# Ca3Ru2O7: Interplay among degrees of freedom and the role of the exchange-
correlation

A.M León<EMAIL_ADDRESS>Max Planck Institute for Chemical
Physics of Solids, 01187 Dresden, Germany.  Faculty for Chemistry, TU Dresden,
Bergstrasse 66c, 01069 Dresden, Germany.  J. W. González  Departamento de
Física, Universidad Técnica Federico Santa María, Casilla Postal 110V,
Valparaíso, Chile.  H. Rosner  Max Planck Institute for Chemical Physics of
Solids, 01187 Dresden, Germany.

###### Abstract

Ca3Ru2O7 is a fascinating material that displays physical properties governed
by spin-orbit interactions and structural distortions, showing a wide range of
remarkable electronic phenomena. Here, we present a density-functional-based
analysis of the interplay among degrees of freedom, such as magnetism, Coulomb
repulsion (Hubbard-U), and lattice degree of freedom, together with clarifying
the role of the exchange-correlation effects (LDA v/s PBEsol functional). Our
goal is twofold: first, to provide a brief overview of the current state of
the art on this compound underpinning to last proposed theoretical models and
experimental research, and second, to give another view to model the
electronic properties compared with the previous theoretical models,
establishing the lower and upper limits of (Hubbard-U). Furthermore,
disentangle the effect of the Coulomb repulsion, structure, and functional
approximation on the electronic band structure. Our findings show that local-
density approximations (LDA) exchange-correlation outperforms the generalized
gradient approximations (GGA-PBE); the Hubbard-U term in range ($0.2\leq U\leq
2.0$ eV) for LDA+U accurately reproduces the structural and electronic
properties of Ca3Ru2O7.

##  I Introduction

Physics driven by spin-orbit interactions is among the most important topics
in contemporary condensed matter physicssoumyanarayanan2016emergent ;
witczak2014correlated . Since the spin-orbit interaction is comparable to the
on-site Coulomb and other relevant interactions create a unique balance
between competing interactions that drive complex behaviors and exotic states.
In particular, compounds involving transition metals are technologically
relevant. However, with their characteristic half-filled $d$ orbitals, they
exemplify this non-trivial competition between interactions of different
natures kosmider2013large ; rejali2020complete ; gonzalez2017complex . The
series Can+1RunO3n+1 compounds exemplify these characteristics; these
$4d$-electrons systems present a perovskite structure, where the relative
rotation and tilting of RuO6 octahedra often affect electronic properties
depending on the number of layers ($n$ in Can+1RunO3n+1). Among these, the
Ca3Ru2O7 compound (CRO) presents several quantum phenomena such as electronic
phase transitions, colossal magneto-resistance, spin density waves, or quantum
oscillations cao1997observation ; colossal-Yoshida2004 ; kikugawa2007ca3ru2o7
; spindensityCa3 . Early works suggested that CRO is a hallmark material
exhibiting both metallic and insulator states. Due to these characteristics,
CRO is cataloged as a peculiar materialcaofrontiers  whose further
comprehension could provide new routes to understand the complex interplay
among spin, orbital, charge, and lattice degree of freedom present in
$4d$-oxides markovic2020electronically ; xing2018existence .

Figure 1: Ca3Ru2O7 in two antiferromagnetic phases, AFM-$a$ and AFM-$b$, with
ferromagnetic coupling within the layers and antiferromagnetic coupling
between the bilayers. The +/- signs represent the relative orientation of the
Ru in-plane magnetic moments: “$+$” for magnetic moments oriented in the axis
direction and “$-$” for those oriented in the opposite direction.

Currently, CRO is recognized as a polar metal holding a strong interplay
between spin-orbit coupling (SOC) and structural inversion
symmetrymarkovic2020electronically ; horio2021electronic , which leads to
their exotic states. Recent experimental research has given new perspectives
to manipulate and continue exploring their quantum states. In this context,
have been observed metamagnetic phases sokolov2019metamagnetic , which may
host mixed textures similar to the chiral magnetic skyrmions, being this an
ideal playground to study diverse antiferromagnetic-ferromagnetic states near
metamagnetic transitions. Also, it was evidenced by the emergence of polar
domains that may be controlled by ferroelectric switching. This last discovery
opens new exciting questions about the mechanism behind supporting such
domains in polar metalsdanilo-2 . Moreover, it was shown that CRO exhibits
extraordinary lattice flexibility because its electronic states can be tuned
by manipulating their crystalline axes via external magnetic fields and under
pressurecolossal-Yoshida2004 ; Lattice2021 . The studies mentioned above open
a new research stage on the CRO system. The appearance of magnetic textures
and polar domains, and the facility for manipulating electronic states under
lattice deformation mechanisms, could give novel perspectives to search for
the next generation of ferroelectric compounds with novel functionalities.

The CRO structure is defined by RuO6 planes intercalated with calcium atoms.
Due to the strong spin-orbit coupling present in the Ru atomsjohansson2010spin
; srnec2008effect , non-collinear magnetism is expected. The interaction
between Ru planes depends on the large coupled rotations and tilts of the RuO6
octahedra that make up the perovskite-like building blocks of this structure,
generating a non-centrosymmetric crystal structure. In particular, one of the
most striking characteristics of CRO is its phase transitions under
temperature, exhibiting a first magnetic transition at the Néel temperature of
56 K (TN). This first transition is associated with the spins aligning
ferromagnetically within each bilayer in the [100] direction but
antiferromagnetically coupled between bilayers (AFM-$a$). A second structural
and magnetic transition happens at 48 K (TS), where occurs a c-axis lattice
parameter compression and the spins are reoriented from [100] to [010]
direction (AFM-$b$) cao1997observation ; kikugawa2007ca3ru2o7 ; yoshida2005 .

Below TS, the in-plane resistivity starts to increase with cooling, showing a
local maximum at T $\approx$ 30 K. The phase transition at Ts has often been
discussed as a metal-insulator transition, and the temperature region between
30 K and TS as an insulating phase of Ca3Ru2O7 (by early
workscao1997observation )); The understanding of the nature of these phases
(either metallic or insulator) has been controversial in the
literaturexing2018existence . In this context, early workskikugawa2007ca3ru2o7
have suggested that different growth batches of Ca3Ru2O7 crystals yield
differing low temperature properties, ranging all the way from the metallic
ground states spindensityCa3 ; kikugawa2007ca3ru2o7 ; baumberger2006nested ;
lee2007pseudogap  to insulating phasescao1997observation ; cao2003quantum ;
karpus2006spectroscopic . Conciliating the different experimental observations
have been an intense topic of research and debate for more than one
decadespindensityCa3 ; colossal-Yoshida2004 ; baumberger2006nested ;
kikugawa2007ca3ru2o7 ; lee2007pseudogap ; cao1997observation ; cao2003quantum
; karpus2006spectroscopic .

Using a theoretical approach based on the density functional theory (DFT), we
explore the electronic and magnetic properties of CRO crystals to evidence the
conditions that lead to either metallic or non-metallic ground state
solutions. In order to gain insight, we discuss the interplay between non-
collinear magnetic configurations, Coulomb repulsion (U-Hubbard), spin-orbit
effects (SOC), lattice degree of freedom, and exchange-correlation. We compare
our results against the latest experimental and theoretical results on CRO.
Furthermore, we contrast the electronic band structure of CRO using two
exchange correlations: PBEsol and LDA approximations and performing a linear
approximation calculation implemented by Cococcioni-Gironcoli
cococcioni2005linear  we establish the upper and lower limits of the optimal
U-Hubbard values. The contrast of our results against the experimental
evidence reveals that the LDA+U description better describes the electronic
and magnetic properties of CRO.

##  II Theoretical advances

As we mentioned before, one of the biggest problems was to conciliate the
metallic or insulator nature of the phase between 30 $\lessapprox$ T $<$ TS,
in which some experimental works suggested a metal/semi-metal
behaviorspindensityCa3 ; kikugawa2007ca3ru2o7 ; baumberger2006nested ;
lee2007pseudogap  and others an insulator phase cao1997observation ;
cao2003quantum ; karpus2006spectroscopic . In contrast, DFT works predicted a
nearly half-metallic phase considering collinear calculations
singh2006-CaRuO-327  and an insulator ground state considering a SOC and
Coulomb repulsion Liu-cro-327 . Recent experimental and theoretical studies
have given new perspectives about the nature of CRO below TS, reaching a
better agreement between theory and experiments. It was revealed that CRO does
not have a real insulator phase since it was evidenced a pseudogap behavior
around $\Gamma$ point and several bands crossing the Fermi level
markovic2020electronically ; horio2021electronic . Despite this advance, the
understanding of the mechanism that causes the phase transitions around TS is
still controversial: on the one hand, it is proposed a Lifshitz-like
transition at Ts together with a new structural phase transition at T
$\approx$ 30K as a result of the cooperative interactions among coulomb
repulsion, SOC and lattice grade of freedom danilo . On the other hand, it is
suggested that the phase transition at Ts arises due to an electronic energy
gain from the pseudogap opening at the Fermi level. The hybridization was
shown to come from a Rashba-type SOC term which develops due to an antipolar
distortion in the crystal structure markovic2020electronically . Currently,
from a theoretical point of view, the role of fundamental interactions such as
Coulomb and SOC has not been clarified yet.

Why is it so complicated to understand the electronic structure of CRO? From a
theoretical point of view, studies based on DFT have observed that the
unconventional magnetic and electronic properties are dictated by the
competition of multiple degrees of freedom (charge, spin, SOC, lattice) that
makes the electronic states highly sensitive to the electronic interactions
and exchange of correlation. Early studies using GGA approximation evidenced a
metallic state and highly anisotropic behavior of CRO. They suggested for the
first time the crucial role of spin-orbit interaction, whereas a metal-
insulator transition happens by the interplay of SOC and Coulomb repulsion (U)
(for U $>$ 3 eV)Liu-cro-327 . Later, other studies considering spin-orbit
interaction with LDA and GGA evidenced that the systems have semi-metallic
states. Here, it was evidenced that GGA fails to describe the band structure
at $\Gamma$ point through comparing with ARPES spectra
markovic2020electronically . Finally, recent work has performed studies using
GGA along with PBEsol approximation and considering the interplay among
SOC+U+lattice degrees of freedom; this study revealed that for small U values,
it is possible to control the hole and electron pocket around $\Gamma$ in
which for U $\geq$ 1.4 give good agreement with the ARPES spectra danilo .
However, it is necessary to propose a new structural phase transition not
reported by experiments so far.

##  III Calculation Details

We perform a theoretical analysis using the density functional theory with
spin-orbit coupling (SOC). We use the plane-wave pseudopotential method
implemented in the Vienna ab-initio simulation package (VASP)Kresse1996
within the PBEsol functional (PS)Perdew2008  and the Local-density
approximations (LDA)ceperley1980ground . PBEsol (PS) belongs to generalized
gradient approximations (GGA), and it gives structural properties closer to
the experimental values when compared to the results of PBEdongho2015density .
We use a plane-wave energy cutoff of 650 eV and set a Regular Monkhorst-Pack
grid of 5$\times$5$\times$3 to perform the atomic relaxation and
7$\times$7$\times$5 to perform the self-consistent calculation. We use a fine
k-grid 17$\times$17$\times$5 dense k-gird within the tetrahedron method for
the density of states. We perform the structural optimization of the unit cell
until a force convergence threshold of at least 10-3 eV/Å per atom.

To take into account the electronic correlation effects in d-orbitals of Ru
atoms, we consider a range of Hubbard on-site Coulomb parameter values (U $=$
0 to 2.0 eV)danilo . Later, we employ the linear response to estimate the
optimal Hubbard-U; we follow the linear approach methodcococcioni2005linear .
The linear response was computed CRO introducing the system’s interacting and
non-interacting occupation response with respect to the localized
perturbations up to $\alpha=\pm 0.2$ eV in the Ru atoms; we compare the Ru
d-orbital occupation upon the perturbation $\alpha$ in LDA/PBEsol, and
with/without spin-orbit coupling.

##  IV Results and Discussion

CRO crystallizes in a orthorhombic structure with space groupyoshida2005
Bb21m (a full description of the structure is available in ref. (32)). The
atomic positions are taken from the neutron diffraction
measurementsyoshida2005  at 8K. To reproduce the different non-collinear AFM
configurations, the unit cell includes 48 atoms; there are 12 Ca atoms, 8 Ru
atoms, and 28 O atoms. Fig. 1 shows the lattice structure of CRO, with two
possible magnetic configurations experimentally reported at T $<$ 48 K and T
$>$ 56 K called AFM-$b$ and AFM-$a$, respectively. Firstly, we carry out
atomic optimizations at the experimental volume given by the
experimentyoshida2005  (V = 580 Å3). We label the ferromagnetic states with
the direction in which the in-plane magnetic moments align. We use AFM-$a$ for
magnetic moments aligned along the $\pm\bf{a}$ direction and AFM-$b$ for
magnetic moments aligned along the $\pm\bf{b}$ direction. In Table 1, we
compare our results for U $=$ 0 eV using LDA and PBEsol (PS) with the
experimental results, showing the energy difference between AFM-$a$ and
AFM-$b$ ($\Delta E=E_{AFM-a}-E_{AFM-b}$), the magnetic moment per Ru-atom
($m$), the volume of octahedra (V(RuO6)), and the length of Ru-O bond
($l$(Ru-O)) for the most stable AFM phase.

###  IV.0.1 Exchange–correlation effects

In the PS approximation, the ground state presents an AFM-$b$ configuration
with a $m$ of $1.36$ $\mu_{B}$ (calculated in a Wigner-Seitz (WZ) radius 1.323
Å). Within the same approximation, the FM phase presents a $m$ of 1.92
$\mu_{B}$; this value overestimates $m$ about a 6% concerning the $m$ measured
by previous experimental reportsyoshida2005 . The energy difference between
the AFM and FM configuration is 1.69 meV/Ru atoms with the magnetic moment
projected in the $b$-direction.

In contrast, the LDA approximation reveals an AFM-$a$ground state with a low
$m$ of 0.84 $\mu_{B}$ (WZ= 1.323 Å). The FM phase shows a total $m$ of 0.83
$\mu_{B}$, being this configuration more stable than AFM-$b$ and AFM-$a$ with
an energy difference of -5.85 meV/Ru and -2.67 meV/Ru, respectively. Despite
the mismatch in the magnetic stability order between PS and LDA, both
approximations reproduce the experimental crystal structureyoshida2005  (see
table 1).

Going further, we perform a complete cell relaxation, including the relaxation
of lattice parameters and cell volume; these values are reported in table S1
of supplementary material, and our results for the most stable phase are shown
in table S2 of supplementary material. Here, we can see that the results have
the same trend at the equilibrium volume as at the experimental volume.

Appr. |  $\Delta$E (meV/Ru) | m (${\mu_{B}}$) | V(RuO6) (Å3) |  $l$(Ru-O) (Å)
---|---|---|---|---
LDA | -2.67 | 0.84 | 10.53 | 1.992
PS | 0.39 | 1.36 | 10.54 | 1.993
Expyoshida2005  | AFM-$b$ | 1.8 | 10.53 | 1.992
Table 1: $\Delta E$ is the energy difference between AFM-$a$ and AFM-$b$
magnetic configurations; negative values correspond to an AFM-$a$ ground state
solution, $m$ the magnetic moment per Ru-atom, V(RuO6) is the RuO6 octahedral
volume (Å3) and $l$(Ru-O) is the Ru-O bond average length (Å) for the most
stable magnetic configuration considering U $=$ 0 eV. The third-row label as
“Exp” corresponds to the experimental datayoshida2005 .

Figure 2: Effect of electron-electron repulsion (Hubbard-U) on the physical
properties of CRO. In (a) magnetic anisotropy Ea-Eb, (b) magnetic moment per
Ru atom (measured in the Wigner-Seitz radius (1.323 Å)), and (c) Volume of
RuO6 octahedra as U increases from 0 to 2.0 eV. Red and green dots correspond
to the crystal phase Bb21a with LDA and PS approximation, respectively, and
blue dots correspond to the Pn21a structure with PS approximation. The right
panel shows a schematic representation of the structural transition from Bb21m
to Pn21a due to the U effect above U $\geq$ 1.4 eV. Note in panels (b) and (c)
for U $\geq$ 1.4 eV and PS approximation, the Ru atoms in the Pn21a phase
present two different crystallographic environments and thus two different
magnetic moments and volumes. The bottom right panel displays the structural
change from Bb21m to Pn21a for U $\geq$ 1.4 eV; the different octahedra color
on the Pn21a structure symbolizes the different environments in this cell.

###  IV.0.2 Electron-electron correlation effects

Now we will study the effect of electron-electron repulsion on the electronic
and structural properties. Based on previous worksLiu-cro-327 ; danilo , we
use Hubbard-U values in the $\left[0,\,2\right]$ eV range. Fig. 2 shows the
effect within the PS and LDA approximations of the Hubbard-U interaction for
the magnetic anisotropy $\Delta E=E_{a}-E_{b}$, the magnetic moment per Ru
atom $m$, and RuO6 octahedra volume V.

On the one hand, we will start with the LDA analysis, represented with red
dots in Fig. 2. The magnetic anisotropy as a function of the Hubbard-U is
presented in Fig. 2 (a). When the Hubbard-U interaction is included, the
AFM-$b$ ground state is restored for U $\geq$ 0.2 eV (this configuration
became more stable as $\Delta$E $>$ 0). In Fig. 2 (b), as expected, we observe
that the U-term increases $m$, from $m=$ 0.84 $\mu_{B}$ at U $=$ 0 to $m=$
1.16 $\mu_{B}$ at U $=$ 0.2 eV, then $m$ increases monotonously to U $=$ 2.0
eV. Finally, in Fig. 2 (c), we found that the octahedra volume decreases as
the U-term increases. To compare with the experimental magnetic moment
measurements, we compute the ferromagnetic (FM) configuration with $m$
alignment in the $b$-direction (FM-$b$) (inset in Fig. 2b). We can observe
that the total magnetic moment per atom agrees with the magnetic moment
evidenced by the inelastic neutron scattering experiments ($m$ = 1.8 $\mu_{B}$
per Ru atom, AFM-$b$ configuration). For a fixed value of Hubbard-U, the
magnitude of the Ru magnetic moments in the FM configuration are close in
value to those of the AFM-$b$ configuration. The magnetic moments are
calculated within a similar Wigner–Seitz radius, in the inset of Fig. 2 (b),
$m$ $=$ 1.32 $\mu_{B}$ (U $=$ 0.5 eV), $m$ $=$ 1.39 $\mu_{B}$ (U $=$ 1.6 eV),
$m$ $=$ 1.41 $\mu_{B}$ (U $=$ 1.8 eV), and $m$ $=$ 1.43 $\mu_{B}$ (U $=$ 2.0
eV). Note that the plane-wave approximation yields minor uncertainties in
predicting s ome localized quantities, like the magnetic moment and the charge
per atomic site.

On the other hand, for the PS approximation represented with blue and green
dots in Fig. 2, we can observe a non-monotonous behavior as the U-term
increases. Here, we identify the critical U-term of $U_{c}=1.4$ eV. When U
increases above the $U_{c}$, a spontaneous structural transition appears,
changing from Bb21m to Pn21a structure (depicted in the bottom-right panel).
This transition could be associated with the interplay of the SOC+U and
lattice degree of freedom (without spin-orbit effects, this phase does not
appeardanilo ). In panel (a), after $U_{c}$ we note a drop in the anisotropy
value, where for U = 2 eV we recover the value obtained for U $=$ 0\. In panel
(b), the magnetic moment after $U_{c}$ shows two different values associated
with the two crystallographic environments observed (see Fig. 2 bottom-right
panel); note that the average between the average magnetic moment follows the
same trend as before $U_{c}$. A similar behavior appears when we look at the
volume of the RuO6 octahedron in panel (c).

As we can observe in our calculations, we have evidenced that for LDA
approximation, it is necessary to consider $U\geq 0.2$ eV interactions to
describe the ground state correctly for the system. The change of AFM
configuration from a-direction to b-direction could be related to the $m$
effect for lower $m$ $<$ 1.0 $\mu_{B}$ the AFM-$a$ phase became more stable,
and above this value, AFM-$b$ became the ground state along with LDA
approximation. Due to LDA approximation at U $=$ 0 going to a low spin state,
as we explained before with AFM-$a$ ground state, it is not helpful to compare
the band structure of PS and LDA at U $=$ 0 in order to discuss these with the
experimental data, we will start discussing the band structure between PS and
LDA considering U $=$ 0 and 0.5 eV, respectively.

Figure 3: (a) The upper and lower panel shows the band structure along PS and
LDA calculation, respectively, for U $=$ 0 and 0.5 eV in each case. Right
panel: total bands and the decomposed orbital bands, in which the orbitals
have a 25-75% of occupation per orbital. b) Surface Brillouin zone of Bb21m
structure. c) diagram of the crystal field splitting of the Ru 4$d$ states in
the octahedral environment.

The band structure and its projection in atomic orbitals for the PS and LDA
approximations are shown in Fig. 3. We will compare systems with similar
magnetic anisotropy and magnetic moments (see Fig. 2 a-b); for the PS
approximation, we will take a value U $=$ 0 eV, and for the LDA approximation,
we choose a U $=$ 0.5 eV. In Fig. 3, we can see that the band structure and
its projection are very similar, independent of the approximation used. The
system exhibits a metallic character with subtle differences at the Fermi
level for both approximations, mainly around the $\Gamma$ point. The valence
band is mainly composed of the contribution of atoms of Ru and O. The Ru ions
are embedded in an octahedral environment formed by the oxygen atoms; the
octahedral crystal field splits the Ru 4$d$ atomic levels, raising the eg and
decreasing the multiple energies t2g, as seen in the right panel of Fig. 3 and
in the partial density of states displayed in the supplementary file Fig. S1.

In the case of PS, we can observe a metallic character at $\Gamma$; however,
this is not observed in the LDA approximation. We observe a local band gap of
65 meV around the $\Gamma$-point in the LDA approximation. For both
approximations, we identify a Dirac-like behavior at Mx, in agreement with the
ARPES measurementsmarkovic2020electronically . Along Mx-$X$ path, a small band
crossing around the Fermi level is found. The projected band structure around
the Fermi level reveals a dominant contribution (between a 25% to 75% per
orbital) of the dxy, dxz, and dyz. The dx2-y2, dz2 orbitals represent less
than 25% of the projection, and therefore they are not displayed. In both
projections, we can see that around the $\Gamma$, the dominant states belong
to the hybridization between dxy-dyz (below the Fermi level) and dyz-dxz
without hybridization (above of Fermi level). Along My-$\Gamma$ the bands are
mainly composed of dxy and dyz orbitals. The Dirac-like bands are
characterized by dxy, dxz orbitals, and finally, along Mx-$X$ path, the most
dominant contribution became from dxy and dyz states. We highlight that Ru
4dxy orbital is not fully occupied as in the case of the Mott-insulator system
Ca2RuO4, besides a small degree of mixing of the pure orbital basis as given
by the octahedral crystal field at the Ru site can be expected due to
rotations and tilts of the RuO6 octahedra.

In the case of LDA for U $=$ 0, we can see a very different trend of the band
dispersion for the ground state (AFM-$a$); such change is related to the low
magnetic moment of the systems (see Fig. S1).

Figure 4: The Upper and lower panel displays the band structure computed with
PS and LDA approximation, respectively. The black band corresponds to the
Bb21m structure and the color band to the Pn21a phase. Calculation considering
SOC+U+ Lattice degree of freedom.

We systematically study the effect of U-Hubbard repulsion on the band
structure; in Fig. 4 we show the evolution of the band structure for PS and
LDA approximations as $U$-term changes in a range from $0.5$ eV to $2.0$ eV.
In general, for the same U-term value, there are similarities between PS and
LDA; below the Fermi level, the bands around the $\Gamma$ point that are
mainly composed by dyz and dxy orbitals, these orbitals are pushed to lower
energies as the U-term increases, favoring the hybridization between these
orbitals (see Fig. S3). Along Mx to X path, we can observe that these states
are slightly pushed below the Fermi level. These bands are rather modified due
to they are mainly composed of dxy-orbitals being close to integer filling
then exhibiting a less grade of hybridization (see Fig. 3 right panel).
Besides, from My-$\Gamma$ as the Hubbard-U increase, the bands are pushed to
higher energies inducing a band distribution aiding the occupation of the dxz
orbitals, which slightly hybridize with the dxy orbitals just below the Fermi
level (see Fig. S3). Although, for PS approximation, we can distinguish three
ranges of $U$-term values. The first region, between 0 $\leq$ U $\leq$ 1.4 eV,
is characterized by a metallic state (with bands crossing the Fermi level at
$\Gamma$, in the first three plots of the top panel in Fig. 4). In the second
region, for 1.4 $\leq$ U $\leq$ 1.8 eV, we can observe that the electronic-
like band at $\Gamma$ and the Dirac-like bands are pushed to high energies
given to the system a semi-metallic character (due to the gap around $\Gamma$
point). These changes are simultaneous with the phase transition when symmetry
changes from Bb21m to Pn21a, as discussed before. In the last region, U $\geq$
1.8 eV, the Dirac-like bands are suppressed due to the orbital occupation
rearrangement favoring the band hybridization, in which the dyz are shifted to
high energies separated by a gap of 0.18 eV of the hybridized dyz-dxz orbitals
below the Fermi level (see Fig. S4). The suppression of the Dirac-like bands
allows the emergence of an insulator phase with Pn21a structure along with PS
approximation.

When examining the band structure along the LDA approximation (lower panels of
Fig. 4), unlike in the PS case for lower U values (U $<$ 1.4 eV), we do not
observe bands crossing the Fermi level in the vicinity of the $\Gamma$ point
and the Bb21m structure continues to be as the most stable configuration.
Although, as in the case of PS, we can observe that for U $\geq$ 1.4 eV, it
favors the semi-metallic character in which the electronic like-bands (around
My) and the Dirac-like bands (at Mx) are pushed to high energies, besides the
bands get closer to the Fermi level near the My for U $\geq$ 1.8 eV.

###  IV.0.3 Effects of the functional

To determine the mechanism promoting the electronic and structural changes
under PS approximation, we perform an additional calculation to distinguish
between the effects in the electronic properties of the functional from the
effects of the structural phase. For this sake, we take the structure obtained
with PS for U $>$ 1.4 eV (Pn21a phase), and with a frozen geometry, we
calculate the electronic structure within LDA approximation. In Fig. 5, we
compare the band structure for PS and LDA, keeping the geometry of the PS
approximation (Pn21a). In the left panel of Fig. 5, we show the band structure
for U $=$ 1.8 eV, the system with Pn21a (PS approximation) remains with a
semi-metallic character. Instead, under LDA approximation, we can observe that
the Dirac-like bands are suppressed, emerging a band gap of around Mx-point.
Furthermore, for U $=$ 2.0 eV, shown in the right panel of fig. 5, both
approximations revels an insulator character with a band gap at Mx of 0.08 eV
for LDA and 0.18 eV for PS approximation. Finally, we perform the band
structure calculation keeping the Bb21m phase without any lattice degree of
freedom (with the atomic positions given by the experimentyoshida2005 ) along
with PS and LDA approximation. We find that the main role of Hubbard-U
interaction is to control the band occupation around the Fermi level, favoring
the semi-metallic character while changing the Hubbard-U from 0 to 2 eV (see
supplementary. Fig. S5).

###  IV.0.4 Comparison with Experimental Evidence

In this section, we aim to get into contact with the experiments; therefore,
we focus on which approximation and parameters best describe the band
structure and magnetic configuration (AFM-$b$) experimentally measured in CRO
samples.

As discussed in section II, understanding physical properties have been a
significant challenge. There are inconsistencies between different
experimental measurements and theoretical predictions. One of the main
inconsistencies between observations and theory was that band structure
calculation performed by DFT calculations predicted that CRO is a semi-metal
with a high carrier concentration. However, on the one hand, single-crystal
measurements characterized CRO as a low carrier-concentration material on the
verge of a metal-insulator. On the other hand, the ARPES measurements reveal
that CRO displays a low-carrier metallicity but is characterized by a semi-
metallic state with a gapped fermi surface. A summary of contradictions around
CRO can be found in ref. (12).

A previous theoretical study (ref. (27)) proposed a mechanism to model the
band structure and Fermi surface of CRO at low temperatures based on DFT
calculations (along with PS approximation). The suggested mechanism aim to
explain the electronic changes between TS $<$ T $<$ 30 K, taking into account
Hubbard-U in the range 0.5 $<$ U $<$ 1.6 eV, in which at U $=$ 1.4 eV
disappears the electronic-like bands at $\Gamma$ accompanied of a spontaneous
electronic and structural transition from metal to semi-metallic and from
Bb21m to Pn21a structure. In such scenario, the emergence of the semi-metallic
state could be related to the gap evidenced below 45 K in the optical
conductivity spectralee2007pseudogap . Besides, the broken translational
symmetry of the Pn21a structure could allow the change (spin-) density wave
transition as been suggested by experimental research spindensityCa3 , this
model offers a natural explanation that supports the mechanism of density
waves formation agreeing with early experimental proves.

Figure 5: Band structure for Pn21a structure performed with PS (color bands)
and LDA (blue bands) approximations with U $=$ 1.8 eV (left panel) and U $=$
2.0 eV (right panel).

However, recently a series of independent experiments have successfully
reconstructed the Fermi surface of CROmarkovic2020electronically ;
horio2021electronic ; xing2018existence  offering a novel approach to
understanding the evolution of the electronic structure as the temperature
decreases. The experimental evidence shows that CRO presents a semi-metallic
character at low temperaturesmarkovic2020electronically ; horio2021electronic
, in which the band structure displays a gap around $\Gamma$ point, Dirac-like
bands at ($\pi$/a,$\pi$/b), and bands intersecting the Fermi level near the
My. Each workmarkovic2020electronically ; horio2021electronic  reveals a key
aspect to consider to identify the mechanism underneath the CRO phase
transition. On the one hand, it was shown that the reconstruction of the Fermi
surface is incompatible with the transnational symmetry-breaking density
waveshorio2021electronic . On the other hand, a band gap magnitude of $\sim$
10 to 15 meV around the $\Gamma$ point was estimated, comparable to the
pseudogap observed in optical spectroscopy spectra markovic2020electronically
. These observations bring us several constraints to impose on our
calculations.

After a systematic study and comparison with the most recent experiments, our
results indicate that the LDA+U approximation is the one that most faithfully
represents the complex behavior exhibited by the CRO. The LDA+U approximation
successfully identifies the magnetic ground state (AFM-$b$ configuration) and
the occupation bands in agreement with the recent experimental
reportshorio2021electronic .

The subtlest region is to model the band spectrum along the high symmetry
lines, specifically near the $\Gamma$ and the My; this region is quite
sensitive to U value; however, these regions are key, as we explained before
the experiment evidences a pseudogap (around $\Gamma$) and hole pocket (near
the My) and Dirac-like bands (at Mx)horio2021electronic , given a semi-
metallic character to the system. In contrast to the PS approach, LDA can
modulate the band occupation at $\Gamma$-point in the studied range of
U-values. For instance, the LDA+U with U $=$ 0.5 evidence a band gap of 96 meV
at $\Gamma$; this value is in the same order of ARPES
measurementsmarkovic2020electronically . Besides, the hole-like bands close to
My show up for U $>$1.8 eV; however, for this range of U, the band dispersion
around $\Gamma$ is pushed down to lower energies, increasing the size gap.
Additionally, we have identified that the main band contribution around the
Fermi level comes from dxy, dxz and dyz orbitals in which dxy does not cross
the Fermi level for U $>$0.5 eV (see Fig. S3) in agreement with the
experimental reports horio2021electronic . In the case of PS, several problems
are found as U increases: Below U $<$ 1.4 eV appears electronic like bands at
$\Gamma$ point and above U $>$ 1.4 eV due to a spontaneous transition toward
the Pn21a structure, being this structural phase unreported until now.

What is the underlying mechanism for the observed structural transition when
considering the PS approximation with spin-orbit coupling (SOC) and Hubbard-U
repulsion (U $\geq$ 1.4 eV) effects?. We have evidenced that without SOC, the
phase transition does not appear in agreement with the previous report danilo
. The combination of the SOC and the Coulomb interactions induces local
changes in the electronic density. Since the magnetization axis of the non-
collinear density now changes from point to point following the magnetization
of the electron, the PS (PBEsol) belonging to the family of functional
generalized gradient approximations (GGA) is more susceptible to convergence
issues and to deviate from experimental results; triggering spontaneous
symmetry broken to stabilize the system desmarais2019spin .

###  IV.0.5 Linear response determination of U-term

Figure 6: Band structure using U computed through the linear response method:
PS with Pn21a structure for UPS $=$ 2.9 eV (left panel) and LDA with Bb21m
phase using ULDA $=$ 2.3 eV (right panel).

Going one step further, we employ a perturbative approach to determine a value
for the U-Hubbard interaction. However, this method tends to overestimate the
value of U. Note that the DFT+U approach also has limitations and should be
considered a pragmatic way to partially eliminate the self-interaction errors
inherent in DFT. In addition, the effective U-value depends on several factors
that limit their transferability, such as atomic coordination and calculation
parameters (pseudopotentials, basis-set). We compute the U-term through a
constrained DFT-based method developed by Cococcioni and
Gironcolicococcioni2005linear  to compute the non-zero second derivative of
energy concerning the local occupancy of the metal ion, in which U is given
byvaspwiki :

$U\approx\left(\frac{\partial n^{\mathrm{scf}}_{I}}{\partial
V_{I}}\right)^{-1}-\left(\frac{\partial n^{\mathrm{nscf}}_{I}}{\partial
V_{I}}\right)^{-1},$ (1)

where $\partial{n_{I}}/\partial V_{I}$ is the occupation response at the site
$I$, and VI is the orbital energy shift in the $I$ site. The first and second
terms are the self-consistent (scf) and non-self consistent (nscf) solutions.
We consider a perturbation $V$ in the $\pm$ 0.2 eV range. As a result of these
calculation we found that $U_{\textrm{PS}}=2.9$ eV and $U_{\textrm{LDA}}=2.3$
eV for PS and LDA respectively.

We plot the electronic structure for the Hubbard-U values found by the linear
response method. In Fig. 6(a), the PS band structure ($U_{\textrm{PS}}=2.9$
eV) and in Fig. 6(b), the LDA band structure ($U_{\textrm{LDA}}=2.3$ eV). As a
result of our calculation, we find that LDA retains the same Bb21m symmetry,
and the PS approximation keeps the Pn21a phase. In the PS approximation, the
larger U-term value ($U_{\textrm{PS}}=2.9$ eV) produces a band gap of 0.80 eV
at Mx (see Fig. 6 left panel). Note that the band gap increases from the 0.153
eV calculated with U = 2 eV (outermost right panel of Fig. 4). The large U
influence in Pn21a structure induces t2g band distribution mainly at Mx, in
which the band structure change from the hybridized bands dxz-dyz for U $=$
2.0 eV to be composed mainly by dxz orbitals for U = 2.9 eV (see Fig. S4).

In the case of LDA approximation, the system presents a Bb21m symmetry;
however, we can observe that to increase to U $=$ 2.3 eV, the Dirac-like bands
are suppressed, emerging a gap of 0.10 eV at Mx similar to the evidence along
with PS approximation at U $=$ 2.0 eV (see Fig. 4). The observed gap emerges
due to the re-arrangement of t2g orbitals around Mx, as under PS approximation
case. Furthermore, in Bb21m structure the large U induces a decreases of the
dyz orbitals (see Fig. S3) and for U $=$ 2.3 eV is favored the band occupation
of dxy-dxz around Mx below the Fermi level (without hybridization, see Fig. S3
at U $=$ 2.3 eV). Additionally, for larger values of the Hubbard U-term (U $>$
2 eV), the magnetic state along the $b$ direction became unstable, and the
magnetic moment direction goes toward the (111) direction in which mb $>$
ma,c. Fig. 7 shows a schematic representation with the electronic and
structural phase transitions as U increases to 2.6 eV and 3.0 eV for LDA and
PS approximations.

Because the structure of the ground state depends on the functional used (PS
or LDA), a direct comparison between band structures is meaningless. To
compare the band structure for a particular value of the Hubbard-U, we fix the
atomic position and lattice vectors to the Bb21m structure (with the atomic
position given by the experimentyoshida2005  at 8 K). Here, we noticed that
regardless of the functional, the gap at Mx appears for $U=2.3$ eV; however,
the system continues metallic (see Fig. S6). For LDA approximation, an
insulator state appears for U above 2.6 eV. However, under PS approximation,
the band structure favors the metallic character (see Fig. S6). The metallic
behavior tendency could be related to a topological band arising as the U-term
increase.

We should point out that the insulator character originates in the strength of
$U$-term independent of the functional choice and structural phase (in the
presence of lattice degree of freedom). The Pn21a structure favors the
insulator state for U $\geq$ 1.8 eV and U $\geq$ 2.0 eV for LDA and PS
approximation, respectively (see Fig.5), whereas the Bb21m favors the
insulator character for large U-term value, as we can observe for U $=$ 2.3
Fig.6 and Fig. S3 under LDA approximation.

Figure 7: A schematic representation with the electronic and structural phase
transitions as a function of Coulomb repulsion (U) in the presence of spin-
orbit coupling (SOC) and lattice degree of freedom (LDF). The black arrows
symbolize the spin direction of the Ru atoms (blue balls). As U increases, we
can distinguish two main regions: among 0 $<$ U $\leq$ 2 eV in which the spins
are along $b$ direction (AFM-$b$), and for U $>$ 2 eV, the spins turn aligned
through (111) direction, in which the magnetic moment along $b$ direction is
bigger than in $a$-$c$ direction (mb $>>$ ma,c).

The large difference in the U-term computed through the linear approximation
with the values chosen a priory reveals could be related to the DFT self-
interaction errors or $d$-ligand interactions between Ru and O atoms. To
develop a U-Hubbard correction for the CRO system could be necessary to
evaluate d-interactions for first principles, then construct a Hamiltonian
that explicitly contains the d-orbitals and the p-orbitals of the system as
interacting degrees of freedom; however, to solve this in practice could be a
tremendous open challengelong2020evaluating ; yu2014communication .

To enclose our last finding, we suggest that the optimum values of CRO lie
among 0.2 $\leq$ U $\leq$ 2.0 eV (under LDA approximation); in this range, the
band structure agrees with the experimental report. However, for U $>$ 1.4 eV,
the band dispersion is strongly pushed to lower around the $\Gamma$ point. The
established range of U are in agreement with the U implemented to modulate the
electronic structure in several ruthenium family such as: Ca2RuO4 (1.0 $\leq$
U $\leq$ 1.5)keen2021ab , Sr2RuO4 (U $=$ 1.5 eV)iwasawa2010interplay , CaRuO3
(1.0 $<$ U $<$ 2.5 eV)zhong20104  in which both electronic and structural
properties depend of the chosen U value.

##  V final remarks

We have evidenced the strong interplay between the electronic and structural
properties and two different functional choices: LDA and PS approximations. We
have focused on the effects of the electronic properties considering spin-
orbit coupling (SOC), U-Hubbard, and lattice degree of freedom.

Depending on the level of theory, we find contradictory results. On the one
hand, the LDA approximation fails to reproduce the correct ground state of
CRO, predicting the AFM-$a$ solution as the most stable configuration;
however, the LDA+U approximation for U $\geq$ 0.2 eV results in the AFM-$b$
alignment as the ground state solution. On the other hand, PS and PS+U
approximations successfully describe the AFM-$b$ magnetic configuration as the
ground state but present a crystallographic phase change from Bb21m to Pb21a
for U $\geq$ 1.4 eV.

Band structure and electronic stability.

  * •

Along with PS and PS+U, it is possible to distinguish three regions as the
Hubbard-U term increases. The system exhibits a metallic character for U $<$
1.4 eV, characterized by electronic-like bands at $\Gamma$ and Dirac-like
bands at My. For U $>$ 1.4 eV, the Bb21m phase became unstable, and the system
relaxes spontaneously into Pn21a structure, exhibiting a semi-metallic
character (gap around $\Gamma$ point). Finally, for U $\geq$ 2.0 eV, a semi-
metal to insulator phase emerges by opening a gap of 0.18 eV at Mx.

  * •

Through the LDA+U approximation, the Bb21m phase is the most stable
configuration, and the band structure presents a semi-metallic character. For
0.2 $\leq$ U $\leq$ 2.0 eV, we have evidenced a gap around $\Gamma$ (gap size
of 96 meV at $\Gamma$ for U = 0.5 eV) and Dirac-like bands at Mx in good
agreement with the band structure reported by the experiments. However, for
above U $>$ 1.4 eV, the band around $\Gamma$ is pushed down to lower energies
increasing the gap size. Moreover, for a large U value, we revealed an
insulator state at the U value suggested by the perturbative approach (ULDA
$=$ 2.3 eV), in which the spins are aligned along (1,1,1) direction.

  * •

When we fix the crystal symmetry to Pn21a and use the LDA+U approximation for
U $\leq$ 1.7 eV, a semi-metallic state characterizes the system, above this
value, the Dirac-like bands are suppressed, in line with the PS approximation
results. Here, a 20 and 80 meV gap appears at Mx for U $=$ 1.8 and 2.0 eV,
respectively. To fix the crystal symmetry to Bb21m, for the lower U-values
(0-2 eV), we identify that the band structure response upon changes on the
Hubbard-U is to modify the gap at $\Gamma$-point. For higher U values (2-2.7
eV), the system has different electronic states in which the system goes
toward an insulator –in the LDA case– and metallic states –in the PS case– for
U $>$ 2.3 eV.

We conclude that the exchange and correlation functional affects the band
occupation mainly at $\Gamma$-point. On the one hand, electronic-like bands
emerge for the PS approximation. On the other hand, a gap bap appears for the
LDA approximation. Additionally, the primary effect of the Coulomb repulsion
(expressed as U-term) is to modify the band occupation around the Fermi level.
For the PS approximation, the electronic-like bands found at $\Gamma$
disappear above for U $\geq$ 1.4 eV independent of the lattice degree of
freedom (compare Fig. 4 and Fig. S5). A spontaneous symmetry change from Bb21m
to Pn21a structure arises U $\geq$ 1.4 eV (only in PS approximation).
Furthermore, Pn21a structure is strongly sensitive at U; for U $\geq$ 1.8 eV
(LDA) and U $\geq$ 2.0 eV (PS), an insulator phase appears due to the opening
gap at My. This insulator state is independent of the functional approximation
(see Fig. 5).

From these findings, we can give keys to model the electronic structure of CRO
system, establishing the effect of Coulomb repulsion and lattice degree of
freedom for two different functional nature. Here we have evidenced that CRO
is very sensitive to these parameters, in which, as a function of U is
possible to get a semi-metallic or insulator state. In this context, we
suggest that to continue exploring the electronic state of CRO system is
necessary to take into account the interplay among the electronic degrees of
freedom, structural symmetry, and functionals of different nature.

##  VI acknowledgment

The authors would like to thank Igor Markovic and Jose Mejía for the fruitful
discussions. We acknowledge the Max Planck Computing Data Facility for
bringing the technical support and Dr. Claudia Felser for providing the VASP
license. Also, AL thanks Dr. Danilo Puggioni for helping with the calculation
setup and discussions in the early stage of this project. JWG acknowledges
financial support from FONDECYT: Iniciación en Investigación 2019 grant N.
11190934 (Chile) and USM complementary research grant.

## References

  * [1] Anjan Soumyanarayanan, Nicolas Reyren, Albert Fert, and Christos Panagopoulos.  Emergent phenomena induced by spin–orbit coupling at surfaces and interfaces.  Nature, 539(7630):509–517, 2016.
  * [2] William Witczak-Krempa, Gang Chen, Yong Baek Kim, and Leon Balents.  Correlated quantum phenomena in the strong spin-orbit regime.  Annu. Rev. Condens. Matter Phys., 5(1):57–82, 2014.
  * [3] K Kośmider, Jhon W González, and Joaquın Fernández-Rossier.  Large spin splitting in the conduction band of transition metal dichalcogenide monolayers.  Physical Review B, 88(24):245436, 2013.
  * [4] Rasa Rejali, David Coffey, Jeremie Gobeil, Jhon W González, Fernando Delgado, and Alexander F Otte.  Complete reversal of the atomic unquenched orbital moment by a single electron.  npj Quantum Materials, 5(1):1–7, 2020.
  * [5] Jhon W González, Tomás Alonso-Lanza, Fernando Delgado, Faustino Aguilera-Granja, and Andrés Ayuela.  Complex magnetic orders in small cobalt–benzene molecules.  Physical Chemistry Chemical Physics, 19(22):14854–14860, 2017.
  * [6] G Cao, S McCall, JE Crow, and RP Guertin.  Observation of a metallic antiferromagnetic phase and metal to nonmetal transition in Ca3Ru2O7.  Physical Review Letters, 78(9):1751, 1997.
  * [7] E Ohmichi, Y Yoshida, SI Ikeda, N Shirakawa, and T Osada.  Colossal magnetoresistance accompanying a structural transition in a highly two-dimensional metallic state of Ca3Ru2O7.  Physical Review B, 70(10):104414, 2004.
  * [8] N Kikugawa, A Rost, F Baumberger, NJC Ingle, MA Hossain, W Meevasana, KM Shen, DH Lu, A Damascelli, AP Mackenzie, et al.  Ca3Ru2O7: Electronic instability and extremely strong quasiparticle renormalisation.  Journal of Magnetism and Magnetic Materials, 310(2):1027–1029, 2007\.
  * [9] Naoki Kikugawa, Andreas Winfried Rost, Clifford William Hicks, Andrew John Schofield, and Andrew Peter Mackenzie.  Ca3Ru2O7: density wave formation and quantum oscillations in the Hall resistivity.  Journal of the Physical Society of Japan, 79(2):024704, 2010.
  * [10] Gang Cao and Lance E Delong.  Frontiers of 4D-and 5D-transition Metal Oxides, (Cap.1-pag.4).  World Scientific, 2013.
  * [11] Igor Marković, Matthew D Watson, Oliver J Clark, Federico Mazzola, Edgar Abarca Morales, Chris A Hooley, Helge Rosner, Craig M Polley, Thiagarajan Balasubramanian, Saumya Mukherjee, et al.  Electronically driven spin-reorientation transition of the correlated polar metal Ca3Ru2O7.  Proceedings of the National Academy of Sciences, 117(27):15524–15529, 2020.
  * [12] Hui Xing, Libin Wen, Chenyi Shen, Jiaming He, Xinxin Cai, Jin Peng, Shun Wang, Mingliang Tian, Zhu-An Xu, Wei Ku, et al.  Existence of electron and hole pockets and partial gap opening in the correlated semimetal Ca3Ru2O7.  Physical Review B, 97(4):041113, 2018.
  * [13] M Horio, Q Wang, V Granata, KP Kramer, Y Sassa, S Johr, D Sutter, A Bold, L Das, Y Xu, et al.  Electronic reconstruction forming a C2-symmetric Dirac semimetal in Ca3Ru2O7.  npj Quantum Materials, 6(1):1–7, 2021.
  * [14] DA Sokolov, N Kikugawa, T Helm, H Borrmann, U Burkhardt, R Cubitt, JS White, E Ressouche, M Bleuel, K Kummer, et al.  Metamagnetic texture in a polar antiferromagnet.  Nature Physics, 15(7):671–677, 2019.
  * [15] Shiming Lei, Mingqiang Gu, Danilo Puggioni, Greg Stone, Jin Peng, Jianjian Ge, Yu Wang, Baoming Wang, Yakun Yuan, Ke Wang, et al.  Observation of quasi-two-dimensional polar domains and ferroelastic switching in a metal, Ca3Ru2O7.  Nano letters, 18(5):3088–3095, 2018.
  * [16] Hengdi Zhao, Hao Zheng, Jasminka Terzic, Wenhai Song, Yifei Ni, Yu Zhang, Pedro Schlottmann, and Gang Cao.  Lattice flexibility in Ca3Ru2O7: Control of electrical transport via anisotropic magnetostriction.  Physical Review B, 104(12), Sep 2021.
  * [17] Erik MJ Johansson, Michael Odelius, Stefan Plogmaker, Mihaela Gorgoi, Svante Svensson, Hans Siegbahn, and Håkan Rensmo.  Spin- orbit coupling and metal- ligand interactions in Fe (II), Ru (II), and Os (II) complexes.  The Journal of Physical Chemistry C, 114(22):10314–10322, 2010\.
  * [18] Martin Srnec, Jakub Chalupský, Miroslav Fojta, Lucie Zendlová, Ludek Havran, Michal Hocek, Mojmir Kyvala, and Lubomír Rulísek.  Effect of spin-orbit coupling on reduction potentials of octahedral ruthenium (II/III) and osmium (II/III) complexes.  Journal of the American Chemical Society, 130(33):10947–10954, 2008\.
  * [19] Yoshiyuki Yoshida, Shin-Ichi Ikeda, Hirofumi Matsuhata, Naoki Shirakawa, CH Lee, and Susumu Katano.  Crystal and magnetic structure of Ca3Ru2O7.  Physical Review B, 72(5):054412, 2005.
  * [20] F Baumberger, NJC Ingle, N Kikugawa, MA Hossain, W Meevasana, RS Perry, KM Shen, DH Lu, A Damascelli, A Rost, et al.  Nested Fermi surface and electronic instability in Ca3Ru2O7.  Physical Review Letters, 96(10):107601, 2006.
  * [21] JS Lee, SJ Moon, BJ Yang, Jaejun Yu, U Schade, Y Yoshida, S-I Ikeda, and TW Noh.  Pseudogap dependence of the optical conductivity spectra of Ca3Ru2O7: A possible contribution of the orbital flip excitation.  Physical Review Letters, 98(9):097403, 2007.
  * [22] G Cao, L Balicas, Y Xin, JE Crow, and CS Nelson.  Quantum oscillations, colossal magnetoresistance, and the magnetoelastic interaction in bilayered Ca3Ru2O7.  Physical Review B, 67(18):184405, 2003.
  * [23] JF Karpus, CS Snow, R Gupta, H Barath, SL Cooper, and G Cao.  Spectroscopic study of the field-and pressure-induced phases of the bilayered ruthenate Ca3Ru2O7.  Physical Review B, 73(13):134407, 2006.
  * [24] Matteo Cococcioni and Stefano De Gironcoli.  Linear response approach to the calculation of the effective interaction parameters in the LDA+U method.  Physical Review B, 71(3):035105, 2005.
  * [25] David J Singh and Sushil Auluck.  Electronic Structure and Bulk Spin-Valve Behavior in Ca3Ru2O7.  Physical Review Letters, 96(9):097203, 2006.
  * [26] Guo-Qiang Liu.  Mott transition and magnetic anisotropy in Ca3Ru2O7.  Physical Review B, 84(23):235137, 2011.
  * [27] Danilo Puggioni, M Horio, J Chang, and James M Rondinelli.  Cooperative interactions govern the fermiology of the polar metal Ca3Ru2O7.  Physical Review Research, 2(2):023141, 2020.
  * [28] Georg Kresse and Jürgen Furthmüller.  Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set.  Computational materials science, 6(1):15–50, 1996.
  * [29] John P Perdew, Adrienn Ruzsinszky, Gábor I Csonka, Oleg A Vydrov, Gustavo E Scuseria, Lucian A Constantin, Xiaolan Zhou, and Kieron Burke.  Restoring the density-gradient expansion for exchange in solids and surfaces.  Physical Review Letters, 100(13):136406, 2008.
  * [30] David M Ceperley and Berni J Alder.  Ground state of the electron gas by a stochastic method.  Physical Review Letters, 45(7):566, 1980.
  * [31] GM Dongho Nguimdo and Daniel P Joubert.  A density functional (PBE, PBEsol, HSE06) study of the structural, electronic and optical properties of the ternary compounds AgAlX2 (X= S, Se, Te).  The European Physical Journal B, 88(5):1–10, 2015.
  * [32] Igor Markovic.  Interplay of spin-orbit coupling and crystal symmetries in the electronic structures of NbGeSb and Ca3Ru2O7. Chapter 5 pages 83-88. PhD thesis, University of St Andrews, 2020.
  * [33] Jacques K Desmarais, Jean-Pierre Flament, and Alessandro Erba.  Spin-orbit coupling from a two-component self-consistent approach. ii. non-collinear density functional theories.  The Journal of Chemical Physics, 151(7):074108, 2019.
  * [34] Calculate U for LSDA+U.  https://www.vasp.at/wiki/index.php/Calculate_U_for_LSDA+U.
  * [35] Olivia Y Long, Gopalakrishnan Sai Gautam, and Emily A Carter.  Evaluating optimal U for 3d transition-metal oxides within the SCAN+U framework.  Physical Review Materials, 4(4):045401, 2020.
  * [36] Kuang Yu and Emily A Carter.  Communication: Comparing ab-initio methods of obtaining effective U parameters for closed-shell materials.  The Journal of Chemical Physics, 140(12):121105, 2014.
  * [37] Harry DJ Keen, Stephen R Julian, and Andreas Hermann.  Ab initio study of pressure-induced structural and electronic phase transitions in Ca2RuO4.  Physical Review B, 104(8):085143, 2021.
  * [38] H Iwasawa, Y Yoshida, I Hase, S Koikegami, H Hayashi, J Jiang, K Shimada, H Namatame, M Taniguchi, and Y Aiura.  Interplay among coulomb interaction, spin-orbit interaction, and multiple electron-boson interactions in Sr2RuO4.  Physical Review Letters, 105(22):226406, 2010.
  * [39] GH Zhong, YL Li, Z Liu, and HQ Lin.  4$d$ electronic and magnetic characteristics in postperovskite CaRuO3.  Journal of Applied Physics, 107(9):09E102, 2010.

Supplemental Material: Ca3Ru2O7: Interplay among degrees of freedom and the
role of the exchange-correlation

Andrea León,1,2 J. W. González,3, H. Rosner1

1 Max Planck Institute for Chemical Physics of Solids, Dresden, Germany.

2 Faculty for Chemistry, TU Dresden, Bergstrasse 66c, 01069 Dresden, Germany.

3 Departamento de Física, Universidad Técnica Federico Santa María, Casilla
Postal 110V, Valparaíso, Chile.

##  SI Equilibrium volume

Table S1 shows the lattice parameter in Å, Ru-O length in Å, octahedra volume
and equilibrium volume in Å3, PBEsol (PS), LDA approximation, and the values
given by the experiment. Here, we can observe that LDA and PS approximation
underestimate the volume a 4% and 1.44%, respectively. Table S2 shows the
energy difference between AFM-$a$ and AFM-$b$ magnetic configuration per Ru
atom, and the magnetic moment per Ru atom ($m$) for each magnetic phase
(calculation considering U $=$ 0).

| a | b | c | Ru-O | V(RuO6) | V0
---|---|---|---|---|---|---
LDA | 5.292 | 5.458 | 19.427 | 1.965 | 10.09 | 556.72
PS | 5.328 | 5.533 | 19.390 | 1.986 | 10.44 | 571.63
Exp.[19] | 5.367 | 5.535 | 19.521 | 1.992 | 10.53 | 580.04
Table S1: Equilibrium lattice parameter computed with LDA and PBEsol (PS) appr. The experimental values are given by [19]. Appr. |  $\Delta$E (meV/Ru) |  $m$ ($\mu_{B}$)
---|---|---
LDA | -3.644 | 0.28
PS | 0.44625 | 1.34
Exp[19] | AFM-$b$ | 1.8
Table S2: $\Delta$E is the energy difference between AFM-$a$ and AFM-$b$
magnetic configuration per Ru atom and $m$ the magnetic moment per Ru atom.

##  SII Band structure

Fig. S1 shows the density of states (DOS) along LDA and PS calculation at U
$=$ 0 for PS and U $=$ 0.5 for LDA at the experimental volume.

Figure S1: Density of states (DOS) for AFM-$b$ configuration along PS and LDA
approximation at U $=$ 0 and U $=$ 0.5 eV, respectively. The Fermi level is
settled at E $=$ 0.

Fig. S2 (a) and (b) show the band structure for LDA calculation with AFM-$b$
and AFM-$a$ configuration considering U = 0 eV at the experimental volume.

Figure S2: (a)-(b) Band structure for LDA approximation with AFM-$b$ and
AFM-$a$ configuration.

In the following the displayed $d$-partial band structure have a percentage of
occupation between 50-75% per each orbital; these correspond to dxy, dxz and
dyz, the dx2-y2 and dz2 represent less than 25% of occupation then they are
not shown. The back point displays the hybridized t2g orbitals representing
between 0-25% of the occupation per band (Fig. S3, S4, S6).

Fig. S3 shows the electronic band structure performed in the phase Bb21m for
several U values. along with LDA approximation. Here we can observe that the
$d$-electronic band distribution changes slightly to increase the U-term.
Along My-$\Gamma$ bands are pushed to high energies inducing a band
distribution, helping the occupation of the dxz orbitals, which slightly
hybridize with the dxy orbitals just below the Fermi level. Around $\Gamma$
point, we can observe that the dxy contribution decreases, and these states
became more hybridized with dyz orbitals below the Fermi level. Additionally,
these bands are strongly pushed toward lower energies. At Mx the Dirac-like
bands move towards the Fermi level, favoring the occupation of dyz orbitals.
Finally, from Mx-X, we can observe that these states are slightly pushed below
the Fermi level. These bands are rather modified due to they are mainly
composed of dxy-orbitals being close to integer filling then exhibiting a less
grade of hybridization. Furthermore, we have included the band structure at U
$=$ 2.3 eV (U computed using the perturbative approach [24]). For this U
value, we can observe that a band gap of 0.25 eV emerges at Mx arising the
transition from semi-metal to insulator. Together with the transition, we can
observe a $d$-orbital re-ordering mainly at Mx in which the dominant bands are
mainly dxz-dxy.

Figure S3: Total and $d$-partial band and structure for AFM-$b$ configuration
along LDA approximation for U = 0.5, 1.3, 1.4, 1.8, 2.0 and 2.3 eV.

Figure S4: Total and partial band structure along PS approximation for U $=$
2.0 and 2.9 eV.

Fig. S4 shows the total bands and partial $d$-band structure along with PS
approximation for the U value in which happens the electronic transition from
semi-metal to insulator (U $=$ 2.0 eV) with Pn21a structure with a gap of 0.18
eV at Mx. Here, we can observe that around Mx appears a grade of hybridization
between dxz and dyz orbitals. To increase the U to 2.9 eV (U computed using
the perturbative approach), we can observe that the gap increases to 0.80 eV
at Mx. The bands around this point are mainly composed of dxz orbitals.

Figure S5: Band structure considering Bb21m structure for U $=$ 0.5, 1.4, 1.8
and 2.0 eV. Black and orange lines correspond to LDA and PS approximation.
Calculation without considering lattice degrees of freedom.

Fig. S5 shows the electronic band structure performed in the Bb21m structure
(with position reported by the experiment at 8K[19]) for several U values
without considering the lattice degree of freedom. Here we can observe that
the role of U is to control the band occupation mainly around the $\Gamma$
point (above and below the Fermi level). Furthermore, we evidence that without
degree of freedom, the band opening at Mx at U $=$ 2.0 eV does not emerge for
PS approximation (see Fig. 4 main text).

Fig. S6 shows the total band and partial $d$-band along with PS and LDA
approximation, with U $=$ 2.3 eV, considering Bb21m structure without
considering lattice degrees of freedom. Here we can observe that for both PS
and LDA approximation emerges, a gap at Mx; however, the system continues in a
metallic state. To increase U to 2.6 eV, we can observe that the system became
an insulator in the case of LDA. Instead, along PS, the system goes toward a
metallic state.

Figure S6: Band structure along PS and LDA approximation with Bb21m structure
for U $=$ 2.3 eV and 2.6 eV (without lattice degree of freedom).