# Probing the Mott-insulating behavior of Ba2MgReO6 with DFT+DMFT

Maximilian E. Merkel  Materials Theory, ETH Zürich, Wolfgang-Pauli-Strasse 27,
8093 Zürich, Switzerland  Aria Mansouri Tehrani  Materials Theory, ETH Zürich,
Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland  Research Laboratory of
Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Claude Ederer<EMAIL_ADDRESS>Materials Theory, ETH Zürich,
Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland

(August 28, 2024)

###### Abstract

We investigate the interplay of spin-orbit coupling, electronic correlations,
and lattice distortions in the $5d^{1}$ double perovskite Ba2MgReO6. Combining
density-functional theory (DFT) and dynamical mean-field theory (DMFT), we
establish the Mott-insulating character of Ba2MgReO6 in both its cubic and
tetragonal paramagnetic phases. Despite substantial spin-orbit coupling, its
impact on the formation of the insulating state is minimal, consistent with
theoretical expectations for $d^{1}$ systems. We further characterize the
electronic properties of the cubic and tetragonal phases by analyzing spectral
functions and local occupations in terms of multipole moments centered on the
Re sites. Our results confirm the presence of ferroically ordered $z^{2}$
quadrupoles in addition to the antiferroic $x^{2}-y^{2}$-type order. We
compare two equivalent but complementary descriptions in terms of either
effective Re-${t_{2g}}$ frontier orbitals or more localized atomic-like Re-d
and O-p orbitals. The former maps directly on a physically intuitive picture
in terms of nominal $d^{1}$ Re cations, while the latter explicitly
demonstrates the role of hybridization with the ligands in the spin-orbit
splitting and the formation of the charge quadrupoles around the Re sites.
Finally, we compare our DFT+DMFT results with a previous DFT+$U$ study of the
tetragonal paramagnetic state. We find good qualitative agreement for the
dominant charge quadrupoles, but also notable differences in the corresponding
spectral functions, underscoring the need for more comparative studies between
these two methods.

##  I Introduction

In materials containing heavy elements with open $4d$, $5d$, $4f$, or $5f$
shells, the interplay of spin-orbit coupling (SOC) and electronic correlations
can lead to the emergence of various exotic phases, such as quantum spin
liquids, Weyl semimetals, topological insulators, superconductors, or states
with magnetic octupolar order and quenched magnetic moments [1, 2, 3, 4, 5].
SOC also affects the appearance of Mott-insulating states and orbital order
[3, 6, 7]. For example, for Sr2IrO4, it has been shown that SOC is crucial for
the formation of the insulating state by reducing the orbital degeneracy [8,
9, 10, 11]. However, the effect of SOC strongly depends on the nominal filling
of the electronic shell, and in general, the Mott-insulating state can be both
stabilized or destabilized [12, 13, 14]. Similarly, in cases where an
electronic degeneracy can be lifted by a lattice distortion, SOC can either
compete or cooperate with this distortion and the corresponding orbital order
[6, 7, 15, 16].

An attractive class of materials to systematically study the interplay between
SOC and electronic correlations are the double perovskites,
$A_{2}BB^{\prime}X_{6}$ [17], with closed-shell ions on the $A$ and $B$ sites
and a heavy $4d$ or $5d$ transition-metal cation on the $B^{\prime}$ site. The
$B$ and $B^{\prime}$ sites are arranged in a 3D-checkerboard pattern (see Fig.
1). Thus the relatively large spacing between the $B^{\prime}X_{6}$ octahedra
leads to fairly narrow transition-metal $d$ bands, such that a Mott-insulating
state can be formed even for a moderately strong local electron-electron
interaction [3, 4, 18, 19].

Specifically, for a $d^{1}$ configuration of the $B^{\prime}$ cation, SOC is
expected to weakly stabilize the Mott-insulating state [12, 13] and to reduce
the tendency towards Jahn-Teller distortions, but without completely
suppressing it [20, 15]. Examples of the corresponding family of $5d^{1}$
double perovskites that have been studied recently are rhenates [21, 22, 23,
24, 25], Ba2NaOsO6 [26, 27, 28, 29, 25, 30, 31], or the more ionic Cs2TaCl6
[32, 33].

Figure 1: (a) Experimental, cubic $Fm\bar{3}m$ crystal structure of Ba2MgReO6
[34]. In the tetragonal $P4_{2}/mnm$ phase, the ReO6 octahedra deform with (b)
rhombic $Q_{2}$ and (c) tetragonal $Q_{3}$ Jahn-Teller distortions that couple
to charge quadrupoles, indicated by green regions with charge excess and red
regions with charge depletion.

Here, we focus on the $5d^{1}$ double perovskite Ba2MgReO6 [35, 21, 36, 37,
34, 38, 39, 40], which exhibits two symmetry-lowering phase transitions. Upon
cooling, the material first transitions from cubic $Fm\bar{3}m$ symmetry to
tetragonal $P4_{2}/mnm$ symmetry at $33\text{\,}\mathrm{K}$. This transition
involves a $Q_{2}$-type Jahn-Teller distortion of the ReO6 octahedra in the
$xy$ plane [see Fig. 1(b)], with the sign of the distortion alternating
between neighboring $xy$ planes along the $z$ direction, and an additional
$Q_{3}$-type distortion, elongating all ReO6 octahedra along the $z$ direction
[see Fig. 1(c)] [41, 34]. At the Néel temperature of $18\text{\,}\mathrm{K}$,
Ba2MgReO6 then develops a canted antiferromagnetic order [37]. Furthermore,
Ba2MgReO6 is insulating at all measured temperatures, and an Arrhenius fit to
electrical resistivity measurements indicates an activation energy of
$0.17\text{\,}\mathrm{e}\mathrm{V}$ [37].

In this paper, we use density-functional theory (DFT) plus dynamical mean-
field theory (DMFT) to establish the Mott-insulating character of both the
cubic and tetragonal paramagnetic phases of Ba2MgReO6. Our calculations
suggest that Ba2MgReO6 can be classified as a regular Mott insulator, where
the insulating character is due to a small band width compared to the Coulomb
repulsion ${\mathcal{U}}$, which we calculate using the constrained random-
phase approximation (cRPA), whereas the SOC only plays a minor role.

We then discuss the spectral properties and charge distribution of
paramagnetic Ba2MgReO6 in terms of local occupations and multipole moments
[42] centered on the Re sites. Thereby, we compare two different sets of
Wannier functions used to represent the Re $d$ states. First, a basis of so-
called frontier orbitals, constructed only from the low-energy bands
immediately around the Fermi level, and second, a more atomic-orbital-like
basis of Re-$d$ and O-$p$ orbitals obtained for a larger energy window. While
the frontier basis is used in our DFT+DMFT calculations and directly maps on a
nominal $d^{1}$ description of the SOC on the Re cations, similar to the
model-based studies in Refs. 12, 13, 43, the atomic-like basis reveals that
both Re and O states experience the SOC and contribute to the orbital
polarization in the tetragonal phase.

The atomic-like basis also allows for a direct comparison to DFT+$U$
calculations for Ba2MgReO6 from Ref. 38, where the paramagnetic structure is
emulated using a large supercell with random magnetic moments on the Re atoms
and zero overall magnetic moment [44, 45]. We find reasonably good agreement
between the two methods for the local quadrupole moments on the Re sites,
whereas the density of states obtained from the DFT+$U$ calculations differs
notably from the spectral function obtained within DFT+DMFT. Even though the
different representations of the local Coulomb interaction in the two
approaches can potentially explain part of this difference, a more systematic
comparison would be of interest in future studies.

##  II Methods

###  II.1 Spin-orbit coupling and local electron-electron interaction

We perform DFT+DMFT calculations for Ba2MgReO6, using a low-energy description
of the material with one electron in the Re-$d$ shell. The crystal field in
the ReO6 octahedra is much stronger than the SOC so that the higher-lying
${e_{g}}$ orbitals are nominally empty and the electron resides in the
${t_{2g}}$ orbitals further split by the SOC.

We first perform calculations with SOC included in the DFT exchange-
correlation potential and obtain the corresponding Kohn-Sham band structure.
To then systematically vary the strength of the SOC in our DMFT calculations,
we compare these Kohn-Sham bands to the ones obtained when excluding SOC from
the exchange-correlation potential and instead adding the local Hamiltonian
[46]

$\displaystyle H_{\mathrm{SOC}}$
$\displaystyle=\lambda\sum_{mm^{\prime}}\sum_{\sigma\sigma^{\prime}}\langle
m\sigma|\bm{l}\cdot\bm{s}|m^{\prime}\sigma^{\prime}\rangle
c_{m\sigma}^{\dagger}c_{m^{\prime}\sigma^{\prime}}$
$\displaystyle=\frac{\mathrm{i}\lambda}{2}\left(c^{\dagger}_{xz\uparrow}c_{xy\downarrow}+c^{\dagger}_{xz\downarrow}c_{xy\uparrow}+\mathrm{i}c^{\dagger}_{yz\uparrow}c_{xy\downarrow}-\mathrm{i}c^{\dagger}_{yz\downarrow}c_{xy\uparrow}\right.$
$\displaystyle\hphantom{=\frac{\mathrm{i}\lambda}{2}(}\left.+c^{\dagger}_{yz\uparrow}c_{xz\uparrow}-c^{\dagger}_{yz\downarrow}c_{xz\downarrow}\right)+\text{h.c.}$
(1)

to the ${t_{2g}}$ Wannier orbitals in a post-processing step, where $\lambda$
parametrizes the SOC strength.

In the presence of SOC, it is also instructive to represent our results in the
eigenbasis $|{J_{\mathrm{eff}}},m_{J_{\mathrm{eff}}}\rangle$ of the
Hamiltonian in Eq. (1), where ${J_{\mathrm{eff}}}$ is the effective total
angular momentum, defined as the sum of the effective orbital momentum $l=1$
of the ${t_{2g}}$ orbitals and the spin, taking values of
${J_{\mathrm{eff}}}=1/2$ and ${J_{\mathrm{eff}}}=3/2$ [47]. These states are
related to the ${t_{2g}}$ orbitals by

$\displaystyle\begin{pmatrix}|1/2,-1/2\rangle\\\ |3/2,-1/2\rangle\\\
|3/2,\hphantom{-}3/2\rangle\end{pmatrix}=\begin{pmatrix}-\mathrm{i}/\sqrt{3}&\hphantom{-}1/\sqrt{3}&-1/\sqrt{3}\\\
-\mathrm{i}/\sqrt{6}&\hphantom{-}1/\sqrt{6}&\hphantom{-}2/\sqrt{6}\\\
-\mathrm{i}/\sqrt{2}&-1/\sqrt{2}&\hphantom{-}0\end{pmatrix}\begin{pmatrix}|xz\uparrow\rangle\\\
|yz\uparrow\rangle\\\ |xy\downarrow\rangle\end{pmatrix}$ (2)

and equivalently for the corresponding time-reversal-symmetric states [18]).
The eigenvalues with respect to $H_{\text{SOC}}$ are $+\lambda$ for all
${J_{\mathrm{eff}}}=1/2$ and $-\lambda/2$ for all ${J_{\mathrm{eff}}}=3/2$
states.

To describe the local electron-electron interactions in the DMFT calculations,
we use the Kanamori Hamiltonian [48]

$\displaystyle H_{\mathrm{int}}=$
$\displaystyle\frac{1}{2}\sum_{m,\sigma}{\mathcal{U}}n_{m\sigma}n_{m\bar{\sigma}}+\frac{1}{2}\sum_{m\neq
m^{\prime},\sigma}{\mathcal{U}}^{\prime}n_{m\sigma}n_{m^{\prime}\bar{\sigma}}$
$\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{m\neq
m^{\prime},\sigma}({\mathcal{U}}^{\prime}-\mathcal{J})n_{m\sigma}n_{m^{\prime}\sigma}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{m\not=m^{\prime},\sigma}\mathcal{J}(c^{\dagger}_{m\sigma}c^{\dagger}_{m^{\prime}\bar{\sigma}}c_{m\bar{\sigma}}c_{m^{\prime}\sigma}+c^{\dagger}_{m\sigma}c^{\dagger}_{m\bar{\sigma}}c_{m^{\prime}\bar{\sigma}}c_{m^{\prime}\sigma})$
(3)

which is exact for real-valued ${t_{2g}}$-type orbitals in cubic symmetry
[49]. The three parameters ${\mathcal{U}}$, ${\mathcal{U}}^{\prime}$, and
$\mathcal{J}$ describe the strength of the screened intra-orbital, inter-
orbital, and Hund’s interaction, respectively.

We calculate these interaction parameters using the cRPA [50, 51, 52], but we
also treat them as free parameters in our DFT+DMFT calculations to analyze the
corresponding effect on the electronic properties. Thereby, the complexity is
reduced by assuming approximate spherical symmetry so that there are only two
free parameters with ${\mathcal{U}}^{\prime}={\mathcal{U}}-2\mathcal{J}$ [49].
In real materials, spherical symmetry is never exactly fulfilled. Therefore,
we verify the quality of this assumption based on the interaction parameters
obtained from cRPA. We can then define averaged parameters

$\displaystyle{\mathcal{U}}_{\mathrm{avg}}$
$\displaystyle=(13{\mathcal{U}}+2{\mathcal{U}}^{\prime}+4\mathcal{J})/15$
$\displaystyle\mathcal{J}_{\mathrm{avg}}$
$\displaystyle=({\mathcal{U}}-{\mathcal{U}}^{\prime}+\mathcal{J})/3$ (4)

that minimize the square of the deviations
${\mathcal{U}}_{\mathrm{avg}}-{\mathcal{U}}$,
$({\mathcal{U}}_{\mathrm{avg}}-2\mathcal{J}_{\mathrm{avg}})-{\mathcal{U}}^{\prime}$
and
$({\mathcal{U}}_{\mathrm{avg}}-3\mathcal{J}_{\mathrm{avg}})-({\mathcal{U}}^{\prime}-\mathcal{J})$,
weighted by their frequency of occurrence in the density-density terms of Eq.
(3).

###  II.2 Computational details

We perform calculations for the experimental cubic $Fm\bar{3}m$ and tetragonal
$P4_{2}/mnm$ structures [34], where we always use a coordinate system with the
axes parallel to the Re-O bonds. The cubic structure has a lattice constant of
$8.08\text{\,}\mathrm{\text{\r{A}}}$ for the $Fm\bar{3}m$ conventional cubic
cell containing four formula units, with a Re-O bond length of
$1.93\text{\,}\mathrm{\text{\r{A}}}$. In the tetragonal phase, the unit cell
distorts very slightly, with $c/a=1.001$ and
$a=$8.08\text{\,}\mathrm{\text{\r{A}}}$$. Furthermore, the Re-O bond lengths
change by $-0.0250.0060.019\text{\,}\mathrm{\text{\r{A}}}$ in the $x$, $z$ and
$y$ direction respectively in one layer, which corresponds to
$|Q_{2}|=$0.044\text{\,}\mathrm{\text{\r{A}}}$$ and
$|Q_{3}|=$0.01\text{\,}\mathrm{\text{\r{A}}}$$ as specified in Ref. 34. All
ReO6 octahedra are symmetry-equivalent, with octahedra in two adjacent $xy$
layers mapped onto each other by a translation combined with a $None$ rotation
around the $z$ axis, thereby swapping the Re-O bond lengths along $x$ and $y$.

For the DFT calculations, we use VASP (version 6.4.1) [53, 54] with the PBE
exchange-correlation potential [55] including the Ba-$5s$, Ba-$5p$, Mg-$2p$
and Re-$5p$ semicore states as valence electrons. To ensure good convergence,
we use a $600\text{\,}\mathrm{e}\mathrm{V}$ plane-wave cutoff and an
electronic energy convergence of ${10}^{-8}\text{\,}\mathrm{e}\mathrm{V}$. To
calculate the electronic bands for the cubic $Fm\bar{3}m$ structure, we use
the primitive cell of the face-centered cubic lattice containing one formula
unit, and a $8\times 8\times 8$ reciprocal $k$-grid. For the DFT+DMFT
calculations, we treat both the cubic and the tetragonal $P4_{2}/mnm$ phase
using the primitive tetragonal unit cell with two formula units on a $7\times
7\times 5$ reciprocal grid.

When SOC is incorporated directly in the DFT exchange-correlation potential,
we preserve time-reversal symmetry by initializing the magnetic moments to
zero and verifying that the bands are always twofold degenerate up to a
precision of $1\text{\times}{10}^{-6}\text{\,}\mathrm{e}\mathrm{V}$ in the
band structure. Without SOC, we only use a single spin channel.

We then define Wannier functions using Wannier90 (version 3.1.0) [56] and
initial projections on atom-centered atomic orbitals with well-defined orbital
character. If applicable, we disentangle the bands with a relative convergence
of the gauge-invariant spread of ${10}^{-9}$ and always perform Wannierization
down to a convergence of the spread of
${10}^{-9}\text{\,}\mathrm{\text{\r{A}}}^{2}$. The only exception is the
calculation with SOC in DFT, where we use the initial projections without
subsequent Wannierization since this would lead to convergence away from the
orbital-spin basis to the ${J_{\mathrm{eff}}}$ basis instead.

To obtain estimates of the screened interaction parameters for the local
electron-electron interaction in this basis of Wannier functions, we perform
cRPA calculations at zero frequency using VASP without SOC. To include enough
empty bands in cRPA, we first obtain the accurate Kohn-Sham energies from
exact diagonalization for 512 bands using the converged charge density.

We perform one-shot DMFT calculations (neglecting charge self-consistency)
with solid_dmft [57] using the quantum Monte-Carlo solver CT-HYB [58] and the
interface to Wannier90 as implemented in DFTTools [59], all based on the TRIQS
library (version 3.1.x) [60]. The calculations are performed at an electronic
temperature of $290\text{\,}\mathrm{K}$ except those discussed in App. A.

We use a basis of $|xz\rangle$, $|-\mathrm{i}yz\rangle$ and
$|\mathrm{i}xy\rangle$, for which the local Hamiltonian, consisting of the SOC
Hamiltonian in Eq. (1) and possibly the local tetragonal splitting, is real.
We enforce a paramagnetic solution by averaging the impurity Green function
with its time-reversed counterpart after every solver run. For all structures,
cubic and tetragonal, we keep the two ReO6 octahedra in the unit cell
symmetry-equivalent by mapping one onto the other with the $None$ rotation
about the $z$ axis and the translation described above. Therefore, in the
following, we only show the occupations of the ReO6 octahedron with the long
bond along $x$ and report all observables per formula unit (f.u.) where
applicable. We obtain the spectral functions from the analytic continuation of
the impurity Green function with MaxEnt [61, 62].

##  III Results

###  III.1 DFT calculations

####  III.1.1 Band structure

Figure 2: DFT band structure obtained for cubic Ba2MgReO6 along a path of the
Brillouin zone of the face-centered cubic lattice. (a) DFT bands with SOC
included, and (b) bands around the Fermi energy for three different cases
where the SOC is incorporated directly at the DFT level (“in DFT”), not at all
(“none”), or is added as a local term in the form of Eq. (1) to the Wannier
Hamiltonian (“local”).

We first discuss the electronic band-structure obtained within DFT for
nonmagnetic, cubic Ba2MgReO6 with SOC included at the DFT level. Fig. 2(a)
shows the corresponding bands along a path in reciprocal space. The band
structure is metallic and features six twofold degenerate bands with a total
width of $1.4\text{\,}\mathrm{e}\mathrm{V}$ around the Fermi energy, which are
separated by large gaps of around $2\text{\,}\mathrm{e}\mathrm{V}$ from bands
at higher and lower energies. Projection of these bands onto local atomic
orbitals reveals their strong Re-${t_{2g}}$ character with an overall weight
of around $5060\text{\,}\%$. The second most dominant contribution comes from
O-$p$ orbitals, consistent with the expected strong hybridization between Re
and O.

We construct a representation of this isolated group of bands in terms of
_frontier_ Wannier orbitals, $|\mathrm{f}_{m}\rangle$, with ${t_{2g}}$
symmetry centered at the Re cations, where $m=yz$, $xz$, and $xy$. The local
part of the Hamiltonian in this Wannier basis has the form of Eq. (1) with a
strength of $\lambda=$0.30\text{\,}\mathrm{e}\mathrm{V}$$, very similar to the
value of $0.3\text{\,}\mathrm{e}\mathrm{V}$ reported for Os systems [18, 29].
In addition, the Wannier Hamiltonian exhibits all intersite terms that are
allowed by cubic symmetry, with the largest hopping of
$-0.11\text{\,}\mathrm{e}\mathrm{V}$ between ${t_{2g}}$ orbitals oriented
along the Re-Re axis of two neighboring ReO6 octahedra. The small hopping
amplitude, due to the large Re-Re spacing, is consistent with the small
bandwidth.

Since Ba2MgReO6 is experimentally known to be insulating in all its phases
(including the cubic one), we investigate the effect of the local electron-
electron interaction within DFT+DMFT on the electronic structure around the
Fermi energy. Given the small bandwidth of the Re-${t_{2g}}$ dominated bands,
this can potentially have a strong effect.

Within our DFT+DMFT calculations, which are presented in the following
sections, we use a slightly different approach to incorporate the SOC,
allowing us to treat the SOC strength as a free parameter. We first perform
the DFT calculation without SOC and again construct ${t_{2g}}$-like Wannier
orbitals centered on the Re sites for the bands around the Fermi level. This
results in a band structure with a higher degeneracy and a slightly smaller
bandwidth of $1.13\text{\,}\mathrm{e}\mathrm{V}$ [blue line in Fig. 2(b)]. We
then add the local SOC in the form of Eq. (1) to the Wannier Hamiltonian and
recalculate the corresponding bands. Indeed, by using
$\lambda=$0.30\text{\,}\mathrm{e}\mathrm{V}$$ as extracted above, we obtain
nearly matching bands from this approach compared to the bands obtained by
incorporating SOC directly in the DFT calculation [red and orange line in Fig.
2(b), respectively], with a maximum deviation of
$0.09\text{\,}\mathrm{e}\mathrm{V}$ at the $\Gamma$ point in the highest-
energy band. Thus, even though the SOC is incorporated in the DFT calculations
on an atomic level, i.e., using projections on atomic-like orbitals [63], its
effect on the bands around the Fermi level in Ba2MgReO6 can be well accounted
for by Eq. (1) applied to the Re-centered frontier orbitals only.

####  III.1.2 Calculation of screened interaction parameters

Before presenting the results of our DFT+DMFT calculations, we calculate the
strength of the local electron-electron interaction in the low-energy basis
with cRPA (without including SOC in the underlying DFT calculations). Thereby,
the bands are separated into a target subspace, containing the set of isolated
bands around the Fermi level, and a screening subspace, which consists of all
other bands. In the cubic phase, the interaction exhibits the exact Kanamori
form, so that we can simply read off the Kanamori parameters from the cRPA
results as ${\mathcal{U}}=$2.82\text{\,}\mathrm{e}\mathrm{V}$$,
${\mathcal{U}}^{\prime}=$2.24\text{\,}\mathrm{e}\mathrm{V}$$ and
$\mathcal{J}=$0.24\text{\,}\mathrm{e}\mathrm{V}$$. The deviation from
spherical symmetry can be quantified by comparing
$({\mathcal{U}}-{\mathcal{U}}^{\prime})/2$ with $\mathcal{J}$, which only
differ by $0.05\text{\,}\mathrm{e}\mathrm{V}$, comparable to other $5d$
materials [49, 30].

Therefore, in the following, we simplify the parametrization by assuming
spherical symmetry with ${\mathcal{U}}^{\prime}={\mathcal{U}}-2\mathcal{J}$
and averaging according to Eq. (4). This results in
${\mathcal{U}}=$2.81\text{\,}\mathrm{e}\mathrm{V}$$ and
$\mathcal{J}=$0.27\text{\,}\mathrm{e}\mathrm{V}$$. These values are very
similar to those reported for Ba2NaOsO6 obtained from cRPA calculations, with
${\mathcal{U}}=$2.9\text{\,}\mathrm{e}\mathrm{V}$$ and
$\mathcal{J}=$0.20\text{\,}\mathrm{e}\mathrm{V}$$ [30], and also from an
estimate based on a cluster model fitted to experimental data [26]. We note
that, due to the large separation between the target bands and the rest, the
known tendency of cRPA for overscreening is expected to be rather small [64].
We also do not expect SOC to strongly influence the interaction parameters,
since the correlated bands do not change much with or without SOC compared to
the large separation from the surrounding screening bands, as shown in Fig. 2.
This is in agreement with Refs. 65, 30.

In the following DFT+DMFT calculations, we always use the value of
$\mathcal{J}=$0.27\text{\,}\mathrm{e}\mathrm{V}$$. In Sec. III.2 and Sec.
III.3, the parameter ${\mathcal{U}}$ is treated as a free parameter to analyze
the effect of the local interaction strength on the electronic structure,
while in Sec. III.4 a more detailed analysis is presented for the cRPA value
of ${\mathcal{U}}=$2.81\text{\,}\mathrm{e}\mathrm{V}$$, which can be
considered as realistic interaction strength for Ba2MgReO6.

###  III.2 Metal-insulator transitions induced by electron-electron
interaction

We first explore the formation of the paramagnetic insulating state and its
dependency on the Coulomb repulsion ${\mathcal{U}}$.

Figure 3: DFT+DMFT results for Ba2MgReO6 as a function of the Coulomb
interaction parameter ${\mathcal{U}}$. (a) The spectral density obtained for
the cubic and tetragonal structures and occupations of the
${J_{\mathrm{eff}}}$ states on one Re site for the (b) cubic and (c)
tetragonal phases.

As shown by the spectral weight, $A(\omega=0)$, as a function of
${\mathcal{U}}$ in Fig. 3(a), there is a metal-insulator transition for the
cubic structure at a moderate
${\mathcal{U}}\approx$1.8\text{\,}\mathrm{e}\mathrm{V}$$. Since this is
significantly lower than the cRPA value of ${\mathcal{U}}=2.81$ eV, this shows
that the electron-electron interaction is strong enough to turn cubic
Ba2MgReO6 insulating, also considering that cRPA tends to underestimate the
screened interaction parameter, as discussed above. The observed critical
value is in good agreement with a simple atomic-limit estimation of the metal
to Mott insulator transition, where a critical
${\mathcal{U}}=$1.85\text{\,}\mathrm{e}\mathrm{V}$$ is obtained by equating
the atomic-charge gap [12] with the bandwidth of
$1.13\text{\,}\mathrm{e}\mathrm{V}$ obtained without SOC. This establishes the
Mott-insulating character of Ba2MgReO6 in the cubic phase.

We also obtain basically identical results from DFT+DMFT calculations where
the SOC is already incorporated in the DFT calculations [gray open circles
shown for some data points in Fig. 3(a)].

In all calculations for the cubic structure, the local density matrix in terms
of Wannier functions is diagonal in the ${J_{\mathrm{eff}}}$ basis defined in
Eq. (2). The evolution of these occupations as a function of ${\mathcal{U}}$
is shown in Fig. 3(b). Note that the total occupation is always 1,
corresponding to the nominal Re-$t_{2g}^{1}$ occupation described by the low-
energy frontier orbitals. All states corresponding to the
${J_{\mathrm{eff}}}=3/2$ quadruplet are always equally occupied, and the same
is true for the ${J_{\mathrm{eff}}}=1/2$ doublet, demonstrating that the
charge density, in this case, does not spontaneously break the cubic symmetry.
In the insulating state, the electron exclusively occupies the
${J_{\mathrm{eff}}}=3/2$ states, whereas in the metallic state, some
occupation is transferred into the ${J_{\mathrm{eff}}}=1/2$ doublet.

Next, we perform analogous DFT+DMFT calculations for the tetragonal structure
of Ba2MgReO6. Here, the main effect of the structural distortions on the
electronic structure is a local splitting of the previously degenerate
${t_{2g}}$ Wannier functions by $-0.050.010.04\text{\,}\mathrm{e}\mathrm{V}$,
while the intersite hopping terms are barely affected.

As shown in Fig. 3(a), the distortions in the tetragonal structure shift the
critical ${\mathcal{U}}$ for the metal-insulator transition to a slightly
smaller value of ${\mathcal{U}}=$1.6\text{\,}\mathrm{e}\mathrm{V}$$ compared
to the cubic case. The distortions also affect the orbital occupations, where
the occupation matrix is not diagonal in the ${J_{\mathrm{eff}}}$ basis
because ${J_{\mathrm{eff}}}$ and $m_{J_{\mathrm{eff}}}$ are not good quantum
numbers anymore.

In Fig. 3(c), we show the diagonal elements of the occupations in the
${J_{\mathrm{eff}}}$ basis for the Re site with the long-bond axis along $x$.
In the insulating state, there is now a polarization within the
${J_{\mathrm{eff}}}=3/2$ quadruplet, and the ${J_{\mathrm{eff}}}=1/2$ states
are also slightly occupied. This is due to the distortion lifting the
degeneracy within the ${J_{\mathrm{eff}}}=3/2$ states and coupling them to the
${J_{\mathrm{eff}}}=1/2$ states. In the metallic state, the occupations become
more similar to the cubic metallic case for decreasing ${\mathcal{U}}$, with
almost equally occupied ${J_{\mathrm{eff}}}=3/2$ states, indicating that the
tetragonal crystal-field splitting is small compared to the kinetic energy.

###  III.3 Effect of spin-orbit coupling strength

We now analyze the effect of the SOC on the metal-insulator transition in
Ba2MgReO6. In the cubic phase, SOC is known to stabilize the insulating state
but has a much weaker effect for the $d^{1}$ case than for other fillings [12,
13]. For example, in the atomic limit, completely turning off SOC would
increase the critical ${\mathcal{U}}$ from the
$1.85\text{\,}\mathrm{e}\mathrm{V}$ discussed before to
$1.94\text{\,}\mathrm{e}\mathrm{V}$, which is still well below the value from
cRPA. We therefore focus on the more complex case of the tetragonal phase with
its competition between tetragonal crystal-field and SOC splittings, again
fixing $\mathcal{J}=$0.27\text{\,}\mathrm{e}\mathrm{V}$$.

Figure 4: (a) Spectral density and (b) occupations in the ${J_{\mathrm{eff}}}$
basis for ${\mathcal{U}}=$1.802.81\text{\,}\mathrm{e}\mathrm{V}$$ as a
function of the SOC strength $\lambda$ obtained for the tetragonal strcuture.
The vertical line corresponds to the realistic value of
$\lambda=$0.30\text{\,}\mathrm{e}\mathrm{V}$$, and the horizontal lines
correspond to the occupations when only the $xz$ orbital is occupied.

Looking at the spectral weight $A(\omega=0)$ as a function of $\lambda$ in
Fig. 4(a), we find that, also in the tetragonal phase, the SOC can indeed turn
Ba2MgReO6 from metallic to insulating, but only in the regime very close to
the Mott transition, e.g., for
${\mathcal{U}}=$1.80\text{\,}\mathrm{e}\mathrm{V}$$. For
${\mathcal{U}}=$2.81\text{\,}\mathrm{e}\mathrm{V}$$, the material is always
insulating, even in the complete absence of SOC, indicating that SOC does not
play a crucial role in stabilizing the insulating state in this case. This is
similar to what has been found for the generic three-orbital Hubbard model
[12] and for the closely related $5d^{1}$ double perovskite Ba2NaOsO6 [30].

The corresponding occupations in the ${J_{\mathrm{eff}}}$ basis, shown in Fig.
4(b), show very similar behavior for the two different ${\mathcal{U}}$ values.
For strong SOC, $\lambda\geq$0.30\text{\,}\mathrm{e}\mathrm{V}$$, the
occupations are essentially constant without any significant changes compared
to the case with $\lambda=$0.3\text{\,}\mathrm{e}\mathrm{V}$$ discussed in
Sec. III.2, with an almost empty ${J_{\mathrm{eff}}}=1/2$ doublet and a
polarized ${J_{\mathrm{eff}}}=3/2$ quadruplet. Upon reducing the SOC strength,
more occupation is transferred into the ${J_{\mathrm{eff}}}=1/2$ states, and
the polarization within the ${J_{\mathrm{eff}}}=3/2$ quadruplet slightly
increases. In the absence of SOC, the results, especially in the insulating
case for ${\mathcal{U}}=$2.81\text{\,}\mathrm{e}\mathrm{V}$$, are relatively
close to only the $xz$ orbital being occupied.

###  III.4 Spectral functions and occupations

Next, we provide further analysis of the spectral features and the orbital
charge distribution for the cubic and tetragonal phase for the most realistic
parameter values, i.e., $\lambda=$0.3\text{\,}\mathrm{e}\mathrm{V}$$ and the
cRPA interaction parameters
${\mathcal{U}}=$2.81\text{\,}\mathrm{e}\mathrm{V}$$ and
$\mathcal{J}=$0.27\text{\,}\mathrm{e}\mathrm{V}$$. We also compare our
DFT+DMFT results to the DFT+$U$ calculations of the tetragonal, paramagnetic
phase from Ref. 38. As stated in Sec. I, in these DFT+$U$ calculations, the
paramagnetic state was emulated by using a large supercell with random
magnetic moments on the Re atoms and zero net magnetization [44, 45], which
can be viewed as a static snapshot of the system in the paramagnetic state.

####  III.4.1 Spectral functions

Figure 5: Spectral function from DMFT for the (a) cubic and (b) tetragonal
phase, with the contribution from the frontier orbitals decomposed into the
eigenstates of the local occupation labeled by their occupation. (c) Density
of states of the tetragonal, paramagnetic DFT+$U$ calculation from Ref. [38],
including the projection onto atomic-like Re states split into a local
majority and minority channel.

We first discuss the spectral function $A(\omega)$ obtained for the cubic
phase, shown in Fig. 5(a), which has a large gap of around
$1.0\text{\,}\mathrm{e}\mathrm{V}$. We note that this gap is much larger than
the activation energy of $0.17\text{\,}\mathrm{e}\mathrm{V}$ obtained from the
Arrhenius fit to the electrical resistivity [37]. However, it is probably
unclear whether this activation energy reflects the electronic gap of the
pristine system. To further analyze the character of the gap, we decompose the
contribution from the frontier orbitals to the spectral function into the
eigenstates of the local occupation. These correspond to the unoccupied
${J_{\mathrm{eff}}}=1/2$ and the quarter-filled ${J_{\mathrm{eff}}}=3/2$
states. One can see that the lower and upper boundary of the gap is defined by
the degenerate ${J_{\mathrm{eff}}}=3/2$ states, while the
${J_{\mathrm{eff}}}=1/2$ states only contribute to higher energies in the
upper Hubbard band, very similar to the DFT+DMFT calculations on Ba2NaOsO6
from Ref. 30.

To align the correlated states relative to the uncorrelated bands in our one-
shot DFT+DMFT calculations, we subtract a double counting correction
$\Sigma_{\text{dc}}$ from the DMFT chemical potential, using the specific form
$\Sigma_{\text{dc}}=({\mathcal{U}}-2\mathcal{J})(n-1/2)=$1.14\text{\,}\mathrm{e}\mathrm{V}$$
[66], where $n=1$ is the total occupation on the Re sites. We note that the
exact form of the double counting correction is ill-defined and thus the
resulting alignment is only approximate. Based on this, the uncorrelated
states lie below $-3\text{\,}\mathrm{e}\mathrm{V}$ and above
$2\text{\,}\mathrm{e}\mathrm{V}$. Thus, while the correlated states move
slightly upward relative to the uncorrelated states compared to the plain DFT
bands shown in Fig. 2, the uncorrelated states are still far away from the gap
and therefore do not play a role for the low-energy behavior of the system.

In the tetragonal phase, there is almost no change in the total spectral
function compared to the cubic case, and the band gap remains
$1.0\text{\,}\mathrm{e}\mathrm{V}$, as depicted in Fig. 5(b). However, the
character of the states around the band gap changes drastically. There is now
only one pair of eigenstates that are partially occupied, i.e., half-filled,
while the other states are essentially unoccupied. This indicates that the
Jahn-Teller distortion has lifted the degeneracy within the
${J_{\mathrm{eff}}}=3/2$ quadruplet (except for the remaining Kramer’s
degeneracy), while also creating some degree of mixing with the
${J_{\mathrm{eff}}}=1/2$ states.

Next, we compare the DFT+DMFT spectral function for the tetragonal structure
with the corresponding density of states from the paramagnetic DFT+$U$
calculations of Ref. 38. To reduce finite-size effects, Ref. 38 used five
configurations with the same ionic positions in the supercell but different
random magnetic moments on the Re atoms. The data depicted in Fig. 5(c)
corresponds to the average over these five configurations. We find very
different features compared to the DFT+DMFT spectral function. The center
bands are split into four distinct peaks, grouped into pairs of two separated
by a gap of about $2\text{\,}\mathrm{e}\mathrm{V}$. The Fermi energy splits
the lower pair of peaks by a ratio of 1:2 in terms of their spectral weights,
which results in a band gap of $0.2\text{\,}\mathrm{e}\mathrm{V}$. Note that
in the DFT+$U$ study of Ref. 38, the effective Hubbard interaction was
specifically chosen to match the gap to the experimental activation energy
[37].

To analyze the nature of the various peaks in the DFT+$U$ spectral function,
we project the density of states from Ref. 38 onto atomic-like functions in
orbital and spin space, and then decompose the density of states in a local
majority- and minority-spin contribution by summing up the spin projections on
the local magnetic moment on a given site for all orbital characters. We then
average over all Re atoms and all independent configurations.

As shown in Fig. 5(c), this reveals that the two lower peaks immediately below
and above the Fermi level represent the local majority spin component, and the
two upper peaks around $2\text{\,}\mathrm{e}\mathrm{V}$ represent the local
minority spin component of the Re-${t_{2g}}$ states. This local spin splitting
roughly coincides with the effective Hubbard interaction of
$1.8\text{\,}\mathrm{e}\mathrm{V}$ used in Ref. 38. The band gap around the
Fermi energy therefore arises from the splitting of the local majority-spin
peaks, which is most likely due to the symmetry breaking by the Jahn-Teller
distortion, the local spin moment, and the strong SOC. Compared to the
DFT+DMFT spectral function, the uncorrelated states lie higher in energy
relative to the Re-$d$ dominated bands around the Fermi level, by about
$2\text{\,}\mathrm{e}\mathrm{V}$. However, also in DFT+$U$, this alignment
depends on the ill-defined double-counting correction [67].

Note that a more systematic quantitative comparison between the paramagnetic
supercell DFT+$U$ calculation and our DFT+DMFT calculations is difficult since
the explicit treatment of the local electron-electron interaction in the two
approaches is applied to different orbitals and using different
parametrizations of the interaction strength. Nonetheless, it is remarkable to
see such significant qualitative differences in the corresponding spectral
functions.

####  III.4.2 Charge distribution in atom-like orbitals

So far, we have employed frontier orbitals constructed from the bands
immediately around the Fermi level, which are suitable for a low-energy
description of Ba2MgReO6, directly map on the nominal $d^{1}$ charge state of
the Re cation, and thus provide a clear and intuitive physical picture. Now,
to allow for a better comparison with the DFT+$U$ results of Ref. [38], we
complement this with an analysis in terms of a more localized atomic-orbital-
like Wannier basis, constructed from a wider energy window. Such a description
in terms of localized atomic-like orbitals also allows us to analyze the
degree of hybridization and quantify the local quadrupoles. Here, the atomic-
like basis might be more suitable for a quantitive comparison to X-ray
absorption and other experimental probes sensitive to the very local
environment around the atoms.

Figure 6: Wannier functions from (left) frontier orbitals and (right)
localized Re-$d$ and O-$p$ orbitals. All functions have ${t_{2g}}$-type
symmetry around the center Re atom.

These localized Wannier orbitals are constructed from all bands in the energy
range between $-13.27.9\text{\,}\mathrm{e}\mathrm{V}$ relative to the Fermi
energy, including all O-$2p$ and Re-$5d$ dominated bands, where the
Re-$5{e_{g}}$ dominated bands need to be disentangled from higher-lying bands.
Per formula unit, we obtain five $d$-like orbitals centered on the Re sites,
denoted by $|\mathrm{l}_{\mathrm{Re},m}\rangle$, and 18 $p$-like orbitals
centered on the O sites. To analyze the hybridization between these orbitals,
we then proceed to combine the O-$p$ orbitals into symmetry-adapted linear
combinations, $|\mathrm{l}_{\mathrm{O},m}\rangle$, which have a clearly
defined symmetry with respect to the Re sites. Five of these linear
combinations have a $d$-type symmetry and therefore hybridize with the Re-$5d$
orbitals. For example, a linear combination with ${t_{2g}}$ symmetry of the
O-$p$ states is shown on the right of Fig. 6.

First, we compute the overlap between a localized orbital
$|\mathrm{l}_{Cm}\rangle$, where $C$ is the atomic character of the localized
state and can thus be Re or O, and a frontier orbital
$|\mathrm{f}_{m^{\prime}}\rangle$ on the same ReO6 octahedron. Using the fact
that the Kohn-Sham states $|\bm{k}i\rangle$ at reciprocal vector $\bm{k}$ with
band index $i$ form a complete ortho-normal basis inside the chosen energy
window, we obtain:

$\displaystyle\langle\mathrm{l}_{Cm}|\mathrm{f}_{m^{\prime}}\rangle=\sum_{\bm{k}i}\langle\mathrm{l}_{Cm}|\bm{k}i\rangle\langle\bm{k}i|\mathrm{f}_{m^{\prime}}\rangle,$
(5)

where the overlaps between Wannier functions and Kohn-Sham states are given by
the $U(\bm{k})$ and $U^{\mathrm{dis}}(\bm{k})$ matrices obtained from
Wannier90 [68]. We note that, due to the gauge freedom of the Bloch functions,
these matrices crucially need to come from the same Kohn-Sham states at the
same DFT iteration for both types of Wannier orbitals.

We find
$\alpha\equiv\langle\mathrm{l}_{\mathrm{Re},m}|\mathrm{f}_{m}\rangle=0.80$ and
$\beta\equiv\langle\mathrm{l}_{\mathrm{O},m}|\mathrm{f}_{m}\rangle=-0.56$ for
all three orbitals with ${t_{2g}}$ symmetry in both the cubic and tetragonal
phase. By symmetry, there is no overlap between orbitals with different
orbital characters, $m\neq m^{\prime}$. The resulting decomposition of the
frontier orbital is schematically visualized in Fig. 6. Thus, the frontier
orbitals consist to $\alpha^{2}=$64\text{\,}\mathrm{\char 37\relax}$$ of
Re-${t_{2g}}$ orbitals and $\beta^{2}=$31\text{\,}\mathrm{\char 37\relax}$$ of
O-$p$ orbitals, with the missing $4\text{\,}\mathrm{\char 37\relax}$ belonging
to other orbitals that we do not explicitly take into account. Such strong
hybridization can be expected from the relatively delocalized nature of the
Re-$5d$ states and is comparable to results from a cluster model for Ba2NaOsO6
fitted to experimental data, which reported a hybridization of
$50\text{\,}\mathrm{\char 37\relax}$ Os and $50\text{\,}\mathrm{\char
37\relax}$ O [26].

We can now compute the occupation in this localized basis from:

$\displaystyle\langle\mathrm{l}_{Cm}|n|\mathrm{l}_{C^{\prime}m^{\prime}}\rangle=$
$\displaystyle\sum_{\mu\mu^{\prime}}\langle\mathrm{l}_{Cm}|\mathrm{f}_{\mu}\rangle\langle\mathrm{f}_{\mu}|n|\mathrm{f}_{\mu^{\prime}}\rangle\langle\mathrm{f}_{\mu^{\prime}}|\mathrm{l}_{C^{\prime}m^{\prime}}\rangle$
$\displaystyle+\sum_{\begin{subarray}{c}\bm{k}i\\\
\epsilon_{\bm{k}i}<\bar{\epsilon}\end{subarray}}\langle\mathrm{l}_{Cm}|\bm{k}i\rangle\langle\bm{k}i|\mathrm{l}_{C^{\prime}m^{\prime}}\rangle.$
(6)

The first term projects the occupations
$\langle\mathrm{f}_{\mu}|n|\mathrm{f}_{\mu^{\prime}}\rangle$ of the central
bands around the Fermi energy from the DMFT calculation onto the localized
orbitals, whereas the second term captures the fully occupied bands below
these correlated bands by summing over all bands with energies
$\epsilon_{\bm{k}i}$ below $\bar{\epsilon}=$-1\text{\,}\mathrm{e}\mathrm{V}$$
[cf. Fig. 2(a)].

The strong hybridization with the oxygen leads to 2.7 electrons in the atomic-
like Re-${t_{2g}}$ and 1.2 electrons in the corresponding ${e_{g}}$ orbitals
and therefore a total of 3.8 $d$ electrons on the Re. This differs
significantly from the nominal $d^{1}$ occupation in the frontier basis and is
quite similar to the 4.0 $d$ electrons found in the paramagnetic DFT+$U$
calculations of Ba2MgReO6 [38]. It is also comparable to the occupation of up
to 6.3 electrons reported in an atomic-like basis for the nominal-$d^{2}$
double perovskites Ba2MOsO6 [69].

We now decompose the occupation matrix in the frontier and localized basis
sets into multipoles as defined in Ref. 42 and implemented in Ref. 70. Here,
we focus on the SOC moment $\langle\bm{l}\cdot\bm{s}\rangle$, which quantifies
the effect of the SOC in Eq. (1) on the spin and charge density, and on the
pure charge quadrupoles centered on the Re sites. We compute the multipoles in
the localized basis from the density matrices
$\langle\mathrm{l}_{Cm}|n|\mathrm{l}_{C^{\prime}m^{\prime}}\rangle$ for the
different $C$ and $C^{\prime}$, and thus obtain a Re-Re, a Re-O, and an O-O
contribution (which all contribute to the multipoles centered around the Re
sites). Note that the multipoles in frontier and localized basis are related
by the approximate decomposition of the frontier into localized orbitals
depicted in Fig. 6. Since the multipole decomposition is a linear operation on
the matrix elements of the density matrix, the considered frontier multipoles
$w$ are related to the corresponding localized ones by

$\displaystyle w_{\text{frontier}}\approx\alpha^{2}w_{\text{Re-
Re}}+2\alpha\beta w_{\text{Re-O}}+\beta^{2}w_{\text{O-O}}.$ (7)

Here, we assume that the bands below $\bar{\epsilon}$ do not contribute to the
frontier multipoles and use the fact that all overlaps and multipoles are
real.

Table 1: Multipoles corresponding to the effect of SOC and charge quadrupoles with ${e_{g}}$ symmetry centered on the Re sites, obtained in the frontier and localized basis sets. The $\pm$ signs indicate the two equivalent Re sites. Note that in the normalization used here [42], a negative $z^{2}$ quadrupole corresponds to excess charge along the $z$ direction.  | frontier | localized
---|---|---
|  | Re-Re | Re-O | O-O
cubic $\langle\bm{l}\cdot\bm{s}\rangle$ | $-0.48$ | $-0.31$ | $0.22$ | $-0.15$
tetra $\langle\bm{l}\cdot\bm{s}\rangle$ | $-0.48$ | $-0.31$ | $0.22$ | $-0.15$
tetra $z^{2}$ | $-0.13$ | $-0.08$ | $0.06$ | $-0.04$
tetra $x^{2}-y^{2}$ | $\mp 0.50$ | $\mp 0.32$ | $\pm 0.22$ | $\mp 0.15$

The resulting multipoles are shown in Table 1. Note that the O-O and Re-O
contributions in the localized basis still correspond to the multipole
centered at the Re site. The SOC moment $\langle\bm{l}\cdot\bm{s}\rangle$ is
close to $-1/2$ in the frontier basis in both the cubic and tetragonal phase,
which is equivalent to the single electron residing in the
${J_{\mathrm{eff}}}=3/2$ states discussed in the context of, e.g., Fig. 3. In
the localized basis, we can see that the effect of the SOC distributes over Re
and O. Therefore, even though the SOC is a local effect caused mostly by the
Re atoms, due to hybridization, the occupations of the O are also affected.

In cubic Ba2MgReO6, the quadrupoles on the Re sites are exactly zero since the
charge density is always cubic in our calculations, as verified for a larger
range of temperatures in App. A. In the tetragonal phase, the $x^{2}-y^{2}$
and $z^{2}$ quadrupoles appear, which are depicted in Fig. 1(b, c) and are the
only quadrupoles allowed in $P4_{2}/mnm$ symmetry. Similar to the SOC moment,
we find that the local quadrupole is close to saturation in the frontier
basis, with a large $x^{2}-y^{2}$ component. In the localized basis, again
both Re and O play a role, with the dominant contributions coming from the Re-
Re and Re-O terms, which enter the total quadrupole moment weighted by
$\alpha^{2}=0.64$ and $2\alpha\beta=-0.90$, respectively, as discussed in Eq.
(7). Since the quadrupole moments are essentially just a specific way to
quantify the local orbital polarization, this shows that an orbital
polarization can also be found on the ligands, and even more pronounced in the
mixed component representing the hybridization between transition-metal ion
and ligand.

Figure 7: Charge quadrupoles on the Re sites in tetragonal Ba2MgReO6,
evaluated in a direct comparison between the DFT+DMFT calculations and the
distributions of quadrupoles on the two different Re sites from the
paramagnetic-supercell DFT+$U$ calculations of Ref. 38.

For the tetragonal phase, we can now directly compare the Re-Re contribution
of the quadrupoles111Note that in Ref. 38 a different coordinate system was
used, where the $x$ and $y$ axes are rotated by $None$ about the $z$ axis away
from the Re-O bonds. We therefore transform the quadrupoles from Ref. 38 to
our choice of axes, i.e. oriented along the Re-O bonds. This swaps $xy$ and
$x^{2}-y^{2}$ and linearly combines $xz$ and $yz$. in the localized basis to
the data from Ref. 38. Due to the random magnetic moments in the paramagnetic
supercell used in DFT+$U$, all Re atoms become symmetry-inequivalent, and in
Fig. 7 we thus show the resulting distribution of quadrupoles (over all five
independent configurations considered in Ref. 38). We show separate
distributions for the two Re sublattices with the $Q_{2}$-related octahedral
elongation along $x$ or $y$ direction, respectively. Then, we compare the
average of these two distributions to the quadrupoles from DFT+DMFT on the
corresponding sites.

In the DFT+DMFT calculations, the quadrupoles with ${t_{2g}}$ symmetry are
zero, as mandated by $P4_{2}/mnm$ symmetry. This agrees with the DFT+$U$
calculations, which show a wide distribution, but with an average always
absolutely smaller than 0.03, so that also from DFT+$U$, Ref. 38 concluded
that these quadrupoles are zero. The $x^{2}-y^{2}$ quadrupole clearly orders
antiferroically in both DFT+DMFT and DFT+$U$, as expected from the symmetry
between the different distortive directions. As discussed before, a
quantitative comparison between the DFT+DMFT and DFT+$U$ results is difficult,
due to the different treatment of the local electron-electron interaction,
which can potentially explain the approximate factor of two in difference
between the predicted $x^{2}-y^{2}$ quadrupoles from the two methods. Finally,
DFT+DMFT confirms the ferroic ordering of the $z^{2}$ quadrupoles, which was
not completely obvious from Ref. 38 due to the rather broad distribution.

##  IV Conclusions and Summary

We have used DFT+DMFT calculations to establish that Ba2MgReO6 is Mott-
insulating both in the cubic and tetragonal paramagnetic phases. In both
cases, the insulating state can be classified as a regular Mott insulator,
originating from the Coulomb repulsion ${\mathcal{U}}$, with a critical
${\mathcal{U}}$ for the metal-insulator transition much smaller than the value
of $2.8\text{\,}\mathrm{e}\mathrm{V}$ we calculated with cRPA. Even though the
spin-orbit coupling is strong, leading to a splitting of around
$0.45\text{\,}\mathrm{e}\mathrm{V}$ between ${J_{\mathrm{eff}}}=3/2$ and
${J_{\mathrm{eff}}}=1/2$ within the low-energy states, it does not
significantly influence the metal-insulator transition. This agrees with the
expected behavior for $d^{1}$ systems [12, 30].

The spectral function of Ba2MgReO6 around the Fermi energy is dominated by two
Hubbard peaks, which form a gap of about $1.0\text{\,}\mathrm{e}\mathrm{V}$ in
both cubic and tetragonal phases. However, the orbital character of the
Hubbard peaks changes across the cubic-tetragonal transition, in line with the
corresponding change in the local Re occupations in the frontier orbital
basis. The insulating, cubic phase can be accurately described in a nominal
$d^{1}$ picture with a quarter-filling of the ${J_{\mathrm{eff}}}=3/2$ states
due to the spin-orbit coupling. In the tetragonal, paramagnetic phase, we find
approximate half-filling of a state determined by the combined effects of the
spin-orbit coupling and the tetragonal crystal field, with a remaining two-
fold degeneracy due to time-reversal symmetry.

We also analyze the local electron distribution around the Re sites in terms
of a more localized atomic-orbital-like basis. These orbitals might be more
suitable for comparison with experimental probes that are sensitive to the
immediate atomic environment such as resonant X-rays, and also allow for a
better comparison with the DFT+$U$ results of Ref. 38. Two complementary but
equivalent pictures emerge when postprocessing the charge and spin density
with either frontier or atomic-like Wannier orbitals. The former directly maps
on the nominal $d^{1}$ picture, with physics comparable to model studies [12,
13, 43], while the latter corresponds to an atomic picture, where both Re and
O orbitals are influenced by the tetragonal distortion and the spin-orbit
coupling. Notably, even though spin-orbit coupling is a local effect coming
mostly from the heavy Re atoms, its signature in the hybridized bands around
the Fermi energy also strongly affects the occupation of the O orbitals,
indicated by a nonzero component of the spin-orbit coupling
$\langle\bm{l}\cdot\bm{s}\rangle$ moment.

Finally, we compare our DFT+DMFT results to DFT+$U$ calculations for the
tetragonal phase of Ba2MgReO6, which have used large supercells with random
magnetic moments to emulate the paramagnetic state [38]. We find good
qualitative agreement between the two methods for the quadrupole moments
around the Re sites. Our calculations even corroborate the existence of a
ferroically ordered $z^{2}$ quadrupole component, which was slightly obscured
in the DFT+$U$ calculation of Ref. 38 by the fluctuations related to the
random magnetic moments but has also been reported experimentally [37, 34,
40]. In contrast, we find noticeable differences in the spectral functions
obtained within DFT+DMFT and DFT+$U$, where in the latter case, the Re-
dominated density of states around the Fermi level splits into four peaks with
well-defined local spin characters, which is rather different from the
DFT+DMFT spectral function. Part of this difference can potentially be
explained by the differences in how the local electron-electron interaction is
treated in the two cases. Nevertheless, this comparison raises the question of
how comparable the spectral features of paramagnetic DFT+$U$ and DFT+DMFT
calculations are, especially in the presence of spin-orbit coupling where the
orbital occupation is strongly influenced by local magnetic moments.

###### Acknowledgements.

We thank Sophie Beck for implementing the real-valued formulation of SOC in
solid_dmft and Andrea Urru for the discussions and help in implementing the
multipole decomposition in Python. This research was supported by ETH Zurich
and a grant from the Swiss National Supercomputing Centre (CSCS) under project
ID s1128.

##  Appendix A Effect of temperature

The critical temperature of $33\text{\,}\mathrm{K}$ for the transition from
cubic to tetragonal is much lower than the electronic temperature of
$290\text{\,}\mathrm{K}$ used in our DMFT calculations. To verify that this
high electronic temperature does not crucially affect the results obtained in
our DFT+DFMT calculations for the tetragonal phase, we now present additional
calculations where we vary the electronic temperature and monitor the effect
on the orbital occupations. We also test whether it is possible to stabilize
spontaneous orbital order in the cubic phase, or whether the charge density
always preserves the $Fm\bar{3}m$ symmetry. Here, we again use the realistic
parameters of $\lambda=$0.3\text{\,}\mathrm{e}\mathrm{V}$$,
${\mathcal{U}}=$2.81\text{\,}\mathrm{e}\mathrm{V}$$, and
$\mathcal{J}=$0.27\text{\,}\mathrm{e}\mathrm{V}$$.

Figure 8: Occupations in the ${J_{\mathrm{eff}}}$ basis for the cubic and
tetragonal phase as a function of the electronic temperature in the DMFT
calculation. The upper temperature axis indicates the points corresponding to
the boundary of the numerical instability region ($170\text{\,}\mathrm{K}$),
room temperature ($290\text{\,}\mathrm{K}$), and the thermal energies equal to
the tetragonal crystal-field splitting ($1100\text{\,}\mathrm{K}$) and the SOC
splitting ($5200\text{\,}\mathrm{K}$). The region of numerical instability is
defined when the average sign in the CT-HYB solver drops below $0.1$.

Fig. 8 shows the occupation as a function of electronic temperature, both for
the cubic and tetragonal case. In the tetragonal case, the polarization of the
${J_{\mathrm{eff}}}=3/2$ quadruplet depends only weakly on the temperature in
the range from $120290\text{\,}\mathrm{K}$. Below $120\text{\,}\mathrm{K}$,
indicated by the gray shaded region in Fig. 8, the calculations become
numerically unstable due to the fermionic sign problem appearing in the
quantum Monte-Carlo solver. Nevertheless, it seems reasonable to assume that,
for all temperatures below $290\text{\,}\mathrm{K}$, the polarization of
paramagnetic Ba2MgReO6 is already more or less saturated. This _a posteriori_
justifies the choice of electronic temperature in the DFT+DMFT calculations
discussed in the previous sections.

Note that at higher temperatures, thermal effects first reduce the
polarization within the ${J_{\mathrm{eff}}}=3/2$ states, and then the
polarization between ${J_{\mathrm{eff}}}=3/2$ and ${J_{\mathrm{eff}}}=1/2$
states, at temperatures of around $11005200\text{\,}\mathrm{K}$, where the
thermal energy is roughly equivalent to the respective energy splittings.

In cubic Ba2MgReO6, for all temperatures above $120\text{\,}\mathrm{K}$ which
are accessible with our DFT+DMFT calculations, we find no tendency towards
spontaneous polarization, in agreement with the low experimental transition
temperature. At higher temperatures, we find the suppression of SOC-induced
polarization, as already observed in the tetragonal phase.

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Combined Higgs boson measurements and their interpretations with the ATLAS
experiment

Presented at DIS2023: XXX International Workshop on Deep-Inelastic Scattering
and Related Subjects, Michigan State University, USA, 27-31 March 2023.

Marc Escalier, on behalf of the ATLAS Collaboration111Copyright CERN for the
benefit of the ATLAS Collaboration. CC-BY-4.0 license
IJCLab, PHE department
15 Rue Georges Clemenceau
91400 Orsay, FRANCE

Abstract: Combining measurements of many production and decay channels of the
observed Higgs boson allows for the highest possible measurement precision for
the properties of the Higgs boson and its interactions. These combined
measurements are interpreted in various ways; specific scenarios of physics
beyond the Standard Model (SM) are tested, as well as a generic extension in
the framework of the Standard Model Effective Field Theory. The latest
highlight results of these measurements and their interpretations performed by
the ATLAS [1] Collaboration are discussed.

##  1 Production modes and decay channels of the Higgs boson, kinematic
properties

The $H\rightarrow\gamma\gamma$, $H\rightarrow ZZ^{*}$, $H\rightarrow WW^{*}$,
$H\rightarrow Z\gamma$, $H\rightarrow bb$, $\rightarrow\tau\tau$,
$H\rightarrow\mu\mu$ analyses are combined [2] in order to probe main
production modes and decay channels. With this Run 2 combined publication, the
$WH$, $ZH$, $ttH+tH$ are observed respectively with a significance of $5.8$,
$5.0$ and $6.4$. An observed $95\ \%$ Confidence Level (CL) limit of $15\times
SM$ is obtained on $tH$. On the decay side, the $H\rightarrow bb$ mode is
observed with a significance of $7.0$. The observed significances of
$H\rightarrow\mu\mu$ and $H\rightarrow Z\gamma$ are respectively of $2.0$ and
$2.3$. Coupling strength modifiers are probed using the $\kappa$ framework
[3], without $Z\gamma$ nor $\mu\mu$ channels, using three scenarios
increasingly reducing the assumptions: (1) boson/fermion symmetry, probing
$Hff$ and $HVV$, (2) independent $\kappa$ parameters, allowing to observe the
compatibility of scaling of coupling with mass of particles, probing also the
specific $\kappa_{c}$ coupling strength modifier by adding the $H\rightarrow
cc$ channel, with a limit of $5.7\times SM$, and (3) a generic parametrization
with or without invisible and undetected particles, which adds the
$H\rightarrow invisible$ channel as input. In this last scenario that allows
non-SM particles, couplings of loops are taken as effective instead of
resolving them as functions of their fundamental constituents. Non-SM Higgs
branching ratios to invisible particles and branching ratios to undetected
particles for which the inputs have no sensitivity are disentangled by a
constraint of Higgs coupling to vector boson below or equal 1. Kinematic
properties of Higgs production are probed using the STXS framework stage 1.2
[4] that splits phase space in so-called truth bins of production modes,
kinematics of the Higgs, kinematics of the jets and vector boson and
multiplicity of jets. All measurements are compatible with the predictions
from the SM.

Using a slightly reduced number of channels in the combination (mostly without
the $H\rightarrow cc$ channel), the coupling strength modifiers are translated
into contour limits in the plane $\tan{\beta}$ vs $\cos{(\beta-\alpha)}$ of
the four 2HDM models that prevent tree-level Flavor-Changing Neutral Currents
(FCNC). Results are compatible [5] with the alignment limit
$\cos{(\beta-\alpha)}=0$ where the coupling of lightest Higgs boson matches
those of the SM one. Non excluded petal regions remain for three types,
corresponding to Yukawa coupling $Hff$ of the same magnitude as in the SM but
with a different sign.

##  2 Interpretation of production and decay of the Higgs boson in SMEFT

The measurement from ATLAS on Higgs production ($H\rightarrow\gamma\gamma$,
$H\rightarrow ZZ^{*}$, $H\rightarrow WW^{*}$, $\rightarrow\tau\tau$,
$H\rightarrow bb$) and electroweak differential cross-section, as well as
results from LEP and SLD experiments on electroweak measurements are combined
[6] to probe either direct Wilson coefficients ($c_{i}$) or eigenvectors of
them. The yields are parametrized as functions of $c_{i}$. It is related to
the perturbative development of the Lagrangian where Wilson coefficients
weight new operators of energy dimension 6. The SM operators correspond to
dimension 4. Dimensions 5 and 7 are ignored from arguments of avoiding
violation of baryonic and leptonic numbers. Linear terms (in $c_{i}$)
correspond to interference between the SM and dimension 6 terms, quadratic
terms correspond to pure dimension 6 terms. Dimension 8 are neglected but
comparison between results from linear and linear+quadratic provides a
qualitative estimation of the validity of neglecting them. In order to reduce
the high number of operators probed, a symmetry is introduced, imposing same
operators within the two first generations of quarks and within the three
generations of charged leptons.

The relative impact of each Wilson coefficient on the cross-section could be
visualized for various STXS truth bins and compared to the corresponding
relative uncertainty of each cross-section measurement. In the yield
parametrization, the signal acceptance times efficiency, signal shape and
background modelisation are assumed to be independent of EFT parameters. This
is a good approximation coming from the similar kinematics behaviour in a
given truth bin. This is no more true for Higgs decays. For example, in the
$H\rightarrow 4\ell$ channel, the subleading dilepton pair distribution varies
a lot from the SM when Wilson coefficients are turned on. Branching ratios are
thus parametrized as functions of $c_{i}$. In the case of $H\rightarrow
4\ell$, it is parametrized after applying the fiducial cuts of the analysis to
take into account the important change.

The parametrization of yields is obtained from a basis of samples with various
values of $c_{i}$. Tree level/LO processes are simulated with Madgraph and
SMEFTSim while loop diagrams in production and decay are treated with
SMEFT@NLO. The specific $H\rightarrow\gamma\gamma$ is parametrized using an
analytic computation. Higher order corrections are obtained and factorized
from the SM. Common systematics are treated correlated among the channels.

The results are presented in the form of measurements on individual and
eigenvector of Wilson coefficients. These last ones are obtained from a
Principal Component Analysis based on the Hessian matrix. Results are provided
for ATLAS only inputs as well as for the further combination with LEP and SLD.
No deviation with respect to the SM is observed.

##  3 Fiducial and differential cross-section

The fiducial and differential cross-sections combination [7] uses
$H\rightarrow\gamma\gamma$, $H\rightarrow ZZ^{*}\rightarrow 4\ell$. The
measured fiducial cross-sections from this $\sqrt{s}=13$ TeV data-taking and
from the former $\sqrt{s}=7$ TeV and $\sqrt{s}=8$ TeV campaigns are in
agreement with the energy depencence prediction. Five differential cross-
sections are measured: multiplicity of jets and $p_{T}$ of the leading jet,
which are sensitive to theory modelization of the QCD radiation and production
modes, rapidity of the Higgs $|y_{H}|$, sensitive to parton density functions
of the proton, transverse momentum $p_{T}^{H}$ of the Higgs boson, sensitive
to perturbative QCD. The measurements are dominated by statistical
uncertainties. The experimental and theoretical uncertainties are correlated
among channels. The measurements are compatible among channels.

$Hcc$ and $Hbb$ Yukawa couplings are sensitive to this last variable by
subleading effects in the loop diagrams, such as gluon fusion and decay of
Higgs to pair of photons, thus affecting more generally branching ratio of
$H\rightarrow ZZ^{*}\rightarrow 4\ell$, but also in quark initiated
production. In the scenario where both the normalization and shape are
exploited, this gives the observed confidence interval on $\kappa_{b}$ and
$\kappa_{c}$ respectively of $[-1.09,-0.86]\ U\ [0.81,1.09]$ and
$[-2.27,2.27]$. The channels are combined further with the channels involving
directly these couplings: $H\rightarrow bb$, $H\rightarrow cc$ with $VH$
production with or without allowing decay to Beyond-Standard-Model (BSM)
particles. This gives most stringent constraints on $\kappa_{c}$.

##  4 Higgs to invisible

In the SM, decay of Higgs to invisible appears at the branching ratio level of
$0.1\ \%$ from the $H\rightarrow 4\nu$ channel. More generally, in BSM
scenario, Higgs could decay to invisible particle and act as a portal between
a dark sector and the SM. The signature of the search is the presence of
missing transverse energy and a tagging particle. This drives to the
combination of $H+j$, $VBF$, $VBF+\gamma$ (where selection of these two last
ones are orthogonal), $ZH$, $ttH$. For Run 2, most systematics are correlated
among channels. Since the performance of object reconstruction and
identification are made from data-driven measurements, while detector layout
and data-taking conditions varied between Run 1 and 2, the systematics on
objects are uncorrelated. The systematics on background modelling are also
uncorrelated to take into account the improvement in the MC simulation and
theory prediction.

The $95\ \%$ CL observed (expected) limit on the branching ratio of Higgs to
invisible is [8] $0.107$ ($0.077$), driven mostly by $VBF$ and $ZH$ channels.
From an effective field theory framework, limits are translated into an spin-
independent scattering cross-section of a Weakly Interacting Massive Particle
(WIMP) to nucleon, giving complementary indirect limits to those from direct
measurement experiments, for scenarios with scalar, majorana, vector of the
WIMP.

##  5 Probing Higgs self-coupling: $HH+H$

The Higgs self-coupling $\lambda$ from the Higgs potential, and more generally
its measured ratio $\kappa_{\lambda}=\lambda/\lambda_{SM}$ with respect to the
SM could be measured directly from double Higgs production but also indirectly
from higher order electroweak corrections to the single Higgs production. In
context of such a double Higgs production, the secondary production mode of
$VBF\ HH$ introduces a coupling $HHVV$, related to the coupling strength
modifier $\kappa_{2V}$, whose double Higgs production is uniquely sensitive
to. The inputs to the combination [9] are made of three double Higgs channels:
$bb\gamma\gamma$, $bb\tau\tau$, $bbbb$ and five single Higgs channels:
$\gamma\gamma$, $ZZ^{*}\rightarrow 4\ell$, $\tau\tau$, $WW^{*}\rightarrow
e\nu\mu\nu$, $bb$. The potential overlap among channels are checked to be
negligible, apart for the $ttH$ category of single Higgs populating the
$HH\rightarrow bb\tau\tau$ channel. For this reason, this specific category is
removed from the $HH+H$ combination. In the statistical model, the systematics
from common physics objects and mass of the Higgs are correlated among
channels. Ingredients from different methodologies are uncorrelated.
Theoretical systematics are correlated, but those on the single Higgs and
double Higgs production are uncorrelated.

The combination of $HH$ channels establishes an observed (expected) $95\ \%$
CL limit on the signal of $2.4\ (2.9)\times SM$. The limits on the rate of
inclusive signal and on $VBF\ HH$ are translated in a limit on the couplings
strength modifiers $\kappa_{\lambda}$ and $\kappa_{2V}$. The combination with
the single Higgs allows to relax assumption on Higgs couplings. In a scenario
with $\kappa_{t}$ floating, the confidence intervals are the same as without
floating it. A generic scenario floating $\kappa_{t}$, $\kappa_{V}$,
$\kappa_{b}$, $\kappa_{\tau}$ is also probed.

##  6 Conclusion

Results presented here improve further those from previous edition of DIS
conference. No deviation with respect to the SM is observed.

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