz/yz}$ orbitals, but may
also contain intensity coming from the $d_{xy}$ band. Their position at the
$S$ point is measured at $-1.7$ eV and $-0.8$ eV respectively. In Fig. 6(a) we
show the comparison between the photoemission spectrum (PES) computed from the
spectral function at the $S$ point by multiplying with the Fermi function for
temperature 290 K, and the experimental data extracted from Fig. 2(c) of Ref.
Sutter  _et al._ , 2017. There is a good agreement between the theoretical and
experimental results concerning both the position of the peaks and their
orbital character, while there is some mismatch in the relative weight of the
peaks. This may have several origins: i) matrix element effects, ii) the
difficulties of maximum entropy analytical continuation in resolving high-
energy spectral features, and ii) the omission of oxygen $2p$ bands, located
below $-3$ eV, whose tails could have an effect on the signal up to $-1.7$ eV.

As a further test of our $GW$+EDMFT scheme, we check the ab-initio results
against recent angle-integrated photoemission experiments,Miyashita  _et al._
(2021) which analyzed how the position of the peaks in the spectra evolve with
hole doping. Experimentally, it has been found that, by increasing doping, a
spectral weight redistribution occurs at low energy and a metallic state
emerges on the sample surface, whereas the bulk remains insulating.Miyashita
_et al._ (2021) In the simulations, we considered a homogeneous system with a
doping of $\delta=0.05$, corresponding to a total density of 15.8 electrons
within the four-site unit cell. In principle this would require us to perform
a separate $G_{0}W_{0}$ and cRPA calculation for the low-energy model of the
non-stoichiometric system, but, because of the small hole concentration, we
used the previous undoped input for simplicity.

In Fig. 6(b) we report the comparison between our bulk calculations and the
experimental PES taken from Fig. 2(c) of Ref. Miyashita  _et al._ , 2021. The
results for the stoichiometric compound are in remarkably good agreement with
the experimentally measured local PES concerning both the positions and
relative weights of the peaks. On the other hand, the doped setup only
reproduces the measured spectral weight transfer at low energy, while the
shift of the higher energy peak (thick blue line), which is due to the
insulating $d_{xz/yz}$ orbitals, is less prominent in the experimental data.
Whether or not this is related to the insulating bulk in the experimentally
investigated compound is an interesting open question.

Figure 6: Comparison between theoretical bulk calculations and experimental
spectra in the $S$-Pbca crystal structure. (a) Results at the $S$ point. Black
thick (dashed) lines show the $d_{xy}$ ($d_{xz/yz}$) spectral functions, the
red line is the theoretical PES profile, while the red dots are experimental
data extracted from Fig. 2(c) in Ref. Sutter  _et al._ , 2017. (b) Local
spectral functions. Black thick (dashed) lines indicate the $d_{xy}$
($d_{xz/yz}$) local spectral function, the red and blue lines are the
theoretical PES profile for the undoped and doped system respectively, while
the red and blue dots are extracted from the PES reported in Fig. 2(c) of Ref.
Miyashita  _et al._ , 2021 (same color code). The theoretical PES profile is
obtained by multiplying the spectral function by the Fermi distribution for
$\beta=40$ eV-1.

###  III.3 Results for the $L$-Pbca structure

In this section, we will analyze in more detail the properties of the high-
temperature $L$-Pbca structure.

####  III.3.1 Self-energies and effective interaction

Figure 7: Frequency-dependent local interactions and related quantities in the
$L$-Pbca crystal structure. The red (black) lines show the results from
$GW$+EDMFT (EDMFT). (a) Imaginary part of the local self-energy on the
Matsubara axis. (b) Strength of the local charge fluctuations in the two
channels on the real frequency axis. (c,d) Density-density interaction
$\mathcal{U}$ and Hund coupling $\mathcal{J}$ on the Matsubara axis. (e)
Frequency dependence of the orbitally-averaged density-density elements of the
interaction. (f) Local bosonic spectral functions showing the energies of
single-particle and collective charge excitations. The analytical continuation
of $\mathcal{U}$ is absent due to the non-causality which prevents maximum
entropy analytical continuation.

Figure 8: Momentum-resolved spectral functions for Ca2RuO4 in the $L$-Pbca
crystal structure at $\beta=20$ eV-1 obtained by applying the maximum entropy
method to the lattice Green’s function computed along the indicated path in
the basis where the orbital character is well defined. Dark colors correspond
to high intensities. The top panels show the (average) result for the
$d_{xz/yz}$ orbitals, while the bottom panels show the results for the
$d_{xy}$ orbital. The left panels show the LDA bandstructure, the middle
panels the correlated electronic structure from $GW$+EDMFT, and the right
panels the local spectra (black for LDA and red for $GW$+EDMFT).

In the $L$-Pbca structure, $GW$+EDMFT yields the expected metallic solution,
while EDMFT produces a wide-gap insulator. Therefore, the comparison between
the local self-energies obtained with the two methods, reported in Fig. 7(a),
shows pronounced differences: in EDMFT the $d_{xy}$ orbital is completely
filled and $\Im\Sigma$ is very small, while the self-energy is much larger for
the half-filled $d_{xz/yz}$ orbitals. The opposite result is obtained in the
metal, where all orbitals contribute to the spectral weight at the Fermi
level. Here, all the self-energies of the $t_{2g}$ manifold are quite similar.
The local charge fluctuations, illustrated in Fig. 7(b), are several orders of
magnitude larger in the metallic state than in the EDMFT case for the same
structure, or for the $GW$+EDMFT solution for the $S$-Pbca setup.
$-\Im\chi_{\mathrm{imp}}/\omega$ now exhibits a large peak at zero energy,
with a relative strength for the different orbitals in agreement with the
relative $\omega=0$ contribution of the orbitals to the local spectral
function shown in Fig. 2(b). We also notice a second peak that roughly matches
the energy separation between the highest occupied state and the lowest
available one.

In panels (c) and (d) of Fig. 7 we report the components of the local
effective interaction as a function of Matsubara frequency: the EDMFT results,
as in the previous structure, are very close to the corresponding cRPA
results, while significant screening effects are found in both the Hund
coupling and density-density interactions in the metallic case. The magnitude
of the total screening in the metal solution is $0.8$-$0.9$ eV, as can be
inferred by comparing the static value of the effective interaction tensor in
the orbital basis,

$\mathcal{U}^{L\text{-Pbca}}\left(0\right){=}\begin{pmatrix}d_{xz}&d_{yz}&d_{xy}\\\
\hline\cr 1.63&1.04&0.89\\\ \cline{1-1}\cr\lx@intercol\hfil
0.29\hfil\lx<EMAIL_ADDRESS>\cline{2-2}\cr
0.26&\lx@intercol\hfil
0.27\hfil\lx@intercol\vrule\lx@intercol&1.43\end{pmatrix},$ (6)

with the corresponding cRPA matrix of Eq. (4).

The frequency dependence of $\mathcal{U}\left(i\Omega_{n}\right)$ reveals the
presence of bosonic modes with non-causal (negative) spectral weight. This is
a known featureNilsson  _et al._ (2017) which may occur in the auxiliary
impurity problem as a result of the local approximation. Modified self-
consistency equations designed to remove this kind of non-causality have
recently been proposed,Backes  _et al._ (2020) but still need to be
systematically investigated in model and materials contexts. Here, we use the
original $GW$+EDMFT self-consistency scheme,Biermann  _et al._ (2003); Nilsson
_et al._ (2017) since the non-causalities in the dynamical mean fields of the
auxiliary impurity problem do not produce any anomalies in physical
quantities. In particular, the fully screened local interaction of the
impurity and lattice system, $W_{\mathrm{loc}}(i\Omega_{n})$, is causal, as
illustrated in Fig. 7(e).

The imaginary part of the analytical continuation of $W_{\text{loc}}$ to the
real axis, $-\Im W_{\mathrm{loc}}\left(\omega\right)$, is reported in Fig.
7(f) and reveals the presence of low-energy charge excitations at $\omega\sim
0.9$ eV. In this plot we did not include $-\Im\mathcal{U}\left(\omega\right)$
because its non-causality prevents us from employing the maximum entropy
method, but looking at the Matsubara axis data, we expect it to be reasonably
close to the spectrum shown in Fig. 4(f).

####  III.3.2 Momentum-resolved spectra and Fermi surface

In Fig. 8 we compare the momentum-resolved LDA spectral functions (left
panels) to the interacting spectral functions for the $L$-Pbca structure from
$GW$+EDMFT (middle panels). All the orbitals remain metallic, but, as
mentioned in Sec. III.1, we find a significant correlation-induced spectral
weight redistribution and the appearance of satellite structures near $-1.2$
eV and $-3.5$ eV.

Near the Fermi level, the $d_{xz/yz}$ bands are significantly renormalized,
especially in the occupied part of the spectrum, while the $d_{xy}$ bands are
mainly broadened, but not much renormalized. A striking feature is the almost
nondispersive band at $-3.5$ eV, which is completely absent in LDA, and beyond
the energy range of the local effective interaction $\mathcal{U}$ in Eq. (6).
This high energy peak most likely originates from physics involving the
higher-energy degrees of freedom, encoded in the frequency dependence of the
interaction parameters. The fact that screening modes between $3$ eV and $4$
eV are present in $-\Im W_{\mathrm{loc}}\left(\omega\right)$ (and likely also
in $\mathcal{U}$), as shown in Fig. 7(f), suggests that this high-energy
spectral feature is a plasmon satellite.

We next discuss the almost dispersionless band at $-1.2$ eV in the $d_{xz/yz}$
spectrum, and an analogous weaker feature in the $d_{xy}$ spectrum. While
there is no spectral weight near $-1.2$ eV in the $d_{xz/yz}$ LDA spectral
function, the $d_{xy}$ orbital contributes some spectral weight, which however
originates from the strong dispersion in the vicinity of the $\Gamma$ point.
The almost dispersionless character of this feature, and its shift relative to
the unoccupied part of the spectrum, which matches the magnitude of the
interactions in Eq. (6), suggests an interpretation as the LHB of the $t_{2g}$
shell. (The upper Hubbard band cannot be easily identified due to strong
damping induced by the large $\Im\Sigma$ in the corresponding energy region.)

Figure 9: (a-d) Orbital resolved DFT Fermi surfaces (spectral weight at
$\omega=0$) of metallic Ca2RuO4 in the $L$-Pbca structure. (e-h) Quasiparticle
Fermi surfaces obtained by correcting the single-particle Hamiltonian with the
static self-energy as described in the text. (i-l) Fully interacting Fermi
surfaces. (m-p) Modulation of the quasi-particle weight $Z_{\mathbf{k}}$ in
the $\left\\{k_{x},k_{y}\right\\}$ plane. Momentum-dependent variations of
$Z_{\mathbf{k}}$ are within $6\%$ of the local value reported in the main
text. In all panels dark colors correspond to high intensities.

The orbital resolved Fermi surfaces (FS) are plotted in Fig. 9 for different
methods and using different procedures: in panels (a-d) we report the FS
resulting from the DFT bandstructure, in the following (e-h) panels we
computed the quasiparticle FS by correcting the single-particle Hamiltonian
with the static component of the self-energy, with the scattering rates
(diagonal imaginary parts) removed. This procedure provides an indication on
the band renormalizations induced by local and non-local interactions, while
suppressing the broadening due to scattering processes. In panels (i-l) the
$\omega=0$ value of the momentum-resolved spectral function is reported. Our
results for the FS are in agreement with the available experimental data,
Riccó  _et al._ (2018) up to small details which may be explainable by the
fact that the experiments were performed on a strained and slightly doped
sample.

In panels (m-p) of the same figure we furthermore provide an estimate of the
non-local correlation effects, i.e. of the $\mathbf{k}$-dependent self-energy,
by plotting the quasiparticle weight

$Z_{\mathbf{k}}=\left(1-\left.\frac{\partial\Im\Sigma\left(\mathbf{k},i\omega_{n}\right)}{\partial\omega_{n}}\right|_{0}\right)^{-1}.$
(7)

$Z_{\mathbf{k}}$ cannot be obtained from standard DFT+DMFT calculations, which
consider only local correlation effects. It may be used to quantify the actual
correlations, which depend on the interaction parameters and on the density,
since the closer the band is to an empty/filled configuration the higher
$Z_{\mathbf{k}}$. From the local quasiparticle weights of the two manifolds we
find that the metallic state is indeed strongly correlated, since
$Z_{\mathrm{imp}}^{xz,yz}\sim 0.328$ and $Z_{\mathrm{imp}}^{xy}\sim 0.271$,
while the momentum-dependent variations are about $6\%$ of these values.
Figure 9(m-p) shows these variations as an intensity plot. One sees that the
$Z_{\mathbf{k}}$ associated to different orbitals are not simply the negative
of the corresponding Fermi surfaces, as one would naively expect by assuming a
high $Z_{\mathbf{k}}$ wherever the spectral weight at the Fermi energy is low
(e. g. close to the $\Gamma$ point). The evolution of $Z_{\mathbf{k}}$ is
indeed highly non-trivial: for instance the $d_{xy}$ behavior in the $Y$-$X$
direction is opposite to the one near the $\Gamma$ point.

##  IV Summary and Conclusions

In this work we have tested the $GW$+EDMFT method on the experimentally well-
characterized layered perovskite compound Ca2RuO4, which exhibits a
paramagnetic metal-insulator transition associated with a structural
transition. We showed that this recently developed ab-initio approach predicts
the physically correct solutions for the two crystal structures, and
electronic properties in remarkably good agreement with the available
experimental data. A similar level of agreement has been obtained in previous
LDA+DMFT studies,Gorelov  _et al._ (2010); Sutter  _et al._ (2017); Riccó  _et
al._ (2018); Hao  _et al._ (2020) but these calculations involve numerous
freely-adjustable parameters (Hubbard $U$, Hund coupling $J$, possible double-
counting corrections, …) which allow to fine-tune the simulation results to
the experimental reference data. In the present scheme, the user merely
selects the strongly correlated subspace (here, the $t_{2g}$ manifold of Ru),
while all the remaining calculations are performed in a fully ab-initio
manner. The level of accuracy demonstrated in this work is therefore a
remarkable outcome and a very encouraging result for the further application
and development of the $GW$+EDMFT framework.

On the technical side, we have explained how materials with distorted
structures and several correlated atoms within a unit cell can be efficiently
treated by transforming the EDMFT impurity problems to the crystal-field
basis. We also demonstrated how the frequency dependence of the interaction
parameters yields significantly stronger correlations than in the static
approximation, while nonlocal screening processes (introduced via the $GW$
polarization) substantially reduce the self-consistently computed interaction
parameters. The subtle balance between these two effects, plus the band-
widening effect of the $GW$ self-energy, are essential for reproducing the
correlation effects in the material. If one of these elements is missing, as
is the case for example in the EDMFT or DMFT treatment, physically incorrect
solutions (e. g. a large gap insulating solution for the $L$-Pbca structure)
are obtained.

The important role of Hund correlations has been revealed by the local state
statistics, in agreement with the current understanding of the Ca2RuO4
phenomenology. In the insulating phase that $GW$+EDMFT predicts for the
$S$-Pbca crystal structure, we obtained momentum resolved and local spectra in
good agreement with available photoemission data. In the $L$-Pbca setup, where
the solution is metallic, the self-consistently determined bosonic fields
provided estimates for the energies associated with plasmonic satellite
structures, while the analysis of the photo-emission spectra and interaction
tensors allowed us to identify a lower Hubbard band feature. Finally, the
Fermi surface of the metallic compound has been mapped out and agrees, up to
small details, with the available photo-emission data. To illustrate the
momentum-dependence of the quasi-particle renormalization introduced by the
$GW$ self-energy, a map of $Z_{\bf k}$ in the Brillouin zone has been
provided.

The results presented in this work demonstrate the reliability of $GW$+EDMFT
for the ab-initio description of complex transition metal compounds, which are
located in the computationally challenging regime of intermediate correlation
strength, where for example small changes in the crystal structure can trigger
a metal-insulator transition. The method provides important insights into the
interplay between correlations and screening, and allows to quantify the
effective interaction strength in solids, which is one of the main challenges
in the theoretical study of correlated materials.

###### Acknowledgements.

We thank Antoine Georges for proposing Ca2RuO4 as a benchmark material for
$GW$+EDMFT, and for insightful discussions. We also thank Fredrik Nilsson and
Ferdi Aryasetiawan for helpful discussions. F.P., V.C. and P.W. acknowledge
support from the Swiss National Science Foundation through NCCR MARVEL and
SNSF Grant No 200021_196966, and from the European Research Council through
ERC Consolidator Grant 724103. The calculations were performed on the
Beo04/Beo05 clusters at the University of Fribourg.

##  Appendix A Tensor Transformation

The most general two-particle scattering term is given by the rank-4 tensor

$\mathcal{H}=\frac{1}{2}\sum_{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}\sum_{abcd}U_{acdb}\hat{c}_{a\sigma_{1}}^{\dagger}\hat{c}_{c\sigma_{2}}^{\dagger}\hat{c}_{d\sigma_{3}}\hat{c}_{b\sigma_{4}},$
(8)

which, by imposing spin conservation $\sigma_{1}\neq\sigma_{2}$,
$\sigma_{3}\neq\sigma_{4}$ and relabeling the orbital indexes, becomes
equivalent to the effective local interaction used in $GW$+EDMFT,

$\mathcal{U}\left(\omega\right)=\sum_{abcd}\mathcal{\mathcal{U}}_{abcd}(\omega)\hat{c}_{a\uparrow}^{\dagger}\hat{c}_{b\uparrow}\hat{c}_{c\downarrow}^{\dagger}\hat{c}_{d\downarrow}.$
(9)

Considering the specific case of a three-orbital system, the product basis
representation of $\mathcal{U}$ at a given frequency can be decomposed in
terms of Kronecker products between $3\times 3$ density-matrix-like operators
$\hat{\rho}_{\left(ab\right)}$:

$\displaystyle\mathcal{\mathcal{U}}_{abcd}\left(\omega\right)\left[\hat{\rho}_{\left(ab\right)}\otimes\hat{\rho}_{\left(cd\right)}\right]=$
$\displaystyle\mathcal{\mathcal{U}}_{abcd}\left(\omega\right)\left(\begin{array}[]{ccc}\rho_{11}\left[\begin{array}[]{ccc}\rho_{11}&\rho_{12}&\rho_{13}\\\
\rho_{21}&\rho_{22}&\rho_{23}\\\
\rho_{31}&\rho_{32}&\rho_{33}\end{array}\right]&\rho_{12}\left[...\right]&\rho_{13}\left[...\right]\\\
\rho_{21}\left[\begin{array}[]{ccc}\rho_{11}&\rho_{12}&\rho_{13}\\\
\rho_{21}&\rho_{22}&\rho_{23}\\\
\rho_{31}&\rho_{32}&\rho_{33}\end{array}\right]&\rho_{22}\left[...\right]&\rho_{23}\left[...\right]\\\
\\\
\rho_{31}\left[...\right]&\rho_{32}\left[...\right]&\rho_{33}\left[...\right]\end{array}\right).$
(20)

In the original (orbital) basis the $\hat{\rho}_{\left(ab\right)}$ operators
are simply given by the sum of $3\times 3$ single-entry matrices
$\delta_{\left[ab\right]}$,

$\hat{\rho}_{\left(ab\right)}=\left[\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right]+\left[\begin{array}[]{ccc}0&1&0\\\ 0&0&0\\\
0&0&0\end{array}\right]+\ldots=\sum_{ab}\delta_{\left[ab\right]},$ (21)

and the nonzero matrix elements of Eq. (20) can be selected by grouping the
indices as $\mathcal{\mathcal{U}}_{(ab)(cd)}(\omega)$. In order to transform
the tensor to a different orbital basis defined by the $3\times 3$ rotation
$\Theta\in\mathrm{SO}\left(3\right)$, one separately rotates all the nine
contributions to $\hat{\rho}_{\left(ab\right)}$ so that

$\tilde{\rho}_{\left(ab\right)}=\sum_{ab}\Theta^{T}\delta_{\left[ab\right]}\Theta.$
(22)

Then the Kronecker product is performed using the rotated
$\tilde{\rho}_{\left(ab\right)}$. This procedure yields the matrix
representation of the interaction tensor in the product basis, which
corresponds to the single-particle basis selected by $\Theta$. We checked that
for the particular case of a rotationally invariant Kanamori interaction our
implementation leaves the representation unchanged for any
$\Theta\in\mathrm{SO}\left(3\right)$.

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