# Breakdown of $J_{eff}$ = 0 and $J_{eff}$ = 3/2 states and existence of large
magnetic anisotropy energy in vacancy ordered 5$d$ antifluorites: K2ReCl6,
K2OsCl6, and K2IrCl6

Amit Chauhan<EMAIL_ADDRESS>B. R. K. Nanda<EMAIL_ADDRESS>1Condensed Matter Theory and Computational Lab, Department of Physics, IIT
Madras, Chennai-36, India  2Center for Atomistic Modelling and Materials
Design, IIT Madras, Chennai-36, India  3Functional Oxides Research Group, IIT
Madras, Chennai-36, India

###### Abstract

Vacancy-ordered antifluorite materials (A2BX6) are garnering renewed attention
as novel magnetic states driven by spin-orbit coupling (SOC) can be realized
in them. In this work, by pursuing density functional theory calculations and
model studies, we analyze the ground state electronic and magnetic structure
of face-centered cubic (fcc) antifluorites K2ReCl6 (KReC, 5$d^{3}$), K2OsCl6
(KOsC, 5$d^{4}$), and K2IrCl6 (KIrC, 5$d^{5}$). We find that KReC stabilizes
in the high-spin $S$ = 3/2 state instead of the expected pseudo-spin $J_{eff}$
= 3/2 state. The former occurs due to large exchange-splitting as compared to
the SOC strength. On the contrary, the KOsC stabilizes in broken $J_{eff}$ = 0
($S$ = 1) simple Mott insulating state while KIrC stabilizes in $J_{eff}$ =
1/2 spin-orbit-assisted Mott insulating state. The presence of an isolated
metal-chloride octahedron makes these antifluorites weakly coupled magnetic
systems with the nearest and next-nearest-neighbor spin-exchange parameters
($J_{1}$ and $J_{2}$) are of the order of 1 meV. For KReC and KOsC, the
$J_{1}$ and $J_{2}$ are estimated to be antiferromagnetic and ferromagnetic,
which leads to a Type-I antiferromagnetic ground state, whereas for KIrC, both
$J_{1}$ and $J_{2}$ are antiferromagnetic, hence, it stabilizes with a Type-
III antiferromagnetic state. Interestingly, in their equilibrium structure,
these antifluorites possess large magnetic anisotropy energy (0.6-4
meV/transition metal), which is at least one-to-two orders higher than
traditional MAE materials like transition metals and multilayers formed out of
them. Moreover, with epitaxial tensile/compressive strain, the MAE enhances by
one order, becoming giant for KOsC (20-40 meV/Os).

††preprint: APS/123-QED

##  I Introduction

Spin-orbit coupling (SOC) driven entanglement of spin and orbital degrees of
freedom has led to the formation of novel quantum phases in transition-metal
(TM) compounds involving chemically active 5$d$ valence electrons. Recently,
TM compounds with heavier $d$ block elements such as Re, Os, and Ir have
emerged as potential candidates due to the realization of topological phases
such as Dirac and Weyl semimetals [1, 2, 3, 4], spin liquid [5, 6, 7], spin-
orbit-assisted Mott insulator [8], etc. in them. Apart from exploring the
aforementioned phases, the current thrust in the research on SOC active 5$d$
quantum materials is to manifest the following two aspects.

Firstly, the competition among crystal field splitting ($\Delta_{cr}$), onsite
Coulomb repulsion ($U$), and SOC ($\lambda$) in an octahedral complex can lead
to the formation of an intriguing electronic and magnetic structure in the
pseudo-spin space. In an octahedral crystal field environment, the five-fold
degenerate $d$ manifold splits into three-fold degenerate $t_{2g}$ and two-
fold degenerate $e_{g}$ states (see Fig. 1). With strong SOC, while the latter
remains unperturbed, the former further splits into spin-orbital entangled
$J_{eff}$ = 3/2 quartet (mj = $\pm 3/2,\pm 1/2$) and $J_{eff}$ = 1/2 doublet
(mj = $\pm 1/2$) with both of them separating energetically by 3/2 $\lambda$.
For $5d^{3}$ electronic configuration, three electrons will occupy the lower-
lying $J_{eff}$ = 3/2 states which will give rise to a weakly spin-polarized
magnetic state due to equal and opposite spin/orbital polarization of each
$m_{j}$ = $\pm 1/2$ or $\pm 3/2$ states.

Resonant inelastic X-ray scattering (RIXS) studies [9] on face-centered cubic
(fcc) double perovskite compounds, Ca3LiOsO6 and Ba2YOsO6, reveal the
stabilization of $J_{eff}$ = 3/2 electronic ground state, however, Ir based
double perovskite Ba2NiIrO6 and Os based pyrochlore member Cd2Os2O7 were found
to stabilize in a $S=3/2$ state rather than $J_{eff}$ = 3/2 state [10, 11].
Recently, the $d^{4}$ family of compounds are intensely explored to realize
the $J_{eff}$ = 0 non-magnetic ground state due to fully occupied $J_{eff}$ =
3/2 manifold (see Fig. 1) [12, 13, 14]. However, contrary to the expected
ground state, recent RIXS and ab-initio studies report magnetic ground state
for 6$H$-hexagonal perovskite iridate Ba3ZnIr2O9, double perovskite iridate
Sr2YIrO6, and quasi-one-dimensional spin chain system Sr3NaIrO6 to be magnetic
[13, 12, 15]. Though there is a lack of literature, the $d^{5}$ configuration
in an ideal octahedron is bound and expected to show a single hole $J_{eff}$ =
1/2 state.

Figure 1: (a) The splitting of five-fold degenerate $d$ manifold in an
octahedral crystal field environment leading to three-fold degenerate $t_{2g}$
and two-fold degenerate $e_{g}$ states. With strong SOC and weak exchange-
splitting ($\lambda$ $>>$ $\Delta_{ex}$), the $t_{2g}$ states further split to
form four-fold and two-fold degenerate spin-orbital entangled $J_{eff}$ = 3/2
and $J_{eff}$ = 1/2 pseudo-spin states. In $\Delta_{ex}$ $>>$ $\lambda$ limit
while the high-, intermediate-, and low-spin configurations stabilizes for
$d^{3}$, $d^{4}$, and $d^{5}$ valence states, in the $\Delta_{ex}$ $<<$
$\lambda$ limit, $J_{eff}$ = 3/2, $J_{eff}$ = 0, and $J_{eff}$ = 1/2 pseudo-
spin ground state forms. (b) The transformation of $J_{eff}$ = 3/2 and
$J_{eff}$ = 1/2 states with $\Delta_{ex}$. The green and blue colors represent
the spin-up and spin-down weights, respectively. (c) The energy eigen spectra
as a function of $\Delta_{ex}$/$\lambda$ in the atomic limit. Increasing
strength of exchange-splitting leads to the complete decoupling of spin-up and
spin-dn channels. The $\lambda$ is chosen to be 0.3 eV which is the SOC
strength of KReC.

Secondly, from the application point of view, since the 5$d$ TM compounds are
expected to demonstrate SOC-led anisotropy, there is a growing interest in
calculating the magnetic anisotropy energy (MAE) in them [16, 17, 18]. Large
uniaxial anisotropy energy can lead to higher coercivity that can enhance the
maximum energy product (maximum of the product of $B$ and $H$ in the second
quadrant of the $B-H$ curve) and hence the stored magnetostatic energy, which
is a standard measure for the performance of a permanent magnet [19].
Moreover, MAE can become a key player in stabilizing “skyrmions” in chiral
magnets [20].

Though synthesized long back [21, 22, 23, 24], vacancy-ordered antifluorite
compounds with chemical formulae K2BX6, X being a halogen and B being a
$5d^{3-5}$ transition metal element, are gaining renewed attention as quantum
materials in recent times [25, 26, 27, 28]. The Ir-based $5d^{5}$
antifluorites, K2IrCl6, K2IrBr6, and (NH4)2IrCl6 are intensely studied
experimentally and many intriguing properties like antiferromagnetic spin
resonance, weak Kitaev anisotropy, non-local cubic distortions, and
enhancement of magnetic couplings with halogen replacement are reported [21,
26, 29, 25, 27, 28]. Earlier neutron diffraction studies on K2ReCl6 (5$d^{3}$)
have revealed Type-I antiferromagnetic ordering with easy axis along [001]
direction. In the case of Os4+ ($5d^{4}$) halides, K2OsCl6, K2OsBr6, Na2OsBr6,
and Na2OsBr6·6H2O, either Kotani-like behavior consistent with a $J_{eff}$ = 0
ground state or a weak magnetic ground state is experimentally proposed [30].

As a whole, the $5d^{3-5}$ antifluorite compounds provide a platform to
examine the complex interplay between various couplings - SOC, electron
correlation, magnetic coupling, crystal field, and local exchange field - so
that non-trivial phases can be explored. So far, most of the studies on
antifluorites have been experimental, and there is a lack of theoretical
analysis of electronic and magnetic structures. Specifically, the role of
competing interactions in obtaining the magnetic ground state is yet to be
explained.

In this work, we adopt three compounds of the antifluorite family, namely,
K2ReCl6, K2OsCl6, and K2IrCl6, and pursued DFT calculations and toy model
study to address the following questions: (i) Do antifluorites possess
expected pseudo-spin ground state? (ii) How the electronic and magnetic
structure evolves due to valence electron count, crystal field, exchange-
splitting, and SOC? (iii) Can a large/giant MAE be realized in these systems
due to possible spin anisotropies? (iv) Can the ground state and MAE be
modified by tuning the crystal field via external stimuli such as strain?

The electronic structure, obtained using density functional theory (DFT)
calculations, and model analysis infers that the breakdown of $J_{eff}$ = 3/2
and $J_{eff}$ = 0 pseudo-spin states in KReC and KOsC is due to the presence
of large exchange-splitting ($\Delta_{ex}$) compared to the SOC strength
($\Delta_{ex}$ $>>$ $\lambda$). On the contrary, the KIrC stabilizes in
$J_{eff}$ = 1/2 state as it possesses weak $\Delta_{ex}$ ($\Delta_{ex}$ $<<$
$\lambda$) so that the SOC led pseudo-spin states remains unperturbed. The
nearest-neighbor magnetic exchange interaction $J_{1}$ is estimated to be $<$
1 meV, indicating that antifluorites are weakly coupled magnetic systems.
While KOsC and KReC stabilize in a Type-I antiferromagnetic ground state with
in- and out-of-plane spin orientations, KIrC stabilizes in a Type-III ground
state with spin orientation along $\hat{x}$. The easy axis/plane of
magnetization is well understood qualitatively as well as quantitatively by
carrying out second-order perturbative analysis of SOC and through the
quantification of magnetic anisotropic constants. With tensile strain, while
the ground state magnetic structure remains robust for KReC and KOsC, it
transforms to Type-II with compressive strain irrespective of B-site ion. The
magnetic anisotropy energy is found to be large $\approx$ 0.6-4 meV/TM and
dominated by single-ion anisotropy energy. Moreover, with strain, the MAE
enhances by one order in magnitude.

##  II Structural and Computational Details

Antifluorites K2ReCl6, K2OsCl6, and K2IrCl6 crystallize in fcc lattice (Fm-3m)
(see Fig. 2) at low temperature, with antifluorite-type A2BX6 crystal
structure. The crystal structure of antifluorites resembles those of double
perovskites (A2BB′X6) due to the presence of vacant B′ sites. As a result, the
BCl6 octahedron is isolated and arranged on a geometrically frustrated fcc
lattice as shown in Fig. 2. The optimized lattice parameters are provided in
Table-1, which are in close agreement with the experimental lattice
parameters.

We have adopted DFT+$U$+SOC formalism to study the electronic and magnetic
structures. The calculations were performed using plane-wave based projector
augmented wave (PAW) [31, 32] method as implemented in Vienna ab-initio
simulation package (VASP) [33] within the Perdew$-$Burke$-$Ernzerhof
generalized gradient approximation (PBE-GGA) for the exchange-correlation
functional. The Brillouin zone integrations were carried out using $4\times
8\times 8$ $\Gamma$-centered $k$-mesh. The kinetic energy cutoff for the
plane-wave basis set was chosen to be $400$ eV. The strong correlation effect
was incorporated via an effective onsite correlation parameter
$U_{\mathrm{eff}}$ = $U-J$ through the rotationally invariant approach
introduced by Dudarev [34]. For the unstrained case, the calculations were
performed on the fully optimized (atomic as well as volume relaxation)
experimental structures, whereas, for the strained case, the atomic positions
were relaxed while keeping the volume fixed. All optimization calculations
were performed by considering onsite correlation $U$ at the TM site.

Figure 2: The crystal structure of K2BCl6. The structure is realized by
introducing vacancies at the B′ sites of the double perovskite A2BB′O6 crystal
structure. Therefore, instead of a corner sharing network of octahedra, in
K2BCl6, the fcc stacked BCl6 octahedra are isolated.

To estimate the Hubbard $U$ parameter, we have employed the linear response
approach [35] wherein the onsite Coulomb repulsion $U$ is computed by
calculating the difference between interacting and non-interacting density
response functions:

$\displaystyle U=\chi^{-1}_{0}-\chi^{-1}=\Big{(}\frac{\partial
n_{i}^{KS}}{\partial\alpha_{i}}\Big{)}^{-1}-\Big{(}\frac{\partial
n_{i}}{\partial\alpha_{i}}\Big{)}^{-1}.$ (1)

The quantity $\alpha$ is the applied perturbation at the TM site where $U$ is
computed. The response functions are obtained by performing non-self-
consistent ($\chi$) and self-consistent ($\chi_{0}$) DFT calculations (see
Fig. 3). The obtained values of response functions and Hubbard $U$ are
provided in Table-1.

Figure 3: (a,b,c) The linear self- and non-self-consistent response functions as a function of the onsite perturbing potential $\alpha$ to estimate the Hubbard $U$ parameter. The solid lines are guide to the eyes. Table 1: The estimated response functions and Hubbard $U$ parameter from linear response theory and experimental, optimized lattice parameters. Compound |  $\chi_{0}^{-1}$ (eV) |  $\chi^{-1}$ (eV) |  $U$ (eV) |  $a^{exp}$ (Å) |  $a^{relax}$ (Å)
---|---|---|---|---|---
KReC | 12.19 | 5.18 | 7.01 | 9.84 | 9.93
KOsC | 5.74 | 2.71 | 3.03 | 9.68 | 9.78
KIrC | 8.13 | 4.76 | 3.37 | 9.77 | 9.92

##  III Electronic and magnetic structure of Antifluorites

###  III.1 Electronic structure

In this subsection, we will analyze the electronic structure of antifluorites
KReC, KOsC, and KIrC wherein the 4+ charge state of TM element leads to
$d^{3}$, $d^{4}$, and $d^{5}$ valence states, respectively.

KReC-$d^{3}$: The spin and orbital resolved partial density of states (pDOS)
of KReC is shown in Fig. 4(a). The strong localization of the states, even in
the absence of onsite repulsion, indicates weak $d-d$ covalent interaction
between neighboring TM ions. This is expected because the nearest-neighbor
hopping channels are broken due to vacant B′ sites, which are otherwise
present in the regular double perovskite materials. The effect of pure crystal
field splitting on $d$ manifold can be understood by analyzing the spin-
polarized GGA pDOS (see Fig. 4(a)).

Figure 4: Left Column: The spin and orbital resolved density of states per
atom with and without considering the effect of onsite Coulomb repulsion in
the absence of SOC. Right Column: The corresponding band structures with SOC
included. With SOC, the band structures are shown instead of the density of
states to elucidate clearly the SOC lead splitting. The $U_{eff}$ values are
used as obtained from the linear response method (see Table-1).

Due to strong octahedral crystal field ($\Delta_{cr}\approx$ 3 eV estimated by
taking out the differences between $t_{2g}$ and $e_{g}$ band centers), the
five-fold degenerate $d$ states split into a higher-energy and unoccupied
$e_{\mathrm{g}}$ doublet ($x^{2}-y^{2}$ and $z^{2}$) and a lower-energy
$t_{2g}$ triplet ($xy$, $yz$, and $xz$). While the majority spin-up channel of
the latter is completely occupied, the spin-down channel is completely
unoccupied, suggesting $t_{2g}^{3\uparrow}$ $e_{g}^{0}$ electronic
configuration and 4+ charge state of Re. However, the hybridization between
Re-$d$ and Cl-$p$ states induces a moment of $\approx$ 0.4 $\mu_{B}$ on the
latter, reducing the effective moment at the Re site (see Table 2). The
complete separation between the spin-up and -down channels is due to the
presence of large exchange-splitting ($\Delta_{ex}$ $\approx$ 1.35 eV), which
also led to the formation of an insulating gap even without the inclusion of
onsite Coulomb repulsion. Therefore, antifluorite KReC possesses a trivial
band-insulating state. With the incorporation of $U_{eff}$ = 7 eV, as obtained
from the linear response method (see Table 1), while the Cl-$p$ states
dominate near the Fermi level ($E_{F}$), the $t_{2g}$ states are pushed deep
in the valence and conduction bands, increasing the band gap remarkably to 3.2
eV (see Fig. 4(b)).

Table 2: The estimated spin and orbital moments, exchange splitting, and SOC strength with or without incorporating $U$ and SOC. | GGA | GGA+SOC | GGA+SOC+$U$
---|---|---|---
Compounds | ms | $\Delta_{ex}$ | ms | ml | $\lambda$ | ms | ml
K2ReCl6 | $2.23$ | $1.35$ | $2.11$ | $-0.15$ | $0.3$ | $2.76$ | $-0.23$
K2OsCl6 | $1.40$ | $0.6$ | $0.93$ | $0.33$ | $0.35$ | $1.42$ | $0.56$
K2IrCl6 | $0.35$ | $0.1$ | $0.20$ | $0.34$ | $0.43$ | $0.28$ | $0.48$

Having understood the effect of $\Delta_{cr}$ and $U$, we now analyze the
effect of SOC on the electronic structure. As shown in Fig. 4(c), the SOC
lifts the three-fold degeneracy of the $t_{2g}$ manifold to form complex
states. As a consequence, the orbital moment (ml) estimated within GGA+SOC
turned out to be 0.15 $\mu_{B}$. Furthermore, with the incorporation of both
$U$ and SOC, the Cl-$p$ states form the valence band maxima and dominate near
the $E_{F}$. The ml/ms ratio is estimated to be 0.07, much less than 1,
indicating the quenched effect of SOC. The large spin magnetic moment
($\approx$ 2.8 $\mu_{B}$) suggests the stabilization of a high spin $S=3/2$
state rather than an ideal $J_{eff}$ = 3/2 state as the ground state of one
hole in the $J_{eff}$ = 3/2 manifold can only exhibit a low-spin state. We
attribute this breakdown to the presence of a strong local exchange field
($\Delta_{ex}>>\lambda$) which suppresses the atomic SOC strength effectively
by $\lambda_{eff}$ = $\lambda_{atomic}/S$ [36].

To gain further insights into the breakdown of the $J_{eff}$ = 3/2 state, we
now analyze how the pseudo-spin orbitals and the splitting between $J_{eff}$ =
1/2 doublet and $J_{eff}$ = 3/2 quartet evolves as $\lambda$ and $\Delta_{ex}$
compete with each other. In the second quantization formalism, the atomic
Hamiltonian is given by,

$H^{at}=\lambda\sum_{\alpha,\beta,\sigma,\bar{\sigma}}\bra{\alpha\sigma}\bm{L}\cdot\bm{S}\ket{\beta\bar{\sigma}}c_{\alpha,\sigma}^{\dagger}c_{\beta,\bar{\sigma}}\\\
+\Delta_{ex}\sigma\sum_{i,\alpha,\sigma}c^{\dagger}_{i\alpha\sigma}c_{i\alpha\sigma},$
(2)

where $\alpha$ ($\beta$) and $\sigma$ represent orbitals ($xy$, $yz$, and
$xz$) and spin ($\uparrow,\downarrow$) indices, respectively. The obtained
results are shown in Figs. 1 (c) and 5 (a) wherein the energy, spin/orbital
moments, and occupation numbers are plotted as a function of
$\Delta_{ex}$/$\lambda$. While in $\lambda$$>>$$\Delta_{ex}$ limit, SOC
entangles the $t_{2g}$ states with respect to spin and orbital to form pseudo-
spin states (see Fig. 1 (b)),

$\displaystyle\ket{\frac{1}{2},\pm{\frac{1}{2}}}$
$\displaystyle=\frac{1}{\sqrt{3}}(\ket{yz,\bar{\sigma}}\pm{\ket{xy,\sigma}}\pm{i}\ket{xz,\bar{\sigma}})$
(3) $\displaystyle\ket{\frac{3}{2},\pm{\frac{1}{2}}}$
$\displaystyle=\frac{1}{\sqrt{6}}(\ket{yz,\bar{\sigma}}\mp
2{\ket{xy,\sigma}}\pm{i}\ket{xz,\bar{\sigma}})$ (4)
$\displaystyle\ket{\frac{3}{2},\pm{\frac{3}{2}}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{yz,\sigma}\pm{i}\ket{xz,{\sigma}})$ (5)

, the increasing strength of $\Delta_{ex}$/$\lambda$ breaks the four-fold and
two-fold degeneracy of $J_{eff}=3/2$ and $J_{eff}=1/2$ states and leads to
complete decoupling of the spin-up and -down states in
$\lambda$$<<$$\Delta_{ex}$ limit (see Fig. 1(b)). The decoupling between the
former and the latter occurs as the occupation numbers of the
$t_{2g}^{\uparrow}$ orbitals increase with $\Delta_{ex}$/$\lambda$ whereas the
occupation numbers of the

Figure 5: The variation in spin (red) and orbital (blue) moments and spin
resolved orbital occupation numbers (black dashed lines) as a function of
$\Delta_{ex}$/$\lambda$ for $d^{3}$, $d^{4}$, and $d^{5}$ fillings in the
atomic limit.

$t_{2g}^{\downarrow}$ orbitals decreases and becomes zero (see Fig. 5 (a)).
This leads to the complete unquenching and quenching of spin and orbital
moments ($m_{s}$ = 3 $\mu_{B}$ and $m_{l}$ = 0). Therefore, $d^{3}$
configuration, which is expected to possess large $\Delta_{ex}$, is bound to
stabilize in a high-spin $S=3/2$ state rather than weakly spin-polarized
$J_{eff}$ = 3/2 state. To understand the vanishing orbital moment with
$\Delta_{ex}$, we analyze the eigen states of $H^{at}$ in $\Delta_{ex}$ $>>$
$\lambda$ limit. As inferred from Fig. 1 (b), while $\ket{3/2,\pm 3/2}$
orbitals remains unperturbed by $\Delta_{ex}$, the $\ket{3/2,\pm 1/2}$ and
$\ket{1/2,\pm 1/2}$ orbitals evolve to $xy\uparrow$, $yz\uparrow-ixz\uparrow$,
$xy\downarrow$, and $yz\downarrow+ixz\downarrow$ orbitals, respectively.
Since, $\ket{3/2,3/2}$ and $yz\uparrow-ixz\uparrow$ orbitals possess equal and
opposite orbital polarization ($\braket{3/2,3/2}{\hat{L_{z}}}{3/2,3/2}$ = 2,
$\braket{yz\uparrow-ixz\uparrow}{\hat{L_{z}}}{yz\uparrow-ixz\uparrow}$ = -2,
and $\braket{xy}{\hat{L_{z}}}{xy}$ = 0), the net orbital moment becomes zero.
However, the spin polarization of lower-lying three orbitals is parallel, as a
result, $S$ = 3/2 state forms.

Invoking an intermediate coupling approach that incorporates both SOC and $U$
on an even footing, RIXS measurements on Ca3LiOsO6 and Ba2YOsO6 (Os-$d^{3}$)
systems report $J_{eff}$ = 3/2 ground state [9]. However, first-principles
studies on these systems found $U$, $\Delta_{ex}$ $>>$ $\lambda$. Therefore,
as established from our analysis for $d^{3}$ configuration in $\Delta_{ex}$
$>>$ $\lambda$ limit, Ca3LiOsO6 and Ba2YOsO6 will exhibit high-spin $S$ = 3/2
ground state rather than $J_{eff}$ = 3/2 pseudo-spin state.

KOsC-$d^{4}$: The $d^{4}$ configuration is expected to give rise to a
nonmagnetic $J_{eff}$ = 0 insulating state due to completely occupied
$J_{eff}$ = 3/2 manifold (see Fig. 1 (a)). However, contrary to the expected
ground state, many candidate materials were found to be stabilized in a broken
$J_{eff}$ = 0 state [13, 15]. To find whether it breaks down or stabilizes in
KOsC, we examine its electronic structure, which is shown in Figs. 4 (e-h).
For the spin-polarized GGA case, the $t_{2g}$ states are completely occupied
in the spin-up channel, whereas, in the spin-dn channel, it is one-third
occupied, leading to a half-metallic state.

Figure 6: (a) The magnetic orderings, namely, FM, Type-I, and Type-III,
considered for the estimation of magnetic exchange interactions $J_{1}$ and
$J_{2}$.

The half-metallic scenario for $U_{eff}$ = 0 occurs due to weak $\Delta_{ex}$
$\approx$ 0.6 eV. The partial occupancy at the $E_{F}$ gets lifted with the
onset of Coulomb repulsion as it separates the $xy$ and $yz$, $xz$ states with
a gap in between (see Fig. 4(f)). Therefore, antifluorite KOsC exhibits a Mott
insulating state with an intermediate spin state $t_{2g}^{3\uparrow
1\downarrow}e_{g}^{0}$. While the gap in KOsC is mediated by strong onsite
Coulomb repulsion, SOC led splitting of states at the $E_{F}$ leads to a
semimetallic state. Within GGA+SOC+$U$, while the orbital moment enhances from
0.33 to 0.56 $\mu_{B}$, the spin moment gets quenched by 0.4 $\mu_{B}$, giving
rise to ml/ms ratio of $\approx$ 0.36. The ml/ms ratio increases as compared
to KReC as the $\lambda_{eff}$ increases with a decrease in the spin moment.
The estimated large spin moment ($\approx$ 1.42 $\mu_{B}$) indicates the
breakdown of the expected $J_{eff}$ = 0 nonmagnetic state in KOsC. Our results
are contrary to a recent experimental study which report a $J_{eff}$ = 0 or a
weakly magnetic ground state for K2OsCl6 [30].

In the atomic limit, as shown in Fig. 5(b), for $\Delta_{ex}$ = 0, the spin
and orbital moments are zero and as a result a $J_{eff}$ = 0 state forms. This
occurs as the $J_{eff}$ = 3/2 states possess equal and opposite spin and
orbital polarization for $m_{J}=\pm 1/2$ or $m_{J}=\pm 3/2$ states which are
completely occupied. With increasing $\Delta_{ex}$/$\lambda$, the spin and
orbital moment increases and saturate at 2 and 1 $\mu_{B}$, respectively. The
former and the latter occurs as yz$\downarrow$ and xz$\downarrow$ orbitals are
half-occupied to form $yz\downarrow$ \- ixz$\downarrow$ orbital which possess
$m_{s}$ = -1, $m_{l}$ = -1 moments while the rest of the three lower-lying
orbitals gives $m_{s}$ = 3 and $m_{l}$ = 0 (see Fig. 1 (b)). As inferred from
Table-2, KOsC possess $\Delta_{ex}$/$\lambda$ ratio of $\approx$ 1.71. As a
consequence, $S$ = 1 state stabilizes instead of the nonmagnetic $J_{eff}$ = 0
state. Therefore, we attribute the breakdown of the $J_{eff}$ = 0 state in
KOsC to the presence of substantial local exchange field.

KIrC-$d^{5}$: The pDOS for KIrC are shown in Figs. 4 (i-j). For the GGA case,
both spin channels are more or less equally occupied and hence a low-spin
state resulting from weak $\Delta_{ex}$ $\approx$ 0.1 eV stabilizes. With the
inclusion of $U$, while the the partially occupied states in the spin-up
channel are now completely occupied, the spin-dn channel remains partially
occupied. Therefore, unlike the case of KOsC, the inclusion of $U$ does not
lead to opening of a gap at the $E_{F}$. With SOC, the $t_{2g}$ states spin
mixes to form completely occupied $J_{eff}$ = 3/2 states and partially
occupied $J_{eff}$ = 1/2 states (see Fig. 4 (k)). Interestingly, a large gap
opens up within GGA+SOC+$U$, as the combined effect of correlation and SOC
lifts the partially occupancy of the $J_{eff}$ = 1/2 states to open up a gap.
Therefore, antifluorite KIrC possesses a non-trivial spin-orbit-assisted Mott
insulating state which is yet not reported in the literature. The opening of a
gap manifests the pronounced effect of SOC as the weak Ir spin moment
$\approx$ 0.28 $\mu_{B}$ enhances the effective SOC with ml/ms ratio of
$\approx$ 1.71.

As shown in Fig. 1 (b), the $J_{eff}$ = 1/2 ($m_{j}$ = 1/2) orbital transform
to $xy\downarrow$ orbital in large $\Delta_{ex}$/$\lambda$ limit. Since $xy$
orbital possess $m_{l}$ = 0, the addition of spin and orbital moments of five
lower-lying states give rise to $m_{s}$ = 1 and $m_{l}$ = -1. However, the
former is the least probable scenario as one hole in the $t_{2g}$ manifold
tends to stabilize the system in a low-spin state. As $\Delta_{ex}$/$\lambda$
in KIrC is very weak ($\approx$ 0.23), the SOC-led pseudo-spin states remain
more or less unperturbed to form a weakly spin-polarized $J_{eff}$ = 1/2
ground state.

###  III.2 Magnetic structure of Antifluorites and its strain engineering

We now proceed to investigate the ground state magnetic structure of
antifluorites and its tuning with strain. Generally, depending on the strength
and nature of the nearest and next-nearest-neighbor spin exchange interactions
($J_{1}$ and $J_{2}$) [37, 38, 39], the spin arrangement on a geometrically
frustrated fcc lattice can be of Type-I, Type-II or Type-III (see Fig. 6)[37,
38, 39].

Figure 7: Upper panel: The variation of magnetic exchange parameters $J_{1}$,
$J_{2}$, and Curie Weiss temperature $\theta_{cw}$ as a function of $U_{eff}$.
Lower panel: The magnetic anisotropy energy/transition metal atom as a
function of $U_{eff}$. The $U_{eff}$ is varied upto the theoretically
calculated value from the linear response theory (see Table 1).

Therefore, in order to examine the magnetic structure of antifluorites, we
estimate the exchange interactions by employing Noodelmann’s broken-symmetry
spin dimer method [40]. In this method, the energy difference between high
spin (HS) and broken symmetry (BS) configurations is given by

$E_{HS}-E_{BS}=\frac{1}{2}S_{max}^{2}J$ (6)

where $J$ is related to the spin-dimer Hamiltonian $H$ =
$\sum_{i<j}J_{ij}S_{i}\cdot S_{j}$ and $S_{max}$ is the maximum spin of the
dimer. For the present case, since each monomer has three (KReC-$d^{3}$), two
(KOsC-$d^{4}$), and one (KIrC-$d^{5}$) unpaired spins, the corresponding
$S_{max}$ values are 3, 2, and 1, respectively. The $E_{HS}$ and $E_{BS}$ are
evaluated by performing the DFT calculations for HS and BS configurations. To
estimate $J$, we have adopted three different spin configurations: FM, Type-I,
and Type-III (see Fig. 6). We have not considered Type-II magnetic ordering
because the minimum supercell size required to design it is
2$\times$2$\times$2 which contains four formulae units of K2BCl6, thereby
making the estimation of $J$ computationally expensive. By employing the spin
dimer Hamiltonian on these magnetic configurations, we have calculated the
spin-exchange energies (per f.u.) in terms of the spin-exchange parameters,
which are expressed as

$\displaystyle E_{FM}=\frac{S^{2}}{2}(48J_{1}+40J_{2}),$ $\displaystyle
E_{I}=\frac{S^{2}}{2}(-16J_{1}+40J_{2}),$ $\displaystyle
E_{III}=\frac{S^{2}}{2}(-16J_{1}+24J_{2}),$ (7)

where $S$ is the spin of each monomer. The spin-exchange energies are mapped
to the total energy obtained from the DFT calculations for the respective
magnetic configuration. Further, using Eqns. 6-7, the $J$ values and Curie-
Weiss temperature ($\Theta_{CW}=-\frac{S(S+1)}{3K_{B}}\sum_{i}Z_{i}J_{i}$
where $Z_{i}$ is coordination number corresponding to each $J_{i}$
interaction) are estimated as a function of $U$ and the results are shown in
the top row of Fig. 7.

As inferred from Fig. 7, the $\Theta_{CW}$ is negative, indicating that the
dominant interaction is antiferromagnetic. The $\Theta_{CW}$ or the Neel
temperature $T_{N}$ are often evaluated as a function of $U$ and matched with
the experimental results. From this matching, a realistic value of $U$ can be
determined to re-validate the $U$ strength obtained from the linear response
method. The experimentally reported values of $\Theta_{CW}$ for KReC [21] and
KIrC [26] are -55 K and -45 K, respectively, which matches close to the
theoretically estimated values $\approx$ -58.15 K and -49.05 K at $U$ = 7 and
3 eV, respectively. These values are in excellent agreement with those
obtained from the linear response method (see Table 1). For KOsC, to the best
of our knowledge, the experimental value of $\Theta_{CW}$ is not available in
the literature. Therefore, its $\Theta_{CW}$ value can be roughly approximated
to -62 K at $U_{eff}$ = 3 eV.

The sharp decrease in $J_{1}$ with increasing $U_{eff}$ manifests the
superexchange interaction between NN spin moments. For KReC, at $U_{eff}$ = 7
eV, the estimated strengths of $J_{1}$ and $J_{2}$ ($\approx$ 0.2 and -0.02
meV) suggest that the NN(NNN) magnetic interactions are weak and
antiferromagnetic(ferromagnetic) in nature. The DFT estimated $J$ strengths
are in good agreement with those estimated using correlated effective-field
theory [41]. The obtained nature of exchange interactions leads to the
stabilization of a Type-I magnetic order [37] consistent with the neutron
diffraction studies [42].

Table 3: The estimated energy differences (meV/TM) with respect to the most stable magnetization direction within DFT+$U$+SOC calculations. The single ion anisotropy energies are given in parentheses. SAXIS | KReC | KOsC | KIrC
---|---|---|---
$[001]$ | 0 | 3.97 (3.8) | 0.18
$[100]$ | 0.20 | 0.44 | 0
$[110]$ | 0.58 | 0 | 0.49
$[111]$ | 0.64 (0.51) | 0.27 | 0.47 (0.28)

To determine the easy/hard axis of magnetization, we perform DFT calculations
with the spin quantization axis (SAXIS) pointing along different directions.
The obtained energy differences are listed in Table-3. As inferred from Table
3, the easy and hard axis for KReC is along [001] and [111] directions,
differing in energy, the so-called magnetic anisotropy energy, by $\approx$
0.64 meV/Re. The underlying origin behind the preferred spin orientation will
be discussed in the next section. Like KReC, KOsC also exhibits Type-I
antiferromagnetic order with weak $J_{1}$ and $J_{2}$ ($\approx$ 0.45 and
-0.02 meV). However, KOsC possesses an easy plane [110] of magnetization which
is stabilized by large MAE of $\approx$ 3.97 meV/Os. For KIC, the $J_{1}$
($\approx$ 1.2 meV) and $J_{2}$ ($\approx$ 0.07 meV) are both
antiferromagnetic and hence KIC exhibits a Type-III order [37] with spin
orientation along $\hat{x}$. Moreover, the ground state remains robust with
increasing $U_{eff}$ as the nature of interactions remains the same.

Often strain is used as an external stimuli to tune the magnetic properties
and transition temperatures. To observe the effect of it on the ground state
of antifluorites, we have applied an epitaxial tensile and compressive strain
and the results are shown in Fig. 8. With tensile strain, while the nature of
$J_{1}$ and $J_{2}$ and hence the magnetic order remains robust for KReC and
KOsC, the easy/hard axis for the former and latter switches to [110]/[001 and
[100]/[001], respectively (see Figs. 8(b-d)). However, for KIrC, spin ordering
transforms from Type-III to Type-I ($J_{2}$$<$0) with easy/hard axis along
[001] and [110] directions. For compressive strain, the $J_{2}$ enhances
significantly as compared to $J_{1}$, therefore, a Type-II order [37]
stabilizes irrespective of the B-site element. Moreover, while the easy and
hard axis remains robust for KIrC, it changes to in- and out-of-plane for KReC
and KOsC, respectively.

##  IV MAE and preferred spin orientation

Magnetic anisotropy energy is an important quantity and can lead to higher
coercivity that can enhance the performance of permanent magnets [19]. It is
estimated by taking the energy difference between easy and hard directions
(MAE = $E^{easy}$ \- $E^{hard}$) of magnetization for a magnetic material.
Since the antifluorites considered in this study involve heavier TM and hence
possess substantial SOC, which is a key player to realize large/giant MAE, it
is worth examining MAE in these compounds. Taking the easy axis as the
reference direction, the calculated values of MAE are listed in Table 3. The
MAE is large and is at least one-to-two orders higher than traditional MAE
materials like transition metals and their multilayers [43].

Figure 8: Upper panel: The variation of magnetic exchange parameters $J_{1}$,
$J_{2}$, and Curie Weiss temperature $\theta_{cw}$ as a function of epitaxial
tensile and compressive strain. Lower Panel: The magnetic anisotropy energy as
a function of strain. The $U_{eff}$ is kept fixed as obtained from linear
response theory.

While we found antifluorites to be promising candidates for the realization of
large/giant MAE, we now examine the SOC-induced interaction that contributes
dominantly to MAE. The SOC-induced interactions include antisymmetric exchange
or Dzyaloshinskii-Moriya (DM) interaction, anisotropic exchange, and single-
ion anisotropy (SIA). Due to the presence of inversion symmetry, the DM
interaction will not contribute to MAE. As the antifluorites possess weak
isotropic $J$, the contribution to MAE from the anisotropic exchange will also
be negligible. Therefore, presumably, SIA will contribute dominantly to the
estimated MAE values. To substantiate it further, we estimate SIA by replacing
the neighboring Ir, Os, and, Re atoms in KReC, KOsC, and KIrC with non-
magnetic Si ions. In this way, the contribution from $J_{1}$ and $J_{2}$
interactions will be killed in the total energy calculations. The obtained SIA
values are provided in the bracketed terms of Table 3. As expected, we found
the SIA to be the dominant contributor to MAE.

Further, to study the strain tunability of MAE, we apply epitaxial strain, and
the results are shown in the lower panel of Fig. 8. The MAE increases linearly
with strain and becomes one-order higher with respect to the ground state
value. Intriguingly, for Os, the MAE becomes giant ($\approx$ 20-40 meV/Os for
2-4 $\%$ strain).

The preferred ground state spin orientation (in or out-of-plane) of
antifluorites can be analyzed by employing second-order perturbative analysis
of SOC Hamiltonian $H_{SO}$ = $\lambda$ $\hat{L}$ $\cdot$ $\hat{S}$. Defining
two independent coordinate systems (x, y, z) and (x′, y′, z′) for the orbital
and spin angular momentum $\hat{L}$ and $\hat{S}$, $H_{SOC}$ can be rewritten
as $H_{SOC}$ = $H_{SOC}^{sc}$ \+ $H_{SOC}^{snc}$, where the spin-conserving
($H_{SOC}^{sc}$) and spin-non-conserving ($H_{SOC}^{snc}$) terms can be
written as [44, 43, 45, 46]

$\displaystyle H_{SOC}^{sc}$ $\displaystyle=\lambda
S_{z^{\prime}}(L_{z}\cos{\theta}+\frac{1}{2}L_{+}e^{-\iota\phi}\sin{\theta}+\frac{1}{2}L_{-}e{{}^{\iota\phi}}\sin{\theta})$
$\displaystyle=\lambda
S_{z^{\prime}}(L_{z}\cos{\theta}+L_{x}\sin{\theta}\cos{\phi}+L_{y}\sin{\theta}\sin{\phi})$
$\displaystyle
H_{SOC}^{snc}=\frac{\lambda}{2}S_{+}^{\prime}(-L_{z}\sin{\theta}-L_{+}e^{-\iota\phi}\sin^{2}{\theta/2}+L_{-}e^{\iota\phi}\cos^{2}{\theta/2})$
$\displaystyle+\frac{\lambda}{2}S_{-}^{\prime}(-L_{z}\sin{\theta}+L_{+}e^{-\iota\phi}\cos^{2}{\theta/2}-L_{-}e^{\iota\phi}\sin^{2}{\theta/2})$
$\displaystyle=\frac{\lambda}{2}(S_{+}^{\prime}+S_{-}^{\prime})(-L_{z}\sin{\theta}+L_{x}\cos{\theta}\cos{\phi}+L_{y}\cos{\theta}\sin{\phi})$

where $\theta$ and $\phi$ determine the magnetization direction with respect
to (x,y,z) coordinate system. It is important to note that while
$H_{SOC}^{sc}$ gives non-zero expectation values for the same spin states due
to $\hat{S_{z}}$, $H_{SOC}^{nsc}$ gives finite contribution for opposite spin
states.

Using perturbation theory, the second-order correction in energy due to SOC is
given by,

$\Delta
E_{soc}=-\lambda^{2}\sum_{o^{\sigma},u^{\sigma^{\prime}}}\frac{|\bra{o^{\sigma}}H_{soc}\ket{u^{\sigma^{\prime}}}|^{2}}{|\epsilon_{o^{\sigma}}-\epsilon_{u^{\sigma^{\prime}}}|}$
(9)

where $u^{\sigma}$ ($o^{\sigma^{\prime}}$) denotes the unoccupied (occupied)
states in spin ($\sigma$, $\sigma^{\prime}$ = $\uparrow$, $\downarrow$)
channels. A qualitative analysis of the spin orientation can be obtained by
analyzing the energy separation and coupling to the SOC Hamiltonian between
$o^{\sigma}$ and $u^{\sigma}$ states. The energy difference
$\epsilon_{u^{\sigma}}$ \- $\epsilon_{o^{\sigma^{\prime}}}$ is roughly
estimated through the spin and orbital resolved density of states shown in the
first column of Fig. 4.

For KReC, the occupied ($t_{2g}\uparrow$) and unoccupied
($t_{2g}\downarrow$+$e_{g}\uparrow\downarrow$) states contributing to MAE are
$xy\uparrow$, $yz\uparrow$, and $xz\uparrow$, and $xy\downarrow$,
$yz\downarrow$, $xz\downarrow$, $x^{2}-y^{2}\uparrow\downarrow$, and
$z^{2}\uparrow\downarrow$, respectively. Since the coupling between $t_{2g}$
and $e_{g}$ states occur in both spin channels
($t_{2g}\uparrow$-$e_{g}\uparrow\downarrow$), the net contribution to MAE
turned out to be zero. The $t_{2g}$ states couple in opposite spin-channel,
hence, the contribution from $t_{2g}\uparrow-t_{2g}\downarrow$ coupling to MAE
comes from the spin-non-conserving term of $H_{SO}$. Using non-vanishing
matrix elements $\bra{xz}\hat{L_{z}}\ket{yz}=1$,
$\bra{xy}\hat{L_{x}}\ket{xz}=1$ and Eq. 9, MAE and is given by,

$\displaystyle
MAE^{[001]-[111]}_{KReC}=\frac{\lambda^{2}}{4}\sum_{o^{\sigma},u^{\sigma^{\prime}}}\big{(}\frac{|\bra{o^{\sigma}}\sqrt{2}\hat{L_{x}}+\sqrt{2}\hat{L_{y}}-2\hat{L_{z}}\ket{u^{\sigma^{\prime}}}|^{2}}{8\epsilon_{o^{\sigma}}-\epsilon_{u^{\sigma^{\prime}}}}$
$\displaystyle-\frac{|\bra{o^{\sigma}}\hat{L_{x}}\ket{u^{\sigma^{\prime}}}|^{2}}{\epsilon_{o^{\sigma}}-\epsilon_{u^{\sigma^{\prime}}}}\big{)}$
$\displaystyle=\frac{\lambda^{2}}{4}\big{(}\frac{1}{4\epsilon_{xy\uparrow-
yz\downarrow}}-\frac{7}{4\epsilon_{xy\uparrow-xz\downarrow}}$
$\displaystyle+\frac{1}{4\epsilon_{yz\uparrow-xz\downarrow}}\big{)}<0.$

Since the energy differences between the band centers of the occupied and
unoccupied $t_{2g}$ orbitals are the same, the MAE of KReC turned out to be
negative. Hence, the [001] magnetization direction is energetically favored.

For KOsC, while the contribution to MAE from
$t_{2g}\uparrow\downarrow$-$e_{g}\uparrow\downarrow$ and
xy$\uparrow$$\downarrow$-yz/xz$\downarrow$ couplings is negligible, the finite
contribution comes from yz/xz $\uparrow$-yz/xz$\downarrow$ interactions and is
given by,

$\displaystyle
MAE^{[001]-[110]}_{KOsC}=\frac{\lambda^{2}}{4}\sum_{o^{\sigma},u^{\sigma^{\prime}}}\frac{|\bra{o^{\sigma}}L_{z}-L_{x}\ket{u^{\sigma^{\prime}}}|^{2}}{\epsilon_{o^{\sigma}}-\epsilon_{u^{\sigma^{\prime}}}}$
$\displaystyle=\frac{\lambda^{2}}{2\epsilon_{yz\uparrow-xz\downarrow}}>0.$

Therefore, KOsC possesses an easy plane of magnetization.

Similarly, for KIrC, the MAE is given by,

$\displaystyle
MAE^{[100]-[111]}_{KIrC}=\frac{\lambda^{2}}{4}(\frac{1}{2\epsilon_{yz\downarrow-
xz\downarrow}}+\frac{1}{4\epsilon_{yz\downarrow-xy\downarrow}}$
$\displaystyle-\frac{3}{4\epsilon_{xz\downarrow-xy\downarrow}})<0,$

and as a result, the easy axis lies along $\hat{x}$.

Qualitatively, the origin of large MAE in KOsC compared to KReC and KIrC can
be understood from the evaluated MAE expressions. For KReC and KOsC, while
terms contributing positively as well as negatively to MAE are present to
lower the MAE magnitude, for KOsC, only positively contributing term is
present. Hence, KOsC exhibits a large MAE.

Table 4: The estimated anisotropic constants to determine the easy axis/plane of magnetization of antifluorites. Compound |  $K_{1}$ (meV) |  $K_{2}$ (meV)
---|---|---
KReC | 1.68 | -1.47
KOsC | -7.32 | 3.72
KIrC | 1.41 | -1.61

A further quantitative approach to identify the easy axis/plane is through the
estimation of anisotropic constants $K_{1}$ and $K_{2}$ involved in the
expression of crystal anisotropy energy $E(\theta$) = $K_{1}$ Sin2 $\theta$ \+
$K_{2}$ Sin4 $\theta$.

Figure 9: The variation in energy difference between spin direction along
$\phi$=0 and $\hat{z}$ as a function of the polar angle $\theta$.

These constants are estimated by fitting the energy differences
$E(\theta)$-$E(\theta=0)$ obtained by rotating the magnetization axis from
[001] to [100] directions from DFT calculations (the blue, red, and magenta-
filled circles in Fig. 9) with the parametric equation (black solid line). The
obtained values of $K_{1}$ and $K_{2}$ are listed in Table-4. The conditions
for easy $c$-axis (basal-plane) are $K_{2}$ $>$ -$K_{1}$ ($K_{2}$ $<$
-$K_{1}/2$) [47], which are very well satisfied by KReC (KOsC and KIrC).
Therefore, the second-order perturbative analysis of SOC and quantification of
anisotropic constants provide good insights into the magnetization direction
and large/giant MAE obtained from total energy DFT calculations.

##  V Summary

To summarize, by pursuing density functional theory calculations and a toy
model, we examine the electronic and magnetic structure of recently emerging
vacancy-ordered antifluorites K2ReCl6 (Re-$d^{3}$), K2OsCl6 (Os-$d^{4}$), and
K2IrCl6 (Ir-$d^{5}$). The competition between spin-orbit coupling $\lambda$
and local exchange field $\Delta_{ex}$ suggests that the breakdown of
$J_{eff}$ = 3/2 and $J_{eff}$ = 0 states in KReC and KOsC is driven by the
presence of large $\Delta_{ex}$ ($\Delta_{ex}$ $>>$ $\lambda$) respectively.
As a consequence, while KReC exhibits high-spin $S$ = 3/2 trivial band
insulating state, KOsC stabilizes in an intermediate-spin $S$ = 1 Mott
insulating state. On the other side, due to weak $\Delta_{ex}$ ($\Delta_{ex}$
$<<$ $\lambda$), the $J_{eff}$ = 3/2 and $J_{eff}$ = 1/2 manifolds remains
unperturbed to stabilize KIrC in $J_{eff}$ = 1/2 spin-orbit-assisted Mott
insulating state. The nearest and next-nearest-neighbor (NN and NNN) spin-
exchange interactions ($J_{1}$ and $J_{2}$) estimated using the spin-dimer
method infers that the antifluorites are very weakly coupled magnetic systems.
For KReC and KOsC, the $J_{1}$ and $J_{2}$ are found to be antiferromagnetic
and ferromagnetic, leading to a Type-I magnetic ground state, whereas, for
KIrC, both $J_{1}$ and $J_{2}$ are antiferromagnetic and hence a Type-III
ground state stabilizes in KIrC. The estimated anisotropic constants $K_{1}$
and $K_{2}$ satisfy the conditions for easy $c$-axis ($K_{2}$ $>$ -$K_{1}$)
and basal-plane ($K_{2}$ $<$ -$K_{1}/2$) of magnetization for KReC and
KOsC/KIrC, respectively.

The experimentally and theoretically estimated Curie-Weiss temperatures are in
good agreement with the theoretically estimated value of onsite Coulomb
repulsion $U$ using linear response theory ($U$ $\approx$ 3 eV for KIrC and
$U$ $\approx$ 7 eV for KReC). Interestingly, the antifluorites exhibit large
magnetic anisotropy energy (MAE) $\approx$ 0.6-4 meV/transition metal which is
one-to-two order higher than traditional MAE materials such as transition
metals and multilayers formed out of them. Furthermore, we find that
compressive/tensile strain can change the easy and hard axes as well as the
magnetic ordering. Most importantly, strain enhances the MAE by one order and
becomes giant for KOsC ($\approx$ 20-40 meV/Os) to make these antifluorites
promising applications in magnetic memory and storage devices. The second-
order perturbative analysis of SOC provides good insight into the origin of
large MAE and easy axis/plane of magnetization obtained from total energy
density functional theory calculations.

##  VI Acknowledgement

This work was funded by the Department of Science and Technology, India,
through Grant No. CRG/2020/004330. We also acknowledge the use of computing
resources at HPCE, IIT Madras.

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# Out-of-plane magnetization oscillation in spin Hall device assisted by
field-like torque

Tomohiro Taniguchi<EMAIL_ADDRESS>National Institute of
Advanced Industrial Science and Technology (AIST), Research Center for
Emerging Computing Technologies, Tsukuba 305-8568, Japan

###### Abstract

An excitation of a large-amplitude out-of-plane magnetization oscillation in a
ferromagnet by the spin Hall effect is of great interest for practical
applications such as microwave generator and neuromorphic computing. However,
both experimental and theoretical works have revealed that only small-
amplitude oscillation around an in-plane easy axis can be excited via the spin
Hall effect. Here, we propose that an out-of-plane oscillation can be excited
due to an assistance of field-like torque. We focus on an in-plane magnetized
ferromagnet with an easy axis parallel to current direction. We notice that
the field-like torque with an appropriate sign provides an additional field
modifying the dynamic trajectory of the magnetization and drives the auto-
oscillation. The condition on the sign of the field-like torque is satisfied
for typical nonmagnet used in spin Hall devices such as tungsten.

Spin-orbit interaction in nonmagnet scatters spin-up and spin-down electrons
to the opposite direction and generates pure spin current, which is known as
the spin Hall effect Dyakonov and Perel (1971); Hirsch (1999); Kato et al.
(2004); Kimura et al. (2007); Ando et al. (2008). The pure spin current
injected into an adjacent ferromagnet excites spin-transfer torque on its
magnetization and induces magnetization dynamics. Whereas the magnetization
switching between two stable states has been mainly studied for the purpose of
three-terminal non-volatile memory Liu et al. (2012a); Lee et al. (2013);
Cubukcu et al. (2014); Yu et al. (2014); Garello et al. (2014); You et al.
(2015); Torrejon et al. (2015); Lau et al. (2016); van den Brink et al.
(2016); Oh et al. (2016), another fascinating dynamics induced by the spin-
transfer torque is an auto-oscillation of the magnetization. It was
demonstrated experimentally that the spin Hall effect can induces the
oscillation in in-plane magnetized ferromagnet Liu et al. (2012b). The result
indicates a possibility to apply the spin-Hall devices to other practical
devices such as microwave generators and neuromorphic devices Demidov et al.
(2014); Awad et al. (2017); Kudo and Morie (2017).

An intriguing challenge in the study of spin-transfer torque induced auto-
oscillation is an excitation of a large-amplitude out-of-plane oscillation
because the emitted power outputted from the oscillator is proportional to the
oscillation amplitude. Unfortunately, however, the oscillation amplitude of
the auto-oscillation by the spin Hall effect was small because the oscillation
occurred around an in-plane easy axis Liu et al. (2012b). Although a large
amplitude oscillation might be possibly excited in the in-plane magnetized
ferromagnet, an external magnetic field will be necessary Kiselev et al.
(2003), which is undesirable for practical purpose. Another proposal requires
time-dependent input and accurate control of the magnetic anisotropy energy,
where both minima and maxima of the energy landscape locate in the in-plane
direction whereas saddle points locate out-of-plane direction Shirokura and
Hai (2020). It was also proved theoretically that the spin Hall effect does
not excite an out-of-plane oscillation in a perpendicularly magnetized
ferromagnet Taniguchi (2015); Taniguchi et al. (2015). Regarding such a
current status, it is of great interest to propose a method to excite an out-
of-plane magnetization oscillation by the spin Hall effect.

The purpose of this work is to propose a geometry and conditions to excite an
out-of-plane magnetization oscillation by the spin Hall effect theoretically.
We focus on an in-plane magnetized ferromagnet, as in the case of previous
work Liu et al. (2012b), however, the geometry is different. The previous work
used a ferromagnet having an easy axis orthogonal to current direction.
According to Ref. Fukami et al. (2016), we call this geometry type Y, for
convention. On the other hand, we focus on a ferromagnet with its easy axis
parallel to the current, which was named as type X Fukami et al. (2016). In
addition, the field-like torque is taken into account in equation of motion of
the magnetization. The field-like torque provides an effective magnetic field
pointing in a direction of the in-plane hard axis and modifies an energy
landscape. As a result of the competition between the spin-transfer torque and
the damping torque including the contribution from the field-like torque, an
out-of-plane magnetization oscillation can be excited when the field-like
torque has an appropriate sign. The condition on the sign of the field-like
torque is satisfied in, for example, tungsten Lau and Hayashi (2017).
Therefore, it will be possible to examine the present proposal experimentally.

Figure 1:  (a) Schematic illustration of the system. Electric current density
$j$ flowing in the $x$ direction in the bottom nonmagnet generates pure spin
current with spin polarization (red arrow) in the $y$ direction, which is
injected into the top ferromagnet and excites spin-transfer torque on the
magnetization $\mathbf{m}$. (b) Time evolutions of $m_{x}$ (red) for the
current density of $j=90$ MA/cm2 and $m_{z}$ (black) for $j=340$ MA/cm2.

Figure 1(a) is a schematic illustration of the system under consideration. The
$x$ axis is parallel to current direction, whereas the $z$ axis points to the
perpendicular direction. The current flowing in the bottom nonmagnet generates
pure spin current injected into the top ferromagnet and excites spin-transfer
torque on magnetization. We use macrospin assumption to describe the
magnetization dynamics excited by the spin-transfer torque because its
validity was experimentally confirmed Fukami et al. (2016). The Landau-
Lifshitz-Gilbert (LLG) equation of the magnetization is given by

$\begin{split}\frac{d\mathbf{m}}{dt}=&-\gamma\mathbf{m}\times\mathbf{H}+\alpha\mathbf{m}\times\frac{d\mathbf{m}}{dt}\\\
&-\frac{\gamma\hbar\vartheta
j}{2eMd}\mathbf{m}\times\left(\mathbf{e}_{y}\times\mathbf{m}\right)-\frac{\gamma\hbar\beta\vartheta
j}{2eMd}\mathbf{m}\times\mathbf{e}_{y},\end{split}$ (1)

where $\mathbf{m}$ and $\mathbf{e}_{y}$ are the unit vectors pointing in the
magnetization and $y$ directions, respectively. The gyromagnetic ratio and the
Gilbert damping constant are denoted as $\gamma$ and $\alpha$, respectively.
The magnetic field consists of shape and interfacial crystalline anisotropy
fields as

$\mathbf{H}=\begin{pmatrix}-4\pi MN_{x}m_{x}\\\ -4\pi MN_{y}m_{y}\\\ (H_{\rm
K}-4\pi MN_{z})m_{z}\end{pmatrix},$ (2)

where $N_{\ell}$ ($\ell=x,y,z$) is the demagnetization coefficient, whereas
$H_{\rm K}$ represents the interfacial contribution to the perpendicular
magnetic anisotropy Yakata et al. (2009); Ikeda et al. (2010); Kubota et al.
(2012). Since we are interested in a ferromagnet having its easy axis along
the $x$ direction, $N_{x}<N_{y}<N_{z}$ and $H_{\rm K}-4\pi MN_{z}<0$. The
saturation magnetization and thickness of the ferromagnet are denoted as $M$
and $d$, respectively. The spin Hall angle, which corresponds to the charge-
to-spin conversion efficiency, is $\vartheta$, whereas the dimensionless
parameter $\beta$ is the ratio of the field-like torque to the damping-like
torque, which correspond to the fourth and third terms on the right-hand side
of Eq. (1). The electric current density flowing in the bottom nonmagnet is
$j$. The values of the parameters are derived from recent experiment Shiokawa
et al. (2021) as $M=1540$ emu/cm3, $H_{\rm K}=14.4$ kOe, $\gamma=1.764\times
10^{7}$ rad/(Oe s), $\alpha=0.04$, and $d=2$ nm. The long and short diameters
of the ferromagnet are 297 nm in the $x$ direction and 122 nm in the $y$
direction. Then, according to Ref. Beleggia et al. (2005), the demagnetization
coefficients are estimated to be $N_{x}=0.008925$, $N_{y}=0.031527$, and
$N_{z}=0.959548$. The spin Hall angle is $\vartheta=-0.338$ for tungsten
Shiokawa et al. (2021). The initial state is set to be an stable state,
$\mathbf{m}=-\mathbf{e}_{x}$. The fourth-order Runge-Kutta method is applied
to solve Eq. (1) numerically. Whereas we focus on the dynamics at zero
temperature in the following, the following conclusions are unchanged even at
finite temperature, as shown in supplementary material.

As discussed below, the parameter $\beta$ of the field-like torque plays a key
role to excite an auto-oscillation. The parameter $\beta$ includes not only
the contributions from bulk spin Hall effect and interfacial spin-orbit
interaction but also the contribution from the Oersted field. Although several
efforts have been made to separate these contributions by using a variety of
experimental techniques Lau and Hayashi (2017); Pai et al. (2012); Kim et al.
(2013); Nakayama et al. (2013); Althammer et al. (2013); Hayashi et al.
(2014); Kim et al. (2016); Takeuchi et al. (2018), the magnetization dynamics
are affected by all of them. Therefore, we regard $\beta$ in this study as the
total value including these contributions. Accordingly, the sign of $\beta$
depends on materials through the bulk and interfacial contributions, as well
as structures. For example, $\beta$ of systems with platinum, tungsten, and
tantalum are positive Lau and Hayashi (2017). In addition, the magnitude of
$\beta$ could be on the order of $1$ Kim et al. (2013). Therefore, we use
$\beta=0.3$ in the following, whereas the results for different values of
$\beta$ are shown in supplementary material.

Figures 1(b) shows the time evolutions of $m_{x}$ (red) for the current
density of $j=90$ MA/cm2 and $m_{z}$ (black) for $j=340$ MA/cm2. The
magnetization slightly deviates from the initial state but stays close to the
easy axis when the current density is small. On the other hand, the
magnetization moves to the $z$ direction when the current is large. In both
cases, the magnetization does not show oscillation. In the middle current
region, however, we find auto-oscillation of the magnetization as follows.

Figure 2(a) shows the time evolutions of $m_{x}$ (red), $m_{y}$ (blue), and
$m_{z}$ (black) for the current density of $j=97$ MA/cm2. The magnetization
finally saturates to an oscillation state after deviating from the initial
state. Figure 2(b) shows the dynamical trajectory of the magnetization in a
steady state. It shows the magnetization oscillates around the negative $y$
axis. Figure 2(c) shows the Fourier transformation $|m_{x}(f)|$ of $m_{x}$.
The spectrum has multiple peaks reflecting a complex trajectory shown in Fig.
2(b). The first main peak appears at $1.3$ GHz, which is higher than that
found in two-terminal spintronics devices without external magnetic field Zeng
et al. (2012). Note that the oscillation occurs around an effective field
associated by the field-like torque. In fact, by taking into the fact that
$\beta\vartheta<0$, the field-like torque in Eq. (1) can be regarded as a
torque due to magnetic field in the negative $y$ direction. Therefore, we can
conclude that the field-like torque is the origin of the auto-oscillation.

Figure 2:  (a) Time evolutions of $m_{x}$ (red), $m_{y}$ (blue), and $m_{z}$
(black), (b) the dynamical trajectory in a steady state, and (c) the Fourier
transformation of $m_{x}$ for the current density of $j=97$ MA/cm2. Similar
plots with different current density, $j=160$ MA/cm2, are shown in (d)-(f).

Although the oscillation in Fig. 2(b) occurs around the $y$ axis, which is
orthogonal to the easy ($x$) axis of the present system, the oscillation
trajectory is similar to a small-amplitude oscillation observed in type-Y
device Liu et al. (2012b), where the magnetization oscillates around its easy
axis parallel to the $y$ axis Taniguchi (2017). We should, however, note that
the output power of the present system becomes larger than that from the
type-Y device due to the following reason. In typical spin Hall devices,
magnetic tunnel junctions (MTJs) are fabricated on the nonmagnetic electrode
Liu et al. (2012b); Fukami et al. (2016), where the ferromagnet in Fig. 1 is a
free layer. The magnetization in the reference layer usually points to the
direction parallel to the easy axis of the free layer to detect the
magnetization switching. The shape magnetic anisotropy also prefers the easy
axis of the reference layer parallel to that of the free layer. Therefore, the
magnetization in the reference layer points to the $x$ direction in the
present system, whereas it points to the $y$ direction in the type-Y device.
The output power is proportional to the projection of the magnetization
$\mathbf{m}$ to the direction of the magnetization in the reference layer. In
the present system, $m_{x}$ varies as $-1\lesssim m_{x}\lesssim 1$, whereas
$m_{y}$ in typical type-Y devices varies as $0\lesssim m_{y}\lesssim 1$
Taniguchi (2017). The difference comes from the fact that the magnetization in
the present system oscillates around the $y$ axis, which is orthogonal to the
in-plane easy axis, whereas the magnetization oscillation in type-Y devices
occurs around the easy axis. As a result, the emission power of the present
system could be larger than that from type-Y devices, although the
magnetization oscillates around the in-plane axis.

Figures 2(d)-(f) show the time evolution of the magnetization, the dynamical
trajectory in a steady state, and the Fourier transformation of $m_{x}$ for
the current density of 160 MA/cm2. The spin-transfer torque due to a
relatively large current moves the magnetization toward the $z$ direction and
excites an out-of-plane auto-oscillation. The oscillation frequency shifts to
higher frequency region and becomes 4.4 GHz. Since $m_{x}$ also varies as
$-1\lesssim m_{x}\lesssim 1$ here, a large output power could be expected. We
also note that an out-of-plane oscillation as such does not occur in type-Y
Liu et al. (2012b) because the field-like torque in this geometry prefers the
in-plane oscillation.

Figure 3:  Dependence of the frequency of the magnetization oscillation on the
current density. The field-like torque is $\beta=0.3$. The small and large
current regions correspond to the in-plane and out-of-plane oscillations shown
in Figs. 2(b) and 2(e), respectively.

Figure 3 summarizes the frequency corresponding to the first main peak in the
Fourier space. It is zero when the magnetization stays near the initial state
($j\leq$ 92 MA/cm2) or saturates to the $y$ direction ($j\geq$ 324 MA/cm2).
The frequency becomes finite when the magnetization shows an auto-oscillation.
In the small current region $93$ MA/cm2$\leq j\leq$ 103 MA/cm2, the frequency
decreases with increasing the current, where the magnetization oscillates
around the $y$ axis, as shown in Fig. 2(b). On the other hand, the frequency
increases with increasing the current in the large current region 104 MA/cm2
$\leq j\leq$ 323 MA/cm2, where the magnetization oscillates around the $z$
axis, as shown in Fig. 2(e). The maximum frequency reaches close to 10 GHz. We
emphasize that no external magnetic field is used in this study, whereas the
previous works on the auto-oscillation in spintronics devices have frequently
used external magnetic field Kiselev et al. (2003); Rippard et al. (2004);
Sankey et al. (2005); Krivorotov et al. (2007); Houssameddine et al. (2007);
Kubota et al. (2013). We also note that the relation between current and
frequency for different values of $\beta$ is shown in supplementary material.
The supplementary material also provides an analytical formula of the current
density to excite an in-plane oscillation.

Figure 4:  Schematic illustration of top view of the system. The red arrow
indicates the direction of an effective field $\mathbf{H}_{\rm FL}$ associated
with the field like torque. The pink arrow indicates the direction of the
damping torque pointing in the direction of $\mathbf{H}_{\rm FL}$, whereas the
blue arrow indicates the direction of the damping-like torque. The dotted
lines represent the shapes of the top ferromagnet and the bottom nonmagnet.

We notice that the auto-oscillation is excited when $\beta$ is positive. In
fact, auto-oscillations have not been observed in this geometry when the
field-like torque is absent Fukami et al. (2016); Taniguchi (2019); Taniguchi
et al. (2020) or $\beta$ is negative Shiokawa et al. (2021). This argument is
also confirmed from the critical-current formula to excite the auto-
oscillation shown in supplementary material, where the critical current is
inversely proportional to $\sqrt{\beta}$, and therefore, becomes finite only
for a finite $\beta$. Let us explain its physical meaning here. For
convention, we assume that $j>0$ and $\vartheta<0$, as used in the above
calculations, although the following discussion is satisfied even when the
sign of $j$ and/or $\vartheta$ are reversed. Auto-oscillations occurs as a
result of the balance between the spin-transfer torque and the damping torque
Bertotti et al. (2009), where the spin-transfer torque refers the damping-like
torque given by $-[\gamma\hbar\vartheta
j/(2eMd)]\mathbf{m}\times(\mathbf{e}_{y}\times\mathbf{m})$ in Eq. (1). The
damping torque induces a relaxation of the magnetization to the direction of
the effective magnetic field. We should remind that the field-like torque
provides an effective field pointing in the negative $y$ direction because
$\beta$ is positive; see Fig. 4 where $\mathbf{H}_{\rm
FL}=[\hbar\beta\vartheta j/(2eMd)]\mathbf{e}_{y}$ extracted from Eq. (1)
represent the effective field associated by the field-like torque. Therefore,
the damping torque acting on the magnetization $\mathbf{m}$ has a projection
to the negative $y$ direction, as schematically shown in Fig. 4. On the other
hand, the spin-transfer torque moves the magnetization to the positive $y$
direction, as shown in Fig. 4. Accordingly, there is a balance between the
spin-transfer torque and damping torque, resulting in an excitation of
sustainable auto-oscillation. If $\beta$ is negative, however, both the
effective field associated by the field-like torque and the spin-transfer
torque points to the positive $y$ direction. In this case, these torques do
not cancel each other, and thus, auto-oscillation cannot be excited.
Therefore, auto-oscillation can be excited only when $\beta$ is positive.

In summary, we showed that an out-of-plane auto-oscillation of the
magnetization can be excited in spin Hall devices with an in-plane magnetized
ferromagnet. The origin of the auto-oscillation is the field-like torque with
an appropriate sign. The field-like torque provides an effective field
pointing in the direction orthogonal to the in-plane easy axis. The damping
torque toward this effective field balances with the damping-like torque and
excites sustainable oscillation. The oscillation found here has the following
advantages for practical applications. First, a large amplitude oscillation
results in a large emission power. Since $m_{x}$ varies in the range of
$-1\lesssim m_{x}\lesssim 1$, the emission power could be maximized in the
present geometry. Second, no external magnetic field is necessary. This is
because the field-like torque provides an effective field to sustain the
oscillations. Third, the oscillation is excited in a ferromagnet with the easy
axis parallel to the current. The type-X structure has a narrower injected-
current cross-section comoared with type-Y devices; thus, it can be used to
reduce power consumption of application applications such as non-volatile
memory Fukami et al. (2016).

See supplementary material where the magnetization dynamics at finite
temperature, the current-frequency relation for a variety of $\beta$, and the
analytical formula of the threshold current density to excite the in-plane
oscillation are summarized.

The author acknowledges to Masamitsu Hayashi, Yohei Shiokawa, Shinji Isogami,
Takehiko Yorozu, Tomoyuki Sasaki, and Seiji Mitani for valuable discussion.
This work was supported by funding from TDK Corporation.

## Data availability

The data that support the findings of this study are available from the
corresponding authors upon reasonable request.

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§ DIAGRAMMATIC COACTIONS FROM HYPERGEOMETRIC FUNCTIONS

At one loop, we know from refs. <cit.> how to construct
a coaction on Feynman graphs with all the features outlined in the previous section.
The situation is very different for higher loops.
As already alluded to previously, there are many obstacles for establishing a
general diagrammatic coaction beyond one loop.
First, while at one loop it is possible to write down a complete basis of master
integrals for any integral, the same is not true beyond one loop,
and one needs to determine
the basis of master integrals from scratch for every family of integrals.[
This should better be done in a consistent manner: if a Feynman graph
$A$ is obtained by pinching the propagators of another graph $B$, and we have
already constructed the basis of masters for the integral associated with $A$,
then this information should be reused in constructing the basis for the master
integral associated with $B$.
Second, the master integrals may not be expressible in terms of multiple polylogarithms.
Indeed, it is known that starting from two loops
a wider class of functions is required.  Finally, little is known about how to find
relations between cuts and build a basis of integration contours at higher loops.

In order to make progress in our understanding of the new features beyond one loop,
it is therefore important to analyse explicit examples of higher-loop integrals.
In the remainder of this paper, we will do so for some classes of two-loop integrals. As anticipated in the introduction,
our strategy will be to use the global coaction on hypergeometric
functions to compute the coaction of the set of master integrals associated with the chosen diagram.
The entries of this coaction will themselves be hypergeometric functions, which can be identified as elements of the diagrammatic coaction: the left entries with a spanning set of master integrals, and the right entries as a spanning set of cuts of the integral under consideration.
By separately computing these (cut) integrals in dimensional regularisation, and then expressing the entries in the
aforementioned global coaction in terms of them, we will explicitly construct diagrammatic
coactions for each case considered.
Before we apply this method at two loops, we review in this section the global coaction
on hypergeometric functions, and how it can be used to reproduce, and even prove, the conjectured
diagrammatic coaction of the one-loop bubble given in eq. (<ref>).

§.§ The coaction on hypergeometric functions

Let us start by defining the class of hypergeometric integrals on which the coaction of ref. <cit.> can be applied. We consider integrands of the form $\omega=\Phi\varphi$, with

\begin{equation}
\label{eq:Phiphi}
\Phi({\bf u}) = \prod_{I} P_I({\bf u})^{a_I \eps}\quad \textrm{and} \quad
\varphi_{n_1\ldots n_K}({\bf u}) = d{\bf u} \prod_{I=1}^K P_I({\bf u})^{n_I}\,,
\end{equation}

where $n_I$ are integers and $P_I$ are polynomials in the kinematic variables $x_j$ and the integration variables $u_i$ with $d\bold{u}=du_1\wedge\ldots\wedge du_n$. In the following we consider the family of integrals defined by letting the integers $n_I$ vary,
with the $a_I$ held fixed. The framework to discuss this type of integrals is that of twisted (co)homology,
where the twist is defined by $\Phi$ <cit.>—see also
refs. <cit.>
for applications of twisted (co)homology to Feynman integrals.
The distinction between
standard (co)homology and twisted (co)homology will not be crucial for most of
the discussion in this paper.
If the structure of the polynomials $P_I({\bf u})$ is simple enough, one can construct
explicit bases for the homology and cohomology groups associated to this integral.
These bases can be used
with eq. (<ref>) to construct a coaction on these hypergeometric functions.
For more details about the construction of this coaction we refer to
ref. <cit.>. Here we simply summarise the case of Gauss' hypergeometric
function $_2F_1$.

Gauss' hypergeometric function admits the Euler representation

\begin{equation}\label{eq:2f1Simp}
\,_2F_1\left(\alpha,\beta;\gamma;x\right)
\int_0^1 u^{\alpha-1}
(1-xu)^{-\beta} du\,.
\end{equation}

We will impose the condition that $\alpha=n_\alpha+a\eps$, $\b=n_\b+b\eps$ and
$\g=n_\g+c\eps$ with integer $n_\alpha$, $n_\beta$ and $n_\g$, so that
Φ(u)    = u^a
(1-xu)^-b ,

φ_n_αn_βn_γ(u)    =
(1-xu)^-n_β du .
Under this condition, the integral in eq. (<ref>) defines a meromorphic
function of $\eps$, and  the Laurent coefficients of the expansion around
$\epsilon=0$ are linear combinations of MPLs with rational coefficients.

It is well known that the
homology and cohomology groups associated with Gauss' hypergeometric function
are two-dimensional. We choose as a basis of the homology group the contours

\begin{equation}
\label{Gamma_basis_2F1}
\gamma_1=[0,1]\,,\qquad
\gamma_2=[0,1/x]\,,
\end{equation}

which go between two zeros of the integrand (strictly speaking, between zeros of the polynomials that define the twist).
As a dual basis for the integrands, we take

\begin{align}
\label{omega_2F1}
\begin{split}
&\omega_1=(c-a)\epsilon\,\Phi(u)\varphi_{101}(u) = (c-a)\epsilon\,u^{a\eps} (1-u)^{-1+(c-a)\eps}
&\omega_2= -b\epsilon x\,\Phi(u)\varphi_{112}(u) = -b\epsilon x\,u^{a\eps} (1-u)^{(c-a)\eps}
\end{split}
\end{align}

The normalisation is chosen to satisfy the duality condition $c_{ij} = \delta_{ij}$, simplifying the form of the coaction.
From eq. (<ref>) we then obtain <cit.>:

\begin{align}
\nonumber\Delta\Big({}_2F_1(\a,\b;\g;x)\Big) &=
{}_2F_1(1+a\eps,b\ep;1+c\ep;x) \otimes {}_2F_1(\a,\b;\g;x) \\
\label{eq:coaction2F1}&- \frac{b\ep}{1+c\ep}\,
{}_2F_1(1+a\ep,1+b\ep;2+c\ep;x) \\
\nonumber& ~~\otimes \frac{\Gamma(1-\b)\Gamma(\g)}{\Gamma(1-\b+\a)\Gamma(\g-\a)}
\end{align}

Let us make a few comments about this result. First, the particular form of the coaction in eq. (<ref>)
depends on the choice of bases made for the contours and integrands. Other
choices would lead to equivalent formulas for the coaction, related
to eq. (<ref>) through standard contiguous and analytic
continuation relations of each of the entries.
Second, as already noted, whenever $n_{\alpha}, n_\beta, n_\gamma$ are integers, then Gauss' hypergeometric function
can be expanded into a Laurent series involving only MPLs. The expansion of the right-hand side in the global coaction in eq. (<ref>)
will be consistent with computing the local coaction on MPLs at each order in the expansion. This was conjectured in ref. <cit.>, and proven
in ref. <cit.>. Note that the right entries in our coaction, both global and local, always have to be interpreted modulo branch cuts, i.e., in the right entries any two expressions that are related by analytic continuation are considered identical. In the case of MPLs, this corresponds to working modulo $i\pi$, which is in practice how we implement this constraint in this paper. Alternatively, one can work with single-valued versions of hypergeometric functions (and MPLs), which is the approach taken in ref. <cit.>.

§.§ The diagrammatic coaction on one-loop integrals reloaded

Let us now discuss how we can relate the global coaction on hypergeometric functions
to the diagrammatic coaction on one-loop integrals, having in mind that we would like
to use the same strategy to construct a diagrammatic coaction beyond one loop.
We illustrate our method in this section on the example of the one-loop bubble integral with one massive propagator.

In eq. (<ref>) we presented
the diagrammatic coaction of the one-loop bubble graph with $m_1\neq 0$ and $m_2=0$ in $D=2-2\eps$ space-time dimensions.
In ref. <cit.> it was checked that when all graphs are
replaced by the first few orders of their Laurent expansion in $\eps$, then
eq. (<ref>) reproduces the (local) coaction on MPLs. Based on this
empirical evidence, we conjectured in refs. <cit.> that this
should hold true to all orders in the Laurent expansion. We will now show how we can
recover eq. (<ref>) from the coaction on Gauss' hypergeometric function.
Since the global and local coactions are proven to be equivalent in the case considered <cit.>,
this will in effect prove the conjecture of refs. <cit.> for this integral to all orders in $\eps$.

We start from the well-known fact that the one-loop bubble integral in $D=2-2\eps$ dimensions can be evaluated in terms of Gauss' hypergeometric function (cf., e.g., ref. <cit.>):

\begin{equation}\bsp\label{bubPM}
J_2(p^2;m_1^2)&\, = \frac{i(p^2-m_1^2)}{2}\,\frac{e^{\gamma_E\eps}}{\pi^{1-\eps}}\int\frac{d^Dk}{(k^2-m_1^2)(k+p)^2}\\
e^{\gamma_E  \epsilon } \Gamma (\epsilon) \left(m_1^2-p^2\right)^{-\epsilon}
\, _2F_1\left(-\epsilon ,1+\epsilon ;1-\epsilon
;\frac{p^2}{p^2-m_1^2}\right)\, .
\esp\end{equation}

Note that we have normalised the integral
so that the coefficients in the Laurent expansion
are pure functions (see appendix B in ref. <cit.> for details).

Using eq. (<ref>) and the representation in eq. (<ref>),
it is straightforward to obtain the global coaction on this integral.
We first need to specialise eq. (<ref>) to the hypergeometric function
in eq. (<ref>), which gives

\begin{align}
\label{2F1_for_one-mass_bubble}
\begin{split}
\Delta\Big({}_2F_1(-\epsilon,1+\epsilon;1-\epsilon;x)\Big)&=
\otimes \frac{2\Gamma^2(1-\epsilon)}{\Gamma(1-2\epsilon)} x^{1+\epsilon}\,.
\end{split}
\end{align}

Then, to account for the prefactor in front of the ${}_2F_1$ function in eq. (<ref>), we recall
that $\Delta(f\cdot g)=\Delta(f)\cdot\Delta(g)$ and that <cit.>

\begin{equation}
\label{grouplikeFactor}
\Delta\left[x^\epsilon\,{e^{\gamma_E\eps}\Gamma(1+\eps)}\right]
\,=
\left[x^\epsilon{e^{\gamma_E\eps}\Gamma(1+\eps)}\right]
\otimes
\left[x^\epsilon{e^{\gamma_E\eps}\Gamma(1+\eps)}\right] \,.
\end{equation}

The representation of the coaction obtained by following these steps, however, has no clear
interpretation in terms of Feynman integrals. To reconcile this result with the coaction of
eq. (<ref>), we recall the expressions for the cuts of this
one-loop integral <cit.>,

\begin{equation}\label{bubPMCutM}
\cutRes_{e_1}
J_2(p^2;m_1^2)=\frac{e^{\gamma_E  \epsilon }}{\Gamma (1-\epsilon )}
\frac{m_1^2-p^2}{2\,p^2 }\left(-m_1^2\right)^{-\epsilon }\,
_2F_1\left(1,1+\epsilon ;1-\epsilon ;\frac{m_1^2}{p^2}\right),
\end{equation}

\begin{equation}\label{bubPMCutP}
\cutRes_{e_1,e_2}
J_2(p^2;m_1^2)=\frac{ e^{\gamma_E  \epsilon } \Gamma (1-\epsilon ) }{\Gamma (1-2 \epsilon )}
\left(p^2\right)^{\epsilon } \left(p^2-m_1^2\right)^{-2 \epsilon }\,,
\end{equation}

and that of the tadpole integral of mass $m_1^2$ in $D=2-2\eps$ dimensions
(and normalised to start as $1+\mathcal{O}(\epsilon)$),

\begin{equation}\label{eq:tad}
\end{equation}

One then needs to rewrite the coaction obtained by using eq. (<ref>)
in terms of these functions. This step might in general be nontrivial.
For the Gauss hypergeometric function, however, all the analytic continuation and
contiguous (or integration-by-parts) relations are known.
Using such relations, we find that eq. (<ref>) can equivalently
be written as

\begin{align}
\label{2F1_for_one-mass_bubble_rewrite}
\begin{split}
\Delta\Big({}_2F_1(-\epsilon,1+\epsilon;1-\epsilon;x)\Big)&=
\otimes \frac{\Gamma^2(1-\epsilon)}{\Gamma(1-2\epsilon)} x^{\epsilon}
\\&\hspace{-100pt}+
\left(\frac{\Gamma^2(1-\epsilon)}{\Gamma(1-2\epsilon)} x^{\epsilon}
\,.
\end{split}
\end{align}

Comparing this form with the results in eqs. (<ref>), (<ref>) and
(<ref>), it becomes clear that we reproduce the diagrammatic
coaction of eq. (<ref>). We emphasise that the formula for the coaction obtained in this section is not conjectural: indeed, it relies only on the coaction of Gauss' hypergeometric function conjectured in ref. <cit.> and proven in ref. <cit.>. We have therefore presented in this section a complete proof of the diagrammatic coaction in eq. (<ref>)
that was conjectured in ref. <cit.>.

Rewriting the coaction of eq. (<ref>) in a form
that made its relation to (cut) Feynman integrals apparent was easy enough in this
case, because we simply needed to show that it reproduced the diagrammatic coaction
we had previously established. Had we not known the form of the diagrammatic
coaction, however, we could have rediscovered eq. (<ref>)
starting from eq. (<ref>).
To do this, one needs to keep in mind the general properties of the diagrammatic
coaction discussed in section <ref>.
Specifically, in this case to obtain a diagrammatic interpretation of eq. (<ref>)
it would have been sufficient to assume that the bubble and tadpole integrals
in eqs. (<ref>) and (<ref>) form a basis for the left entries of the coaction
and that the cuts in eqs. (<ref>) and (<ref>) form a basis for the right entries.
This assumption is motivated by the form of eq. (<ref>) and the interpretation of its left entries as
spanning master integrands and its right entries as spanning cuts of the original integral considered.
Going beyond one loop, this exercise provides us with a roadmap for constructing a coaction on
specific multi-loop Feynman integrals, as we illustrate in a first example in the next section.

We thank Aris Ioannou for useful discussions.
This work was supported by the U.K. Royal Society through Grant URF\R1\201607 (SA),  by the
European Research Council under the European Union's Horizon 2020 research and innovation programme through grants
647356 (RB) and  637019 (CD), and the STFC Consolidated Grant “Particle Physics at the Higgs Centre” (EG, JM).