# Simulating Calibration and Beam Systematics for future CMB space mission
with TOAST package

Giuseppe Puglisi  Computational Cosmology Center, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  Space Sciences Laboratory at University
of California, 7 Gauss Way, Berkeley, CA 94720  Department of Physics,
University of California, Berkeley, CA, USA 94720  Reijo Keskitalo
Computational Cosmology Center, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  Space Sciences Laboratory at University of
California, 7 Gauss Way, Berkeley, CA 94720  Department of Physics, University
of California, Berkeley, CA, USA 94720  Ted Kisner  Computational Cosmology
Center, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  Space
Sciences Laboratory at University of California, 7 Gauss Way, Berkeley, CA
94720  Department of Physics, University of California, Berkeley, CA, USA
94720  Julian D. Borrill  Computational Cosmology Center, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA  Space Sciences Laboratory at
University of California, 7 Gauss Way, Berkeley, CA 94720  Department of
Physics, University of California, Berkeley, CA, USA 94720

###### Abstract

We address in this work the instrumental systematic errors that can
potentially affect the forthcoming and future Cosmic Microwave Background
experiments aimed at observing its polarized emission. In particular, we focus
on the systematics induced by the beam and calibration, which are considered
the major sources of leakage from total intensity measurements to
polarization. We simulated synthetic data sets with TOAST, a publicly
available simulation and data analysis package. We also propose a mitigation
technique aiming at reducing the leakage by means of a template fitting
approach. This technique has shown promising results reducing the leakage by 2
orders of magnitude at the power spectrum level when applied to a realistic
simulated data set of the LiteBIRD satellite mission.

cosmology, cosmic microwave background, systematics uncertainties,
simulations, surveys

##  1 Introduction

The Cosmic Microwave Background (CMB) emission provides one of the most
favorite channels for probing the Universe at large scales. In particular, CMB
forthcoming experiments are increasingly focusing on measuring the primordial
CMB polarization B-mode ( expected to peak at the degree scales (Seljak and
Zaldarriaga, 1997; Wayne and White, 1997)) which is directly linked to a
stochastic gravitational wave background emitted at the time of
inflation(Guth, 1981; Starobinsky, 1982). The amplitude of the primordial
B-modes is quantified by the tensor-to-scalar ratio, $r$. To date, the best
constraints on $r$ have recently been set to $r<0.07$ by BICEP/Keck
Collaboration et al. (2018); BICEP/Keck and SPTpol Collaborations et al.
(2020).

A critical piece of this framework will necessarily be the ability of
generating synthetic mission data sets of sufficient realism, both in their
complexity and their size, to be truly representative of the data set that
would be gathered by a given mission configuration. In this research note, we
aim at showing preliminary results of simulations encoding systematic effects
injected with the Time-Ordered Astrophysics Scalable Tools (TOAST) package,
namely calibration errors, gain fluctuations and optical beam asymmetries.

Figure 1:  Angula power spectra of B-modes. Spectra obtained from the residual
maps encoding noiseless simulations with (dashed thick blue) calibration
systematics and (solid red) mitigated systematics with the template fitting
procedure outlined in Section2.1. Spectra obtained with noisy simulations
encoding no-mitigation and mitigation cases are shown respectively in (dashed
thick orange ) and (solid green). As a reference, we include the theoretical
angular spectra for primordial $r=0.01$ (dashed thin purple ) and $r=0.001$
(solid thin purple) and from the gravitational lensing B-modes (thick solid
grey).

##  2 Calibration systematics

We consider two source of calibration systematics: calibration uncertainties
and a long term fluctuation of the calibration gain (commonly referred as
_gain drift_) due to thermal instabilities in the focal-plane. The former
essentially quantifies the error affecting the calibration measurements
performed between two consecutive observations, e.g. for the LiteBIRD
satellite, (Sugai et al., 2020), we assume the calibration to happen every 24
hours. Calibration errors can be thus simulated by drawing Gaussian random
values centered around $1$ and with a width (hereafter assumed to be $1\%$).

We also simulate the instrumental thermal instability which can lead into
changes of the gain calibration during one observation time scale, injected as
a slowly varying drift with a $1/f$ frequency spectrum, typical for the CMB
measurements. Moreover, the amplitude of the drift fluctuations can be
directly linked to an effective temperature related to fluctuations in the
focal-plane unit.

The gain drift signal is parametrized by the following power spectrum density:

$PSD(f)=\delta_{g}\left(\frac{f_{knee}}{f}\right)^{\alpha},$

with $\alpha=1$, $f_{knee}=20$ mHz, $\delta_{g}=10\mu\mathrm{K}$.

Both calibration errors and gain drift are simulated independently for each
detector and observation.

###  2.1 Template fitting mitigation

In order to mitigate the systematic residuals induced by calibration
systematics described in the previous Section, we outline below a technique
aimed at further reduce the leakage.

We assume that both calibration errors and gain drifts can be approximated as
a linear combination of _Legendre polynomials_ up to a certain order,
$n_{poly}\lesssim 4$, within each observation:

$g(t)=\sum_{i}^{n_{poly}}\hat{a}_{i}\mathcal{L}^{(i)}(t).$ (1)

The template fitting mitigation proposed in this study consists in estimating
the weights $\hat{a}$ that minimize a $\chi^{2}$ problem provided we have an
approximated estimate of the underlying signal via a _template map_ (a similar
approach can be found in (Planck Collaboration, 2020; Keihänen et al., 2010)):

$(F^{t}C_{n}^{-1}ZF)\hat{a}=F^{t}C_{n}^{-1}Zd,$ (2)

with $C_{n}$ being the noise covariance matrix, $F$ the matrix built from the
template signal, $Z$ encodes the scanning strategy informations and $d$ the
time-ordered data acquired by the detector encoding astrophysical signal,
noise and (eventually) systematics errors. The linear problem in eq.2 is
solved iteratively by means of the Preconditioned Conjugate Gradient (PCG)
algorithm.

Note that this mitigation relies on how much the template signal is a good
approximation of the simulated one $d$. Furthermore, the more _redundant_ the
scanning strategy, i.e. the more a give line of sight is observed with
different scanning orientation, the higher the signal-to-noise will be in
determining the amplitudes $\hat{a}$.

###  2.2 LiteBIRD Calibration systematic simulations

We implement in TOAST the injection of calibration errors and gain drifts as
well as the template fitting mitigation111For further details see
https://github.com/hpc4cmb/toast/tree/master/src/toast, DOI Zenodo
10.5281/zenodo.4270476. We then perform simulations for one LiteBIRD frequency
channel, i.e. 48 polarization sensitive detectors 140 GHz observing for 1 year
with the nominal observing strategy. We adopt the parameter values for
calibration errors and gain drifts as reported in this note and inject both
calibration systematics.

The simulated signal encodes: i) the CMB polarized anisotropies (including
Solar dipole), ii) realistic polarized and unpolarized Galactic emissions at
the sub-mm wavelengths. On the other hand, we built a template signal encoding
only unpolarized Galactic emission and the dipole, to assess the quality of
the mitigation in the presence of an incomplete template signal.

The output maps are estimated with libmadam map-maker (Keihänen et al., 2010).
The residuals in Figure 1 are only due to the systematic errors pre- and post-
mitigation, the other set includes also instrumental noise. We note that the
template fitting procedure is able to reduce the leakage by two orders of
magnitude at the angular power spectrum level indicating that even without a
representative template map the LiteBIRD scanning strategy is optimized in
such a way that we are able to estimate with good signal-to-noise ratio the
gain amplitude. Moreover, as expected, this procedure is not affected by
instrumental noise. In fact, we do not see any effect in the post-mitigation
power spectra with noisy simulations.

##  3 Conclusions

We implemented into TOAST package several modules to inject systematic
uncertainties due to beam and calibration. Although not mentioned in this
work, TOAST can also simulate frequency bandpass mismatches, correlated noise
and half-wave plate non-idealities, making it a promising simulation and data
analysis framework for the forthcoming CMB experiments (both ground-based and
space experiments). We propose a mitigation technique that aims at further
reducing the leakage induced by gain drifts and calibration errors. The
template fitting approach can be thus extended to other kind of systematics
effects, e.g. bandpass mismatches, beam imperfections.

We acknowledge financial support from the APRA grant: “Overcoming Systematic
Effects in Cosmic Microwave Background Satellite Missions”, Grant number
80NSSC19K0697. These calculations were performed on the NERSC supercomputer
facility.

## References

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# Emergence of a potential charge disproportionated insulating state in SrCrO3

Alberto Carta  Anwesha Panda  Claude Ederer  Materials Theory, ETH Zürich,
Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland

###### Abstract

We use a combination of density functional theory (DFT) and dynamical mean-
field theory (DMFT) to investigate the potential emergence of a charge-
disproportionated insulating phase in SrCrO3, whereby the Cr cations
disproportionate according to $3\text{Cr}^{4+}\rightarrow
2\text{Cr}^{3+}+\text{Cr}^{6+}$ and arrange in ordered planes perpendicular to
the cubic [111] direction. We show that the charge disproportionation couples
to a structural distortion where the oxygen octahedra around the nominal Cr6+
sites contract, while the octahedra surrounding the Cr3+ sites expand and
distort. Our results indicate that the charge-disproportionated phase can be
stabilized for realistic values of the Hubbard $U$ and Hund’s $J$ parameters,
but this requires a scaling of the DFT+DMFT double counting correction by at
least 35 %. The disproportionated state can also be stabilized in DFT+
$\mathcal{U}$ calculations for a specific magnetic configuration with
antiparallel magnetic moments on adjacent Cr3+ sites and no magnetic moment on
the Cr6+ sites. While this phase is higher in energy than other magnetic
states that can arise in SrCrO3, the presence of an energy minimum suggests
that the charge disproportionated state represents a metastable state of
SrCrO3.

##  I Introduction

The electronic properties of many materials with partially filled $d$ or $f$
shells are characterized by strong electronic correlations, which can lead to
peculiar phenomena such as, e.g., metal-insulator transitions,
superconductivity, or non-Fermi-liquid behavior [1, 2, 3]. The basic low-
energy physics of these systems can often be understood in terms of an
effective multi-orbital Hubbard model, where electronic correlations emerge as
a consequence of the local Coulomb repulsion $U$ and the Hund’s interaction
$J$ [4, 5, 6]. A strong $U$ generally favors the Mott-insulating state, where
the electron number fluctuations on any given site away from the nominal
integer filling are suppressed by the strong Coulomb repulsion [7, 1, 8]. The
Hund’s coupling $J$ also plays an important role by favoring a local spin
polarization. While, at half-filling, $J$ cooperates with $U$ in the formation
of the Mott insulator, away from half-filling, $J$ can compete with $U$ and
give rise to strongly correlated metallic phases [9, 10, 6].

For strong enough $J$ and not too large $U$, the Hund’s coupling can also
favor the formation of a charge disproportionated insulating phase, where the
initially symmetry-equivalent correlated sites separate into two or more
sublattices with different valences [11, 12, 13, 14, 15]. Such charge
disproportionation can be observed in several transition metal perovskite
oxides, for instance, in rare-earth nickelates [16, 17], CaFeO3 [18], and
other ferrites [19, 20]. The charge disproportionation is usually accompanied
by a structural “breathing” distortion of the oxygen octahedra surrounding the
transition metal cations, where the oxygen octahedra surrounding the more
occupied cations expand and those around the less occupied sites contract [16,
17, 21]. This strong coupling between the electronic and structural degrees of
freedom can play a crucial role in stabilizing the disproportionated state
[12, 22, 23, 13, 24, 15, 25].

The physics of charge disproportionation in the rare-earth nickelates, and to
some extent also in CaFeO3, can be understood in terms of an effective model
considering only the $e_{g}$ orbitals, which disproportionate according to
$2e_{g}^{1}\rightarrow e_{g}^{2}+e_{g}^{0}$ [12]. While for the nickelates,
the lower-lying $t_{2g}$ orbitals are completely filled, they are half-filled
in the case of CaFeO3, but can to some approximation be treated as localized
$S=3/2$ spins [15]. In both cases, the strong hybridization between transition
metal $d$ and O $p$ states, and the small or even negative charge transfer
energy in materials containing late $3d$ cations with high oxidation states
have been proposed as being favorable, or even crucial, for the charge
disproportionation [26, 27, 28].

A recent study of the three-orbital Hubbard model by Isidori et al. [14]
suggests that in principle similar physics can also emerge in systems with
partially filled $t_{2g}$ orbitals. In particular, for an average filling of
two electrons per site, a disproportionation according to $3d^{2}\rightarrow$
${2d^{3}+d^{0}}$ can occur. However, it is unclear whether specific materials
in the required parameter regime can be identified. Furthermore, the model
studied in Ref. [14] was based on a featureless density of states without a
specific underlying lattice structure. So the question on how the spatial
arrangement of inequivalent sites might affect the stability of the
corresponding charge disproportionated state remains open.

The perovskite chromates, $A$CrO3, where $A$ can be Ca, Sr, Ba or Pb, contain
nominal Cr4+ with two electrons in the ${t_{2g}}$ manifold. This and the high
oxidation state of the Cr makes this family of compounds ideal candidates to
explore a potential charge disproportionation in the ${t_{2g}}$ shell. In the
case of PbCrO3, there have indeed been reports of charge disproportionation
accompanied by insulating behavior [29, 30, 31]. However, the fact that Pb can
also disproportionate into Pb2+ and Pb4+, where Pb2+ hosts a steorechemically
active lone-pair, makes this case significantly more complex and results in
partially conflicting reports [32, 33].

For the Ba compound, there is experimental evidence showing insulating
behavior in thin films [34], which, however, has been suggested to arise as a
consequence of orbital and magnetic order, either in combination with a Jahn-
Teller structural distortion or driven by spin-orbit coupling [35, 36].

Partially conflicting experimental reports are available regarding the
metallic or insulating character of both the Ca and Sr compounds [37, 38, 39,
40, 41, 42], and a strain-induced metal-insulator transition has been reported
for SrCrO3 thin films [43]. In a previous work [44], we have shown that
tensile strain promotes a Jahn-Teller distorted and orbitally-ordered
insulating state in SrCrO3, which is compatible with this reported strain-
induced metal-insulator transition.

Here we explore whether charge-disproportionation can provide an alternative
mechanism for the formation of an insulating state in SrCrO3. We mainly use
density functional theory in combination with dynamical mean-field theory
(DFT+DMFT) [45, 46, 47, 48, 49], which provides a straightforward way to
analyze the effect of local Hubbard-type interactions in the paramagnetic
state, and also allows for a systematic comparison with the model study from
Ref. [14]. We find that a charge disproportionated insulating state can indeed
be stabilized in SrCrO3 for realistic interaction parameters, but its
stability depends on a subtle balance between the local interaction treated in
DMFT and the DFT response to the DMFT charge-density correction, and requires
a scaling of the double counting correction for the local interaction included
both at the DFT and DMFT level. Further DFT+ $\mathcal{U}$ calculations
suggest that the charge disproportionated state is not the ground state of the
system, with different magnetically and orbitally ordered states exhibiting
lower energies. Nevertheless, charge-disproportionation in SrCrO3 could
potentially be stabilized by appropriate tuning of suitable parameters such
as, e.g., strain, pressure, ionic substitution, etc.

The remainder of this work is organized as follows: in Sec. II we first
specify the specific spatial arrangement of inequivalant sites and the
resulting unit-cell used to represent the charge disproportionated state, then
we describe the corresponding structural breathing mode distortion. After
that, we give a brief summary of the DFT+DMFT method and specify all
parameters used in our calculations. We then start Sec. III by discussing
results of simple one-shot DFT+DMFT calculations, followed by an analysis of
the significant changes introduced by charge-self consistency. We subsequently
explore the interplay of both lattice distortion and the DFT+DMFT double-
counting (DC) correction on the stability of the charge disproportionated
insulator. Finally, we also present results of DFT+ $\mathcal{U}$
calculations, which allow to perform full structural relaxations and provide
alternative insights in the stability of charge disproportionation in SrCrO3.

##  II  Methods

###  II.1 Choice of unit cell and distortion mode

Figure 1: (a) Structural arrangement of inequivalent Cr sites in our unit cell
allowing for the emergence of charge disproportionation into nominal $d^{3}$
and $d^{0}$ Cr sites in otherwise cubic SrCrO3. (b) Depiction of the
$\Lambda_{1}$ distortion: the octahedra around the nominal $d^{0}$ sites
contract and the octahedra around the $d^{3}$ sites expand and distort. The
thin black lines in both panels indicate the hexagonal unit cell containing
three formula units of SrCrO3.

To explore the possibility for charge disproportionation in SrCrO3, we need to
use a unit cell that can accommodate possible arrangements of inequivalent Cr-
sites with a ratio of two nominal Cr3+ ($d^{3}$) sites per one nominal Cr6+
($d^{0}$) site. The optimal spatial arrangement of such a disproportionated
phase is not clear a priori. Here, we use the arrangement shown in Fig. 1(a),
which consists of two potential Cr $d^{3}$ layers followed by one potential Cr
$d^{0}$ layer arranged perpendicular to the cubic $[111]$ direction and
repeated periodically. This results in a tripling of the conventional cubic
perovskite cell, which can be described in a hexagonal setting with the cubic
$[111]$ direction defining the hexagonal axis. This specific arrangement is
motivated as the configuration that minimizes the nearest neighbor Coulomb
interaction, since each $d^{0}$ site is surrounded by six $d^{3}$ nearest
neighbors, whereas each $d^{3}$ site is surrounded by three nearest neighbors
of the same kind and three (i.e., the maximum possible amount given the 2:1
ratio) of the other ionic species. Furthermore, the same disproportionation
pattern has also been observed in La1/3Ca2/3FeO3 [20], where the Fe cations
disproportionate according to $3\mathrm{Fe}^{3.67+}\rightarrow
2\mathrm{Fe}^{3+}+\mathrm{Fe}^{5+}$. The space group symmetry in the charge
ordered phase then reduces from cubic $Pm\bar{3}m$ to rhombohedral
$P\bar{3}m1$.

As mentioned in the introductory section, the transition to a charge-
disproportionated insulating phase in other perovskite oxides is generally
accompanied by a structural “breathing mode” distortion consisting of an
expansion and contraction of the oxygen octahedra surrounding the cation sites
with higher and lower $d$ occupation, respectively [16, 17]. Furthermore, it
has been shown that the coupling between electronic degrees of freedom and
this octahedral breathing mode is crucial for stabilizing the charge
disproportionated insulating state in the rare earth nickelates [12, 22, 23,
13, 24, 25].

Since we expect similar physics to be relevant for a potential charge
disproportionation in SrCrO3, we also study the coupling to an analogous
octahedral breathing mode. However, in the proposed arrangement, there are
always two $[111]$ planes of $d^{3}$ sites adjacent to each other. Since the
corresponding oxygen octahedra are connected, the $d^{3}$ octahedra cannot
expand uniformly without distorting. As a result, the oxygen octahadra
surrounding the nominal $d^{3}$ sites exhibit three longer and three shorter
Cr-O bond lengths, whereas the $d^{0}$ octahedra can exhibit a simple uniform
contraction.

The resulting mode with symmetry label $\Lambda_{1}$ is shown in Fig. 1(b),
and can be understood as a simple displacement of the oxygen atoms in between
the $d^{3}$ and $d^{0}$ sites along the Cr-O bond direction. The amplitude of
this distortion can thus be parameterized by specifying only one degree of
freedom describing the contraction of the octahedra around the $d^{0}$ sites:

$Q=\frac{\ell^{0}_{\text{Cr-O}}-\ell_{\text{Cr-O}}}{\ell^{0}_{\text{Cr-O}}}\quad.$
(1)

Here, $\ell_{\text{Cr-O}}$ and $\ell_{\text{Cr-O}}^{0}$ are the Cr-O distances
of the $d^{0}$ octahedra in the distorted and undistorted states,
respectively, where the latter is equal to half the Cr-Cr distance
$\ell^{0}_{\text{Cr-O}}=\ell_{\text{Cr-Cr}}/2$.

The $\Lambda_{1}$ mode leads to a symmetry lowering compatible with the
$P\bar{3}m1$ space group. It can thus be assumed that this mode is the
energetically dominant structural distortion coupling to the charge
disproportionation, and we investigate the effect of the $\Lambda_{1}$ mode in
Sec. III.3 using DFT+DMFT calculations. However, the $P\bar{3}m1$ space group
symmetry also allows for a number of other distortion modes relative to the
ideal cubic perovskite structure, which could in principle lead to a further
energy lowering of the charge disproportionated state. Therefore, to verify
the dominant role of the $\Lambda_{1}$ distortion mode for the potential
charge disproportionation in SrCrO3, we also perform full structural
relaxations using the DFT+ $\mathcal{U}$ method [50, 51], which are presented
in Sec. III.4.

DFT+ $\mathcal{U}$ calculations have been applied to rare earth nickelates and
charge disproportionated ferrites, but these calculations require a local spin
splitting in order to provide realistic crystal structures for these systems
[23, 52, 53]. The simplest way to introduce such a local spin splitting
without any further symmetry-lowering is to impose a ferromagnetic (FM) order
in the material. However, superexchange between two neighboring $d^{3}$ Cr
sites is expected to lead to an antiferromagnetic coupling, whereas the
nominal $d^{0}$ Cr has no magnetic moment. Therefore, we also explore a
peculiar magnetic pattern, where the two nominal $d^{3}$ Cr sites are
initialized with opposite spins and the nominal $d^{0}$ Cr is initialized
without a magnetic moment, leading to a $[\,\cdots\uparrow 0\downarrow\uparrow
0\downarrow\cdots\,]$ arrangement of spins along $[111].$ We refer to this as
“d-AFM”, which stands for disproportionated antiferromagnetic phase.

###  II.2 DFT+DMFT for SrCrO3

Within DFT+DMFT (and DFT+ $\mathcal{U}$), the electronic degrees of freedom
are separated into the _correlated subspace_ , for which the strong local
electron-electron interaction is incorporated explicitly, and the rest, which
is treated within the usual semi-local approximations of DFT. The Hamiltonian
then takes the following form [47, 48, 49]:

$\hat{H}=\hat{H}_{\text{KS}}+\hat{H}_{\text{int}}-\hat{H}_{\text{DC}},$ (2)

where $\hat{H}_{\text{KS}}$ is the effective single particle Kohn-Sham
Hamiltonian, which contains the kinetic energy as well as the interaction with
and among the uncorrelated electrons and the ionic cores,
$\hat{H}_{\text{int}}$ describes the local electron-electron interaction
within the correlated subspace only, and $\hat{H}_{\text{DC}}$ is a double
counting correction.

In the case of SrCrO3, the correlated subspace for our DMFT calculations is
spanned by the partially occupied bands around the Fermi level with dominant
Cr-$t_{2g}$ character (see Sec. III.1), expressed in a basis of maximally
localized Wannier functions [54] with three $t_{2g}$-like orbitals on each Cr
site. The local interaction $\hat{H}_{\text{int}}$ is then expressed in the
fully rotationally invariant Slater-Kanamori form [6], including spin-flip and
pair hopping terms:

$\displaystyle\hat{H}_{int}$
$\displaystyle=U\sum_{m}\hat{n}_{m\uparrow}\hat{n}_{m\downarrow}$ (3)
$\displaystyle+\sum_{m\neq
m^{\prime}}\Big{(}(U-2J)\hat{n}_{m\uparrow}\hat{n}_{m^{\prime}\downarrow}+\sum_{\sigma=\uparrow,\downarrow}(U-3J)\hat{n}_{m\sigma}\hat{n}_{m^{\prime}\sigma}\Big{)}$
$\displaystyle-\sum_{m\neq
m^{\prime}}J\hat{c}_{m\uparrow}^{\dagger}\hat{c}_{m\downarrow}\hat{c}_{m^{\prime}\downarrow}^{\dagger}\hat{c}_{m^{\prime}\uparrow}+\sum_{m\neq
m^{\prime}}J\hat{c}_{m\uparrow}^{\dagger}\hat{c}_{m\downarrow}^{\dagger}\hat{c}_{m^{\prime}\downarrow}\hat{c}_{m^{\prime}\uparrow}\quad.$

Here, $U$ and $J$ denote the inter-orbital Coulomb repulsion and Hund’s
interaction, respectively, while $\hat{c}_{m\sigma}^{\dagger}$ and
$\hat{c}_{m\sigma}$ are the creation and annihilation operators for electrons
with spin $\sigma$ in orbital $m$, and
$\hat{n}_{m\sigma}=\hat{c}_{m\sigma}^{\dagger}\hat{c}_{m\sigma}$ is the
corresponding number operator.

The double counting correction, $\hat{H}_{\text{DC}}$, is supposed to subtract
the effect of the local electron-electron interaction within the correlated
subspace that is already included in $\hat{H}_{\text{KS}}$. However, the
explicit form of $\hat{H}_{\text{DC}}$ is not uniquely defined [55]. In this
work we follow [47] where the double counting correction is based on the
average interaction energy in the atomic limit. This results in a constant
shift of the self-energy on the Cr sites that can be written as:

$\Sigma_{\text{DC}}=\bar{U}(n-1/2)\quad,$ (4)

where $n$ is the total occupation of the corresponding site and $\bar{U}=U-2J$
is the average interaction strength for the three-orbital Kanamori interaction
(considering only density-density terms).

Within DMFT, each correlated site $i$ is mapped on an effective impurity
model, thereby neglecting the $k$-dependence of the self-energy
$\Sigma({\mathbf{k}},\omega)\rightarrow\Sigma(\omega)$ in the local basis
[45]. The effective impurity model is determined self-consistently by
requiring that the corresponding impurity Green’s function,
$G_{\text{imp}}(\omega)$, is equal to the local part of the full lattice
Green’s function, $G({\mathbf{k}},\omega)$ for all correlated sites [45].

In a fully charge self-consistent DFT+DMFT calculation, the charge density
determining the Kohn-Sham potential is recalculated from the local occupations
obtained within the DMFT step, and then the procedure is iterated until self-
consistency is achieved. However, sometimes, for computational efficiency,
such full charge self-consistency is neglected and so-called “one-shot”
calculations, i.e., without recalculating the charge density and Kohn-Sham
potential from the DMFT occupations, are performed. In the latter case, there
is an ambiguity on whether the double counting correction in Eq. (4) is
evaluated using the site occupations computed at the DFT level,
$n=n_{\text{DFT}}$, or the corresponding DMFT occupations, $n=n_{\text{DMFT}}$
[56]. While the DMFT occupations should be considered as results of the
DFT+DMFT calculations and thus as the actual physically relevant occupations,
$n_{\text{DFT}}$ are the occupations that give rise to the charge density
entering the Kohn-Sham potential within a one-shot calculation. In this case
it might therefore be more appropriate to use $n_{\text{DFT}}$ to account for
the electron-electron interaction contained at the DFT level. Note that in
charge self-consistent calculations no ambiguity arises, since then
$n_{\text{DMFT}}$ are the physically relevant occupations that also determine
the charge density from which the Kohn-Sham potential is calculated.

In the case of of a charge disproportionated system, the double counting
correction in Eq. (4) directly contributes to the difference of the one-
particle energies between sites with different total $d$ orbital occupations.
Since $\Sigma_{\text{DC}}$ is subtracted from the Kohn-Sham energies, the
double counting correction lowers the energy of the more occupied site
relative to the less occupied site, and thus a difference in these site
occupations favors the charge disproportionation. Therefore, we explore both
choices for $n$ in Eq. (4) in our initial one-shot calculations presented in
Sec. III.2, while in all our charge self-consistent calculations we always use
$n=n_{\text{DMFT}}$.

###  II.3 Computational Details

We perform all DFT, DFT+ $\mathcal{U}$, and DFT+DMFT calculations employing
the QuantumESPRESSO package [57] (version 7.2) with ultrasoft pseudopotentials
(GBRV library [58]) and the PBEsol exchange correlation functional [59]. All
results are obtained for the 15 atom hexagonal unit cell described in Sec.
II.1, with lattice vectors fixed to a lattice constant of 3.77 Å of the
underlying cubic perovskite structure, which corresponds to the relaxed PBEsol
value for cubic SrCrO3. We use a plane wave cutoff of 70 Ry for the
wavefunctions and 840 Ry for the charge density. Brillouin zone integrations
are performed on $\Gamma$-centered $6\times 6\times 6$ Monkhorst-Pack k-point
grid and the Marzari-Vanderbilt “cold-smearing” scheme [60] with a broadening
parameter of 0.001 Ry.

The correlated subspace for the DFT+DMFT calculations is constructed using
maximally localized Wannier functions generated with Wannier90 [61]. The
DFT+DMFT workflow is implemented using solid_dmft [62, 63] based on
TRIQS/DFTTools [64, 65], where the three Cr sites in the cell are mapped on
three effective impurity models, which are then solved independently using the
continuous time quantum Monte Carlo solver CT-HYB included in TRIQS [66]. We
use 8000 warmup cycles followed by 107 Monte-Carlo cycles of 120 moves each at
an inverse temperature of $\beta=40$ eV-1 (approximately 300 K). The resulting
Green’s functions are represented using 35 Legendre coefficients [67].

All DFT+DMFT calculations are performed for the spin-degenerate paramagnetic
case, while spin polarized DFT+ $\mathcal{U}$ calculations are performed for
the FM and d-AFM configuration, with the Hubbard correction applied onto
orthonormalized atomic projectors. We also perform full cell relaxation within
DFT+ $\mathcal{U}$ using a threshold of 10-3 Ry/Å for the forces and 10-4Ry
for the change in energy.

As a final note, we point out that the Hubbard parameters used for the
DFT+DMFT and for the DFT+ $\mathcal{U}$ calculations differ, since different
projectors as well as different parameterizations of the local interaction are
used by the two methods (three ${t_{2g}}$ frontier Wannier functions with
Kanamori interaction in DFT+DMFT compared to five ortho-normalized atomic $d$
orbitals with Slater parameters in DFT+ $\mathcal{U}$). Therefore, we denote
the Hubbard parameter in the DFT+DMFT calculations as $U$, while we use the
notation $\mathcal{U}$ for the DFT+ $\mathcal{U}$ calculations. Finally, we do
not use an explicit Hund’s interaction parameter in our DFT+ $\mathcal{U}$
calculations, since its effect is implicitly taken into account by the
effective $\mathcal{U}$ parameter in the simplified form of the $+\mathcal{U}$
correction according to Ref. [68]. While we performed calculations for several
$\mathcal{U}$ values ranging from 2 eV to 4 eV, in Sec. III.4 we present only
the results for the representative value of $\mathcal{U}=3$ eV.

##  III  Results

###  III.1  DFT band structure and choice of correlated subspace

Figure 2: Kohn-Sham band structure of cubic SrCrO3 obtained within spin-
degenerate DFT (blue lines). The orange lines show the dispersion of the
Wannier bands corresponding to the Cr $t_{2g}$ frontier orbitals.

Fig. 2 shows the spin-degenerate Kohn-Sham band structure obtained for cubic
SrCrO3 represented in our 15 atom hexagonal supercell. The bands around the
Fermi level, from slightly above $-2$ eV to about 1 eV, have predominant Cr
$t_{2g}$ character, consistent with the formal $d^{2}$ configuration of the
Cr4+ cation.

We construct nine maximally localized Wannier functions from these bands using
an energy window between $-2$ eV and 2 eV, starting from initial projections
on three $t_{2g}$ orbitals on each Cr site. The resulting Wannier functions
still resemble such atomic $t_{2g}$ orbitals centered on the Cr, but also
contain $p$-like tails on the surrounding oxygen ligands, representing the
$\pi^{*}$-type anti-bonding character of the hybridization with the O-$p$
orbitals in the underlying bands. The dispersion recalculated from these
Wannier functions is highlighted in orange in Fig. 2.

###  III.2  Phase diagrams for cubic SrCrO3 from one-shot DFT+DMFT

We first describe results obtained within one-shot DFT+DMFT calculations using
the DFT occupations to evaluate the double counting correction, i.e.,
$\Sigma_{\text{DC}}(n)=\Sigma_{\text{DC}}(n_{\text{DFT}})$. Since, in these
one-shot calculations, all Cr sites are equivalent at the DFT level, the
corresponding occupations are also identical and thus the double counting
correction just leads to an overall shift of the chemical potential, with no
effect on the result. These calculations are therefore very similar to the
model calculations of Ref. [14], except that we use the bandstructure of a
real material and treat the local interaction within DMFT.

Fig. 3(a) shows the value of the averaged spectral weight at zero frequency,
$\bar{A}(0)=-\frac{\beta}{\pi}\mathrm{Tr}[G(\beta/2)]$ as function of the
interaction parameters $U$ and $J$. Non-zero $\bar{A}(0)$ denotes a metallic
solution, while a value close to zero indicates an insulating state.

Figure 3: Scatter plot of the value of the spectral function $A(\omega=0)$
evaluated at the Fermi level, for two different choices of the occupations in
the double counting correction. Yellow dots correspond to a metallic solutions
while dark blue ones denote the presence of a gap. Using DFT occupations (a)
in the double counting formula qualitatively reproduces the results obtained
for the three-orbital Hubbard model [14], while employing DMFT occupations (b)
shifts the onset of the Hund insulator phase to much lower values of the
exchange coupling $J$.

The phase diagram exhibits the qualitative features that have been reported in
the literature for a generic three orbital Hubbard model with an average
filling of two electrons per site [10, 6, 14]. A Mott-insulating region, where
all Cr sites remain equivalent, is found for large $U$ in the region $U>3J$.
The corresponding phase boundary as function of $J$ exhibits the “Janus
effect” [10], i.e., increasing $J$ from zero first favors the Mott-insulating
state, while for $J>0.5$ eV a further increase of $J$ favors the metallic
state. In the region $U<3J$ we observe a second insulating phase, where the Cr
sites indeed disproportionate according to ${2d^{3}+d^{0}}$, corresponding to
the “Hund’s insulator” reported in Ref. [14]. Finally, along the line with
$U=3J$, where the Mott and Hund’s insulators are energetically degenerate in
the atomic limit, the system remains metallic up to high $U$, in agreement
with the persistent Hund’s metal discussed in [14].

We quantify the degree of disproportionation in the Hund’s insulating region
by defining the _site polarization_ $\Delta n$ as the occupation difference of
the more and less occupied Cr sites. We observe a strong site polarization of
$\Delta n>2.6$ for essentially all data points within the Hund’s insulating
region, indicating a relatively abrupt change across the phase boundary to the
metallic phase.

On a technical note, in the absence of the $\Lambda_{1}$ mode ($Q=0$) and
without considering charge self-consistency or using $n_{\text{DMFT}}$ in the
double counting, all three Cr sites remain symmetry equivalent at the single
particle level, and thus the specific spatial arrangement of $d^{0}$ and
$d^{3}$ sites does not provide a significant bias to the input of the
effective DMFT impurity problems. This leads to convergence issues in the
Hund’s insulating regime, where we observe that sites with $d^{0}$ and $d^{3}$
filling can arbitrarily switch during DMFT iterations. We do not consider this
as problematic, since the calculation always shows insulating behavior in the
corresponding regime.

Fig. 3(b) shows the analogous phase diagram (with a different scale for $J$)
obtained by using the DMFT occupations for the double counting correction,
i.e., $\Sigma_{\text{DC}}(n)=\Sigma_{\text{DC}}(n_{\text{DMFT}})$. While the
Mott insulating phase is unaffected by the different double-counting
correction, the most striking difference to the previous case is a strong
reduction of the minimal $J$ necessary for the onset of the Hund’s insulating
phase. For $U=2.5$ eV, the disproportionated phase emerges around $J=0.35$ eV.
For comparison, using
$\Sigma_{\text{DC}}(n)=\Sigma_{\text{DC}}(n_{\text{DFT}})$ results in a
critical $J$ of 1.25 eV at the same $U$ value.

This massive reduction of the critical $J$ is due to the occupation dependent
double counting correction, which, as discussed in Sec. II.2, results in a
lowering of the energy levels on the more occupied relative to the less
occupied Cr site, and thus provides a strong positive feedback loop that
stabilizes the charge disproportionated phase for much smaller $J$ values than
for $\Sigma_{\text{DC}}(n)=\Sigma_{\text{DC}}(n_{\text{DFT}})$. The difference
in the double counting correction on the different Cr sites,
$\Delta\Sigma_{\text{DC}}=\bar{U}\Delta n$, reaches up to 5 eV for $U=2.5$ eV
and $J=0.4$ eV.

We note that previous calculations of the interaction parameters for SrCrO3
using the constrained random phase approximation (cRPA) obtained $U=2.7$ eV
and $J=0.42$ eV [69]. This would potentially put SrCrO3 in the region of the
parameter space where the material would spontaneously disproportionate.
However, the results presented so far do not include the effect of charge
self-consistency, i.e., the DFT response to the charge correction obtained
from DMFT is absent. Indeed, when performing such fully charge self-consistent
DFT+DMFT calculations for SrCrO3, we find that the entire Hund’s insulating
region in the phase diagram disappears and the metallic region extends over
the whole region with $U<4.5$ eV in Fig. 3(b).

To better understand this strong difference between the one-shot and the
charge self-consistent calculations, we now analyze how the charge
disproportionation is affected by the structural distortion ($\Lambda_{1}$
mode) described in Sec. II.1.

###  III.3 Effect of the structural distortion

In general, to understand the interplay between electronic and structural
degrees of freedom, one may consider a simplified picture where the free
energy of the system contains three contributions [24, 25]:

$\mathcal{F}(\Delta n,Q)=\mathcal{F}_{el}[\Delta n]-\frac{g}{2}Q\Delta
n+\frac{K}{2}Q^{2}\quad.$ (5)

Here, $\mathcal{F}_{el}$ depends only on the electronic degree of freedom,
which, for the present case, can be represented in terms of the site
polarization $\Delta n$. The second term describes a linear coupling between
$\Delta n$ and the structural distortion mode $Q$, while the last term
represents the structural mode stiffness. In the case of SrCrO3, $Q$ denotes
the normalized amplitude of the $\Lambda_{1}$ mode shown in Fig. 1(b).

By minimizing the free energy in Eq. (5) with respect to $Q$, one obtains
[24]:

$\frac{2K}{g}Q=\Delta n(Q)\quad,$ (6)

where $\Delta n(Q)$ is the site polarization obtained by minimizing the free
energy for fixed $Q$. Thus, if the electronic response to the distortion
described by $\Delta n(Q)$ is linear in $Q$, Eq. (6) has only one solution for
$Q=0$ and the structural distortion is unfavorable. However, if the response
of the system is non-linear and, in particular, if there is a strong increase
of $\Delta n(Q)$ followed by an approach to saturation, the functions
described by the left and right hand sides of Eq. (6) can intersect at a value
$Q\neq 0$ and the system exhibits an energy minimum at non-vanishing
distortion.

Figure 4: (a) Energy difference per Cr atom between distorted and undistorted
SrCrO3 and (b) site polarization $\Delta n$ as a function of the $\Lambda_{1}$
distortion amplitude $Q$ obtained from fully charge self-consistent DFT+DMFT
calculations for different $J$ (and $U=2.7$ eV).

In Fig. 4 we plot the total energy as well as the site polarization as a
function of the normalized mode amplitude $Q$ obtained from fully charge self-
consistent DFT+DMFT. Data are shown for the cRPA values of the interaction
parameters, $U=2.7$ eV and $J=0.42$ eV [69], as well as for two higher values
for the Hund’s coupling $J$. In all cases, the only minimum of the total
energy occurs at $Q=0$, i.e., the structural distortion is energetically
unfavorable.

Simultaneously, the site polarization $\Delta n(Q)$ shows only a linear
response to the distortion. Even though this linear response is relatively
strong and leads to $\Delta n\approx 1$ for 5 % distortion, the system always
remains metallic so that there is no energy gain related to a gap opening in
the spectral function. This behavior persists even if the Hund’s coupling is
significantly raised beyond the cRPA value, although $\Delta n$ increases with
$J$. We note that $J=0.8$ eV is much higher than the unscreened value,
$J_{\textrm{bare}}=0.55$ eV, obtained in [69], and thus can be considered as
unrealistically high.

Figure 5: a) DMFT charge correction (isosurface level: 0.00644
$\mathrm{\AA}^{-3}$) after converging the one-shot calculations for the
disproportionated phase ${2d^{3}+d^{0}}$. Dark blue (red) spheres represent
the Cr (O) atoms. (b-c) Schematic picture of the effect of charge self-
consistency for the charge disproportionated phase of undistorted SrCrO3. (b)
In the initial DFT calculation, all Cr sites are equally occupied. In the
undistorted case, all sites are symmetry equivalent so that also the on-site
energy is the same. (c) One-shot DMFT converges a charge disproportionated
state. This changes the on-site energies within the subsequent DFT step,
opposing the tendency towards charge disproportionation.

Thus, our results so far indicate no tendency towards charge
disproportionation in SrCrO3.

In Fig. 5(a), we plot the DMFT correction for the charge density in the
disproportionated state obtained from the converged one-shot calculation for
$U=2.7$ eV, $J=0.42$ eV, and $Q=0$ (using
$\Sigma_{\text{DC}}(n_{\text{DMFT}})$). One can clearly see how the ${t_{2g}}$
orbitals around the nominal Cr $d^{0}$ sites get depleted while more electrons
accumulate mostly on the Cr $d^{3}$ sites and, to a lower extent, also on the
oxygen atoms that are shared between two $d^{3}$ octahedra.

The regions of accumulated and depleted charge lead to significant changes in
the Kohn-Sham potential, which translates to strong shifts of the local
atomic-like levels (the on-site term of the one-particle Hamiltonian defined
by the Wannier projections). In particular, charge accumulation pushes the
corresponding local level up in energy, while depletion does the opposite.
This is schematically illustrated in Fig. 5(b) and (c). Thus, the
recalculation of the Kohn-Sham potential based in the DMFT-corrected charge
density strongly disfavors the charge disproportionation, whereas, as
discussed in Sec. III.2, the use of the more physical DMFT occupations for the
double counting correction strongly supports it. The balance between these two
effects therefore determines whether the final state is indeed charge
disproportionated.

One can assume that these DFT level shifts are dominated by the strong local
Hartree interaction, even though inter-site Coulomb interactions also
contribute. This local component of the $d$-$d$ interaction is exactly what
should be compensated by the double-counting correction. However, since the
available forms for $\Sigma_{\text{DC}}$ are based on rather heuristic
arguments, these may easily overcompensate or undercompensate the DFT Hartree
shift.

In the present case, the double counting correction is constructed from the
atomic limit, which might not be very accurate for the relatively delocalized
frontier Wannier orbitals used to define the correlated subspace. Furthermore,
the use of frontier orbitals leads to essentially nominal $d$ occupations and
thus maximizes the resulting site polarization $\Delta n$. While this is
attractive for an intuitive physical interpretation, it also amplifies any
uncertainties in the double counting correction with respect to the resulting
relative local energy levels.

Figure 6: (a) Energy difference per Cr atom between distorted and undistorted
SrCrO3 and (b) site polarization $\Delta n$ as function of the $\Lambda_{1}$
distortion mode amplitude for different scaling of the double counting
correction ($U$ = 2.7 eV and $J$=0.42 eV).

We note that there have been many instances reported in the literature, where
a scaling of the double counting was necessary to recover the insulating
behavior of the studies materials [55, 70, 71, 72]. Therefore, to investigate
how changes in the double counting correction affect the tendency towards
charge disproportionation in SrCrO3, we now perform charge self-consistent
DFT+DMFT calculations where we scale the double counting correction by a
constant factor denoted by $f_{\textrm{DC}}$ while keeping $U$ and $J$ fixed
to 2.7 eV a nd 0.42 eV, respectively. The scaling of the double counting
correction can also be viewed as a scaling of the average interaction strength
$\bar{U}$ in Eq. (4). The results are shown in Fig. 6.

It can be seen that $f_{\text{DC}}=1.35$ indeed leads to the formation of an
energy minimum at a nonzero distortion amplitude of around 4-5 %.
Simultaneously, the system shows a strong nonlinear increase of the site
polarization around $Q=1.5$ %, which marks the onset of the electronic
instability towards formation of the charge disproportionated insulating state
(cf. Ref. [24]). Scaling the double counting correction to 150 % deepens the
energy minimum, and even leads to a spontaneously charge disproprtionated
state for zero distortion amplitude with almost saturated $\Delta n$. A
further scaling of the double counting correction thus has negligible effect.

These results indicate that SrCrO3 is at least close to an instability towards
a charge disproportionated insulating phase. However, due to the fundamental
uncertainty regarding the size of the double counting correction the
predictive capability of the DFT+DFT approach is limited in this case. It is
thus not possible to draw a final conclusion whether the charge
disproportionated state is indeed a stable state for SrCrO3 in ambient
conditions.

###  III.4  DFT+ $\mathcal{U}$: structural relaxations and the influence of
other distortion modes

In the previous sections, we explored the impact of the $\Lambda_{1}$
distortion mode on the energetics using DFT+DMFT. The observed energy lowering
for the scaled double counting correction suggests that the choice of the
distortion mode is indeed reasonable. Nevertheless, it is important to also
consider other symmetry-allowed distortion modes that might lead to an
additional energy lowering. To investigate this, we employ DFT+ $\mathcal{U}$,
which allows us to perform systematic structural relaxations and thus explore
a broader range of distortion modes as well as provide further insight into
the stability of a charge disproportionated state in SrCrO3.

Figure 7: Total energy (per Cr) as function of the $\Lambda_{1}$ distortion
mode amplitude $Q$ obtained from DFT+ $\mathcal{U}$ ($\mathcal{U}=3$ eV) for
d-AFM and FM order for different unit cell volumes, indicated by the
corresponding pseudocubic lattice constant. The red diamond corresponds to the
energy and distortion amplitude obtained from a full structural optimization.

We first perform analogous calculations as for the DFT+DMFT case, where we fix
the lattice constant to $a=3.77$ Å, corresponding to the equilibrium lattice
constant of cubic SrCrO3 obtained from a spin-degenerate PBEsol relaxation,
and then vary the amplitude $Q$ of the $\Lambda_{1}$ distortion mode. As
described in Sec. II, we impose the d-AFM magnetic order in these DFT+
$\mathcal{U}$ calculations to allow for an insulating solution. The resulting
energy as function of $Q$ for $\mathcal{U}=3$ eV is shown as blue line in Fig.
7. Indeed, we see that the $\Lambda_{1}$ mode causes an energy lowering of the
system, with a minimum around $Q=3$%.

The d-AFM structure is insulating, as shown in Fig. 8, where we plot the
corresponding density of states (DOS) and band structure for $Q$ = 3%. The
projected DOS shows that the top of the valence band region is primarily
formed by $d$ states of the nominal $d^{3}$ Cr, while the unoccupied low-lying
conduction band is dominated by states corresponding to the nominal $d^{0}$
Cr. Thus, even though the difference in the integrated charge on the two
inequivalent sites remains relatively small, there is a clear signature
consistent with the formal $d^{3}$ and $d^{0}$ charge states in the electronic
structure. Furthermore, the local magnetic moments are $\sim
2.6~{}\mu_{\text{B}}$ on the magnetic $d^{3}$ Cr and close to zero on the
nonmagnetic $d^{0}$ Cr site, consistent with a high degree of
disproportionation between the inequivalent sites.

Figure 8: Bandstructure and density of states of SrCrO3 (lattice constant
$a=3.77$Å) with d-AFM ($[\,\cdots\uparrow 0\downarrow\uparrow
0\downarrow\cdots\,]$) magnetic order for $\mathcal{U}=3$ eV and $Q$ = 3%. The
right panel also shows projections on the two inequivalent Cr sites: the
magnetic (Mag, magenta) $d^{3}$ Cr and the non-magnetic (NM, blue) $d^{0}$ Cr.

Next, we perform a full structural optimization (within the $P\bar{3}m1$ space
group) for the d-AFM order. The resulting energy and $\Lambda_{1}$ mode
amplitude is marked by the red diamond in Fig. 7. The relaxation further
lowers the energy by about 50 meV and leads to a somewhat larger mode
amplitude of $Q\sim 3.8$%. While additional symmetry-allowed distortion modes
become non-zero in the relaxed structure, their amplitudes remain negligible
compared to the $\Lambda_{1}$ mode. The main effect leading to the energy
lowering is a change in cell volume, while the underlying lattice remains
cubic to a good approximation with a pseudocubic lattice constant of
$a=3.81$Å.

To double check the influence of the additional modes, we perform an
additional calculation where we fix the lattice vectors to those obtained from
the fully relaxed structure, but discard all other distortions except for the
$\Lambda_{1}$ mode. The resulting energy as function of $Q$ is shown by the
orange line in Fig. 7, and exhibits a minimum that is indeed very close to the
energy and distortion obtained from the full relaxation. This confirms that
the effect of other distortion modes is essentially negligible.

Finally, we also perform a full structural relaxation for the FM case, and
then calculate the corresponding total energy as function of the $\Lambda_{1}$
distortion amplitude (green line in Fig. 7). The relaxed cell volume (with
$a=3.81$ Å) for the FM case is essentially identical to that obtained for the
d-AFM order. However, the FM calculations always lead to a metallic band-
structure and the $\Lambda_{1}$ distortion is energetically unfavorable in
this state. Furthermore, the energy of the undistorted FM state is lower than
the energy minimum of the d-AFM state. This indicates that, while a distorted
and disproportionated insulating state can be stabilized in these
calculations, it is clearly not the energetically preferred ground state of
the system. The d-AFM state also competes with other magnetic phases. For
example, the Jahn-Teller distorted C-AFM state explored in Ref. [44] is around
0.15 eV per Cr lower in energy compared to the minimum of the FM phase, and
0.2 eV lower when compared to the minimum of the d-AFM case.

##  IV Conclusions and outlook

To summarize, we have employed both DFT+DMFT and DFT+ $\mathcal{U}$
calculations to explore the possible emergence of a charge disproportionated
insulating phase in SrCrO3. We observed that, compared to calculations based
on a simple multi-band Hubbard model, the effect of full charge-self
consistency in the DFT+DMFT cycle significantly alters the phase diagram
obtained as function of the local interaction parameters $U$ and $J$, which
can even result in a full suppression of charge disproportionation. Thereby,
the stability of the charge disproportionated state is controlled by a subtle
balance between the local interaction treated on the DMFT level, which drives
the charge disproportionation in particular for $U<3J$, and the DFT response
to the DMFT charge correction, leading to a strong Hartree-shift of the local
Cr levels that opposes disproportionation. In practice, this balance is
regulated by the double counting correction, for which the explicit form is
not clearly defined, therefore limiting the predictive power of the
corresponding DFT+DMFT calculations.

Nevertheless, we have shown that a paramagnetic charge disproportionated
insulating state can be stabilized in SrCrO3 for realistic interaction
parameters via coupling to a structural breathing distortion, if the DFT+DMFT
double counting correction is enhanced by about 35 %. The breathing distortion
and disproportioned state can also be stabilized in DFT+ $\mathcal{U}$
calculations with an appropriate magnetic (d-AFM) order. However, comparison
with other magnetically ordered states indicates that this d-AFM state is not
the ground state configuration of bulk SrCrO3 at ambient conditions.
Nevertheless, the existence of such a low-energy metastable state, and the
existing reports of charge disproportionation in the sister compound PbCrO3
[29, 30, 31], suggest that it might be possible to stabilize charge
disproportionation also in SrCrO3 by appropriate tuning of parameters such as
strain, pressure, ionic substitution, or others.

Our DFT+DMFT calculations also show that a $U$ of at least 4.5 eV is needed
for SrCrO3 to reach the “normal” paramagnetic Mott-insulating region. This is
significantly higher than the corresponding values obtained from cRPA
calculations [69], and suggests that, similar to what has been discussed for
BaCrO3 in Ref. [35], a further symmetry breaking, such as, e.g., orbital order
or charge disproportionation, is required for SrCrO3 to become insulating.

The partially conflicting experimental reports on either metallic or
insulating SrCrO3 [37, 40, 41, 42, 43], together with our previous work
indicating a C-AFM Jahn-Teller distorted ground state favored by tensile
epitaxial strain [44], suggests that SrCrO3 is close to many potentially
competing structural and electronic instabilities. Further work is thus needed
to shed more light on the interplay of magnetism, lattice distortions, and
electronic correlations in SrCrO3 and other members of this family of
perovskite chromates.

###### Acknowledgements.

This research was supported by ETH Zurich and a grant from the Swiss National
Supercomputing Centre (CSCS) under project ID s1128. Calculations were
performed on the “Eiger” cluster of the Swiss Supercomputing Centre (CSCS) and
the “Euler” cluster of ETH Zurich. We thank Maximilian Merkel, Simon Jöhr and
Marta Gibert for valuable discussions.

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# Entity Linking in Tabular Data Needs the Right Attention

Miltiadis Marios Katsakioris
School of Mathematical and Computer Sciences
Heriot-Watt University
Edinburgh, UK
<EMAIL_ADDRESS>
Yiwei Zhou
<EMAIL_ADDRESS>
Daniele Masato
Amazon Alexa
Cambridge, UK
<EMAIL_ADDRESS>

###### Abstract

Understanding the semantic meaning of tabular data requires Entity Linking
(EL), in order to associate each cell value to a real-world entity in a
Knowledge Base (KB). In this work, we focus on end-to-end solutions for EL on
tabular data that do not rely on fact lookup in the target KB. Tabular data
contains heterogeneous and sparse context, including column headers, cell
values and table captions. We experiment with various models to generate a
vector representation for each cell value to be linked. Our results show that
it is critical to apply an attention mechanism as well as an attention mask,
so that the model can only attend to the most relevant context and avoid
information dilution. The most relevant context includes: same-row cells,
same-column cells, headers and caption. Computational complexity, however,
grows quadratically with the size of tabular data for such a complex model. We
achieve constant memory usage by introducing a Tabular Entity Linking Lite
model (TELL ) that generates vector representation for a cell based only on
its value, the table headers and the table caption. TELL achieves 80.8%
accuracy on Wikipedia tables, which is only 0.1% lower than the state-of-the-
art model with quadratic memory usage.

##  1 Introduction

Tabular data, such as web tables and databases, provides invaluable
information about the world. According to [1], by 2008, there are 14.1 billion
HTML tables from Google’s general-purpose web crawl, and 154M of them are high
quality relational data. There has been various efforts to leverage rich
factual information contained in tabular data for Knowledge Base Augmentation
[2, 3], Question Answering [4], etc. These applications require to
automatically interpret and understand the semantic meaning of tabular data at
scale. Entity Linking in tabular data, which target at linking a cell value
(“Titanic”) in tabular data with its corresponding real-world entity reference
in a knowledge base (Q44578 in Wikidata111https://www.wikidata.org/), is an
important step for semantic table interpretation.

Comparing with Entity Linking in unstructured text [5, 6], Entity Linking in
tabular data needs to tackle some additional challenges. First, each cell
value is an entity mention to be linked, the environment it appears is not a
complete sentence, but other entity mentions in short texts, real numbers, or
dates. Second, besides content information in the form of rows and columns,
tabular data is usually associated with metadata information, such as HTML
page titles, table captions and column headers. For each cell in the table,
this metadata information can contain both signals and noises. Third, sizes of
tables can vary significantly. Once the table schema is determined, the size
of a table can grow infinitely.

In this work, we focus on end-to-end Entity Linking solutions in tabular data,
the main contributions are as follows:

  * •

Besides entity disambiguation, we also consider the scenarios that a cell
value is not an entity mention and the correct entity is not included in the
list of candidate entities, so that the solution is robust to errors
introduced during candidate entities retrieval.

  * •

We reduce the required prior knowledge for existing entities in the targeted
KB to entity names and entity descriptions, which increases the solution’s
generalisation capability to unseen entities.

  * •

We experiment with various models to generate a vector representation for each
cell value to be linked, and verified the importance of applying an attention
mechanism and an attention mask to regulate the interactions between cell
values and table metadata.

  * •

We propose a simple yet effective model, Tabular Entity Linking Lite (TELL),
which generates cells’ vector representations only based on cell values and
metadata. TELL reduces the computation complexity of tabular structure aware
models from quadratic to linear while sacrificing the accuracy by 0.1%, from
80.9% to 80.8%.

##  2 Related Work

Most former works [2, 3] on Entity Linking in tabular data requires fact
lookup during training and inference time. These solutions can only link cell
values with entities that are well populated with facts in the targeted KB,
and their entity linking capability is restricted to only cell values in the
subject column of a table. There has been some learning based Entity Linking
approaches for tabular data. However, they are either based on an assumption
that the correct entity is included in the list of retrieved candidate
entities, and only focus on the disambiguation part of the problem [7, 8]; or
they are dependent on some additional knowledge about the entities in the KB,
such as prior probabilities of entity mentions [9], entity embeddings [8], or
entity types [7], which limit their generalisation capability to unseen
entities during training.

Figure 1: Overview of our EL pipeline. First, all entity mentions $T_{i,j}$
are passed through the candidate generator. The resulting list of candidates
$E$ and the $T_{i,j}$ with the corresponding metadata are encoded by BERT and
the entity linker, ranks and makes the final decision for each $T_{i,j}$,
whether it has a link from $E$ or not.

##  3 Entity Linking in Tabular Data

Without loss of generality, we assume that each table $T$ consists of $M$ data
rows and $N$ data columns, and $T_{i,j}$ represents the cell value in the
$i^{th}$ row and $j^{th}$ column. Each table can be associated with additional
metadata, which includes: (1) Table caption $C$, a short text description
summarizing the content of the table; (2) Page title $P$, the title of the web
page the table was in; (3) Table headers $H=[h_{0},\dots h_{N-1}]$, one for
each column to define the table schema.

Same as [8, 7, 9], we consider each table cell $T_{i,j}$ is a potential
mention for real world entities in the target KB. Specifically, we define the
Entity Linking in tabular data problem as follows:

Definition 1. Given a table $T$ and a target KB, link cells $T_{i,j}$ in $T$
with their corresponding real-world entity references in the target KB if
possible, while automatically detect and ignore the ones that cannot be
linked.

To increase the entity linking solution’s generalisation capability to unseen
entities [5] and any target KB, we only assume the existence of names and
descriptions for entities in the target KB, which is minimal comparing with
former works.

Figure 1 presents an overview of our EL pipeline which consists of two stages:
(1) Candidate Generation, which generates candidate entities $E$ for all cells
in each table $T$, and (2) Entity Disambiguation, which ranks and selects the
best $e\in E$ for each $T_{i,j}$ if there is any (refer to as NIL otherwise).

###  3.1 Candidate Generation

Most existing work [9, 10, 11, 7] directly use exact string matching-based
lookup services provided by the target KB to generate candidate entities for
each cell, which cannot tackle the variations of entity mentions in web
tables. A more sophisticated method for this stage, we can increase the
probability of including the right entity in the candidate entities ($P_{E}$)
while introducing minimal noises. Following [12], we used an gazetteer
constructed based on Wikidata entity names and alias, as well as Wikipedia
article titles, hyperlinks and redirects. Additionally, we applied BM25 to
measure the similarity between a cell value and a KB entity rather than exact
string matching. According to our analysis, the combination of gazetteer and
BM25-based similarity search has increased $P_{E}$ from 77% to 88%.

###  3.2 Entity Disambiguation

We use a shared BERT [13] encoder to encode all the textual inputs, which
include: cell value $T_{i,j}$, metadata $(C,P,H)$, entity name $e_{name}$ and
entity description $e_{desc}$.

For a KB candidate entity, its vector representation $e$ is achieved by adding
its averaged name token embedding and its averaged description token
embedding:

$e=mean(BERT(e_{name}))+mean(BERT(e_{desc}))$ (1)

When encoding the cells of a table we have to take into account how to model
the tabular structure, if and how to use metadata and the role of attention
for the optimal representation. We compare different ways of encoding in a
top-down approach. At the top, in terms of complexity and information load is
$MaskAttEnc$, an encoder based on TURL [7], a state-of-the-art framework for
relational table understanding that consists of a structure-aware Transformer
encoder to model the row-column structure of the table and capture the textual
information and relational knowledge of each cell. During the self-attention
calculation, a “hard-coded” attention mask limits the aggregation of
information from one entity cell to other structurally related entity cells,
such as cells in the same row/column. In our ablation study, when we are not
using the hard-coded attention mask, we refer to the model as $AllAttEnc$.

We remove the hard-coded attention mask and we treat each cell value as a
separate entity. Instead of encoding the metadata separately using extra
attention heads and fusing them with the encoded cells, we concatenate them to
each cell. We refer to the resulting module as TELL . $MaskAttEnc$ makes
predictions on the table level whereas TELL can make predictions on the
individual cell level, allowing for extra flexibility.

We simplify the entity mention representation of TELL further by removing all
BERT attention heads. We refer to these models as $SingleEnc$, with the cells
being treated separately as in TELL. However, instead of attention, we encode
the sentence embeddings either by an LSTM, $SingleLSTMEnc$ or by averaging the
embeddings and passing them through a linear feedforward layer,
$SingleLinearEnc$.

After getting the vector representation for a cell, we calculate the matching
score between $T_{i,j}$ and $e$ by,

$P(e)=\frac{exp(T_{i,j}\cdot e)}{\sum_{e^{{}^{\prime}}\epsilon
E}exp(T_{i,j}\cdot e^{{}^{\prime}})}$ (2)

and we select the candidate with the highest probability.

##  4 Dataset

For our experiments we pre-processed and generated our own data splits of the
WikiTables corpus [9]. WikiTable corpus contains 1.65M tables extracted from
Wikipedia pages, and most of the tables contain hyperlinks between cell values
and Wikipedia entity articles labelled by Wikipedia contributors.

We cleaned the table cells by lower-casing their content, removing HTML tags
and removing special characters. For cells containing multiple hyperlinks, we
retained the first link only. In order to fit training batches (25 batches) in
memory, we discarded tables with more than 500 cells (entity mentions). We
also discarded tables with no linked entities, and tables with either no
candidates or more than 1800 candidates overall. For computational efficiency
we remove duplicate cell values from the tables. In addition, to remove noises
in Wikipedia hyperlinks, we compute the difference in lengths between each
cell value and its corresponding linked entity’s name. As long as the
difference is bigger than 10, we will ignore the hyperlink because it is very
unlikely to happen.

In order to map hyperlinked Wikipedia entities in WikiTables with Wikidata
entities, we use Wikimapper222See https://github.com/jcklie/wikimapper..

Splits | # Tables | # NIL | # Entities
---|---|---|---
Train | 554,239 | 16.6M | 26.6M
Validation | 4,738 | 142K | 317K
Test | 4,660 | 139K | 312K
Table 1:  Dataset statistics. In the ‘NIL’ column we show an estimate of the
entity mentions that have no link and in the last column the total entity
mentions $T_{i,j}$ from all tables.

Statistics of our data splits are summarized in Table 1. The table shows that
the mean percentage of NIL entities in all splits is around 50%. The average
number of NIL entities is consistent for both the dev and test splits.

##  5 Experimental Results

We evaluated our solution end-to-end using accuracy and F1 score. We first
established a strong baseline, by encoding each cell, only with the cell value
using BERT. This baseline, which does not use any extra context, achieves an
accuracy of 77.5%. This baseline is simply learning mappings from cell value
embeddings to entity embeddings, we expect any approach that uses additional
context to outperform the baseline.

Table 2 shows increasing levels of ablation. It starts with $MaskAttEnc$, a
model using both the whole table content and the metadata, then transitions to
the aforementioned baseline model, a model using only separated cell values.
The proposed solution, TELL , uses separated cell values as well as metadata,
achieves 80.8% accuracy and 79.3% F1 score, and it only requires a linear
number of parameters. The comparison between TELL and the baseline shows the
importance of the metadata and the attention that happens in each cell between
its text and the metadata.

On the other hand, $MaskAttEnc$ attends to the whole table and achieves 80.9%
accuracy with the help of the attention mask. This mask regulates that each
cell only attend to structurally related cells from the entire table and the
metadata. We see the importance of the attention mask when we compare it with
$AllAttEnc+meta$ with a drop in accuracy of 2.3%. Without the attention mask
as a regularization method, the model fails to automatically learn the most
relevant information for each cell.

A marginal improvement (0.1%) in accuracy compared to TELL , for the sacrifice
of scalability since computation complexity of the $MaskAttEnc$ solution grow
exponentially $O(n^{2})$. This is because $MaskAttEnc$ needs to model the
relationship between any pair of cell values, rather than treating them
separately as TELL. This also makes $MaskAttEnc$ able to only work on small
tables. For TELL and its ablations, table size is not an issue, as long as the
metadata are being passed on together with each cell. Aside from the type of
attention and the structured relatedness, we see that the metadata are crucial
and always contribute to the performance.

For tabular data is critical to apply the right attention in order to attend
to the right cells when generating the cell representations. Otherwise,
information dilution from irrelevant cells will greatly impact the performance
and it might be better to simply pass each cell individually.

Method | Acc | F1 | Comp.
---|---|---|---
MaskAttEnc + meta (TURL) | 80.9 | 80.3 | $O(n^{2})$
SingleAttEnc + meta (TELL ) | 80.8 | 79.3 | $O(n)$
AllAttEnc + meta | 78.6 | 78.1 | $O(n^{2})$
SingleLinearEnc + meta | 77.9 | 76.1 | $O(n)$
SingleAttEnc (baseline) | 77.5 | 76.3 | $O(n)$
AllAttEnc | 76.8 | 76.0 | $O(n^{2})$
SingleLSTMEnc + meta | 69.7 | 69.1 | $O(n)$
Table 2:  Evaluation of all variants on test set. In the Big $O$ notation, $n$
symbolises the number of input cells.

##  6 Conclusion

The state-of-the-art framework for relational table understanding (TURL)
performs EL for a given cell using a Deep Neural Network with attention that
aggregates information from the whole table, including the content and
location of surrounding cells, and additional table metadata such as title,
caption and headers. TURL achieves 80.9% accuracy on Wikipedia tables, but its
computation complexity grows quadratically with the number of cells in a
table.

In this paper we presented a lightweight approach for EL on tabular data that
can achieve almost state-of-the-art accuracy with linear computation
complexity. We consider both challenges of candidate retrieval and entity
disambiguation, whilst trying to find a balance between the two. We showed
that metadata are crucial. In order to avoid a noisy cell representation it is
important to apply the right attention to avoid information dilution. In
future work, we will focus on improving the candidate entity retrieval
mechanism, by applying some embedding based approach. Another option is to
develop solutions to transform tabular data into unstructured text, in order
to leverage EL solutions for unstructured text to tackle the problem.

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