# Electronic and optical properties of the fully and partially inverse CoFe2O4
spinel from first principles calculations including many-body effects

Shohreh Rafiezadeh  Department of Physics and Center for Nanointegration
Duisburg-Essen (CENIDE), University of Duisburg-Essen, Duisburg, Germany
Vijaya Begum-Hudde  Department of Materials Science and Engineering,
University of Illinois, Urbana-Champaign, Urbana, IL 61801, USA  Department of
Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University of
Duisburg-Essen, Duisburg, Germany  Rossitza Pentcheva  Department of Physics
and Center for Nanointegration Duisburg-Essen (CENIDE), University of
Duisburg-Essen, Duisburg, Germany

###### Abstract

Using density functional theory (DFT) calculations and state-of-the-art many-
body perturbation theory, we investigate the electronic and optical properties
of the inverse spinel CoFe2O4, a common anode material for photocatalytic
water splitting. Starting with different exchange-correlation functionals, at
the independent particle level we obtain a direct band gap of 1.38 eV
(PBE+$U$, $U$ = 4 eV) and 1.69 eV (SCAN+$U$, $U$ = 3 eV), whereas HSE06
renders an indirect band gap of 2.02 eV. Including quasiparticle effects
within $G_{0}W_{0}$, a larger and indirect band gap is obtained for all
functionals: 1.78 eV (PBE+$U$, $U$= 4 eV), 1.95 eV (SCAN+$U$, $U$ = 3 eV) and
2.17 eV (HSE06) which is 29%, 15% and 5% higher than the independent particle
(IP) band gap, respectively. Excitonic effects, taken into account by solving
the Bethe-Salpeter equation (BSE) lead to a redshift of the optical band gap
to 1.50 (SCAN+$U$, $U$ = 3 eV) and 1.61 eV (HSE06), in good agreement with the
reported experimental values 1.50$-$2.0 eV. The lowest optical transitions in
the visible range, identified by means of oscillator strength, are at 2.0,
3.5, and 5.0 eV, consistent with experimental observations at 2.0, 3.4, and
4.9 eV. We also explored the effect of the degree of inversion: the band gap
is found to decrease from 1.69 ($x=1$) to 1.45 ($x=0.5$), and 1.19 eV ($x=0)$
within the IP approximation with SCAN+$U$, $U$=3 eV. This trend is reversed
after the inclusion of excitonic effects, resulting in a band gap of 1.50,
1.57, and 1.64 eV for $x$ = 1.0, 0.5, and 0.0, respectively. The oscillator
strength analysis of the BSE calculations indicates that both $x$ = 0.0 and
$x$ = 0.5 exhibit transitions below 1 eV with extremely small oscillator
strengths that are absent in the inverse spinel. This corroborates previous
suggestions that these transitions are due to the presence of Co2+ cations at
the tetrahedral sites.

††preprint: APS/123-QED

##  I Introduction

The high demand for large-scale green energy production has led to an
increased necessity for photocatalysts with optimized performance [1, 2]. Iron
and cobalt oxides such as Fe2O3, Co3O4, NiFe2O4 and CoFe2O4(CFO) are promising
candidates due to their desirable properties, such as abundance, low cost,
high chemical stability under reaction conditions and optical transitions in
the visible range [3, 4, 5, 6, 7]. Therefore, a fundamental knowledge of their
optical properties is essential for the catalyst selection process and opens
up possibilities for further optical applications.

In this work, we investigate the electronic and optical properties of the
inverse spinel cobalt ferrite CoFe2O4. In this compound equal amounts of Fe3+
ions occupy octahedral (Oct) and tetrahedral (Tet) sites, with antiparallel
orientation of the spins on the two sublattices. The ferrimagnetic nature of
this material with Curie temperature $T_{\rm C}=790$ K [8] stems from the
ferromagnetically ordered Co2+ ions which are located at octahedral sites, as
shown in Fig. 1. However, in real samples some degree of disorder can occur
which is described by the degree of inversion $x$, quantifying the fraction of
divalent Co cations occupying octahedral sites. The chemical formula,
describing the partially inverted spinel, is (Co1-xFex)Tet(CoxFe2-x)OctO4
where $x$ = 1 presents the perfect inverse spinel structure. The magnitude of
$x$ can significantly influence the electronic, magnetic, and optical
properties of spinels [9, 10, 11].

Despite numerous experimental and theoretical studies [8, 12, 13, 4, 14], the
optical properties of CFO, as well as the effect of the degree of inversion,
are still debated. For example, Holinsworth et al. [15] obtained a direct gap
of 2.80 eV at 4.2K and 2.67 eV at 800K using the Tauc plot approach. By using
the same method Himcinschi et al. [16] reported a direct gap of 1.95 eV,
whereas, Kalam et al. [17] and Ravindra et al. [18] obtained an optical band
gap of 2.5-2.65 eV. In addition, Dileep et al. [19] reported a direct optical
gap of 2.31 eV by employing spatially resolved high resolution electron energy
loss spectroscopy. Singh et al. [20] measured an optical gap of 1.65 eV. which
reduces to 1.55 eV and 1.43 eV upon applying a magnetic field of 400 and 600
Oe, respectively. Sharma and Khare [21] reported an optical gap of 1.58 eV
($T$ = 500∘) and 1.41 eV ($T$ = 700∘) for CoFe2O4 films deposited on quartz
substrates. Recently, Singh et al. [22] showed that the optical band gap
decreases from 1.9 eV to 1.7 eV with increasing nanoparticle size. The larger
nanoparticles were found to have a cation distribution similar to bulk CFO
with respect to the degree of inversion. Overall, the reported experimental
direct optical gaps show a wide variation between $\sim$0.55 - 4.1 eV.
Moreover, optical transitions below 1 eV have been related to crystal field
transitions of tetrahedrally coordinated Co2+ which is not present in the
fully inverse spinel [23].

Figure 1:  Crystal structure of the inverse spinel CoFe2O4: (a) conventional
unit cell, containing 56 atoms, with octahedral (CoOct and FeOct) and
tetrahedral (FeTet) sites (b) primitive unit cell with two formula units, 14
atoms, and (c) high symmetry points in the Brillouin zone of CoFe2O4, adopted
from AFLOW [24].

Besides the variation in the measured band gaps, theoretical calculations also
show a wide range of values from 0.52 to 1.90 eV [25, 19, 15, 26, 27],
depending on the method and exchange-correlation functional used. Density
functional theory (DFT) calculations within the generalized-gradient
approximation (GGA) fail to render the insulating state and predict a half-
metallic behavior instead [28]. Dileep et al. [19] calculated a total indirect
band gap of 0.80 eV in the minority channel using the modified Becke-Johnson
local density approximation. Using GGA in the parameterization of Perdew,
Burke, and Enzerhof (PBE) [29] with an on-site Hubbard Coulomb term
$U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV on the Co and Fe $3d$ electrons an indirect
band gap in the minority spin channel of 0.52 eV was obtained at the GGA
lattice parameter[30] using the VASP code [31, 32] and 0.80 eV [26] with the
QUANTUM ESPRESSO (QE) code [33]. For the GGA+$U$ lattice parameter, a larger
band gap of 0.90 eV [25] was reported. On the other hand, a direct gap of 1.08
eV between the minority valence band and majority conduction band was found by
Pemmaraju et al. [34], employing an atomic self-interaction correction (ASIC)
scheme. In contrast, an indirect band gap of 1.60 eV [30] was obtained with
the hybrid functional(HSE03) [35]. While most theoretical studies have focused
on the inverse spinel, Sharma et al.[11] reported recently that the band gap
of 1.09 eV (PBEsol+$U$,$U_{\rm{Co}}$=$U_{\rm{Fe}}$= 4 eV) for the inverse
spinel is reduced by 6% for $x$ = 0.5. Much smaller band gaps, decreasing from
0.72 eV ($x$ = 1.0) to 0.1 ($x$ = 0.0), were found by Hou et al. [36] using
PBE+$U_{\rm{eff}}$ ($U_{\rm{Co}}$= 4.08, $J_{\rm{Co}}$ = 0.79 eV and
$U_{\rm{Fe}}$= 4.22, $J_{\rm{Fe}}$ = 0.80 eV).

An improved description of the electronic structure beyond DFT can be achieved
by considering many-body effects, e.g. by calculating the quasiparticle
energies by means of the self-energy as a product of the single-particle
Green’s function $G$ and the screened Coulomb interaction $W$, in the $GW$
approximation introduced by Hedin [37]. The single shot $G_{0}W_{0}$ was shown
to yield a good description of the band gap of other spinels such as Co3O4
[38, 39] and ZnFe2O4 [40]. An important aspect is the starting point of the
$GW$ calculation. In particular for transition metal oxides as well as rare
earth compounds, adding an on-site Coulomb term within LDA(GGA)+$U$ renders a
better description than LDA or GGA [41, 42, 43, 44, 45, 40]. For example, Lany
[44] showed that employing a Hubbard $U$ term significantly improves the $GW$
band structure for a series of nonmagnetic, antiferromagnetic and
ferrimagnetic transition metal compounds.

Electron-hole interactions can significantly influence the optical spectrum.
These can be taken into account by solving the Bethe-Salpeter equation (BSE)
[46, 46]. This generally improves the agreement with experiment regarding the
spectral features and energetic positions of the peaks in a wide range of
(transition) metal oxides such as ZnFe2O4 [40], MgAl2O4 [43], LiCoO2 [47],
$\alpha$-Fe2O3 [45], SrTiO3 [48, 49] and MgO [50, 51].

To our knowledge, $G_{0}W_{0}$ and BSE have not been applied previously to
CFO. In this work, starting from different exchange correlation functionals,
we have calculated the optical spectrum of CFO including quasiparticle
corrections within the single-shot $G_{0}W_{0}$ and excitonic corrections by
solving BSE. To evaluate the effect of different exchange-correlation
functionals on the electronic and optical properties of bulk CFO, we have
employed the GGA (PBE) and the strongly constrained and approximately normed
meta-GGA (SCAN) [52] functionals with different Hubbard $+U$ values, as well
as the hybrid functional HSE06 [53]. To gain a deeper understanding of the
impact of cation distribution and in particular the degree of inversion on the
optical properties of CoFe2O4, we calculated the optical spectra additionally
for $x$ = 0.5 and 0.0. Due to the high computational demand of
$G_{0}W_{0}$+BSE calculations, we also tested a model BSE scheme (mBSE) with
lower computational cost for the treatment of static screening [54, 55, 56,
57]. This approach has been applied previously to SrIrO3 (also Sr2IrO4 and
Sr3Ir2O7) [56], SrTiO3 [49], MgO [51] and to a set of transition metal oxide
perovskites like SrTiO3, SrMnO3 and LaVO3 [58] with an overall good agreement
between the mBSE and $G_{0}W_{0}$+BSE spectra.

The paper is structured as follows: In Sec. II the computational details are
presented. The results are discussed in Sec. III. Specifically, Sec. III.1 is
dedicated to the ground state structural and electronic properties of CFO,
whereas Sec. III.1.2 presents the quasiparticle (QP) band structure. Sec.
III.2 discusses the optical properties of CFO, in particular, we present and
analyze the real and imaginary parts of the dielectric function and the
absorption coefficient at different levels of treatment with different
starting exchange-correlation functionals. Finally, in Sec. III.3 we assess
the effect of the degree of inversion and cation distribution on the
structural, electronic, and optical properties of CFO. The results are
summarized in Sec. IV. In Appendix A, the spectrum of the inverse spinel
obtained with model BSE is compared to the $G_{0}W_{0}$+BSE spectrum.

Table 1:  The calculated lattice constant $a$ (Å), magnetic moment ($\mu_{\rm{B}}$) and band gap of CoFe2O4 in the independent particle (IP) and quasiparticle (QP) approximation, using PBE+$U$, SCAN+$U$, and HSE06 functionals with a Hubbard $U$ term applied on the Co and Fe 3$d$ states. $E_{\rm{gap}}^{\rm{TOT}}$ is the total band gap in eV. $E_{\rm{gap}}^{\rm{dn}}$ and $E_{\rm{gap}}^{\rm{up}}$ are the band gaps in the minority and majority spin channels, respectively. Index ”(i)” and ”(d)” denotes an indirect and direct band gap, respectively. |  | magnetic moments ($\mu_{\rm{B}}$) | IP | $G_{0}W_{0}$ |
---|---|---|---|---|---
Functional/$U$ in eV |  $a$ (Å) | CoOct | FeTet | FeOct | $E_{\rm{gap}}^{\rm{up}}$ | $E_{\rm{gap}}^{\rm{dn}}$ | $E_{\rm{gap}}^{\rm{TOT}}$ | $E_{\rm{gap}}^{\rm{up}}$ | $E_{\rm{gap}}^{\rm{dn}}$ | $E_{\rm{gap}}^{\rm{TOT}}$ |
PBE+$U$, $U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 | 8.39 | 2.65 | -4.08 | 4.21 | 1.74(i) | 0.92(i) | 0.92(i) | 2.54(i) | 1.32(i) | 1.32(i) |
PBE+$U$, $U_{\rm{Co}}$=$U_{\rm{Fe}}$= 4 | 8.40 | 2.70 | -4.29 | 4.41 | 2.05(i) | 1.53(i) | 1.38(d) | 2.68(d) | 1.78(i) | 1.78(i) |
PBE+$U$, $U_{\rm{Co}}$= 5, $U_{\rm{Fe}}$= 4 | 8.39 | 2.78 | -4.30 | 4.42 | 2.28(i) | 2.48(i) | 1.49(d) | 2.83(d) | 2.75(i) | 2.07(d) |
SCAN+$U$, $U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 | 8.33 | 2.70 | -4.22 | 4.32 | 2.45(i) | 1.72(i) | 1.69(d) | 3.08(i) | 1.95(i) | 1.95(i) |
SCAN+$U$, $U_{\rm{Co}}$=$U_{\rm{Fe}}$= 4 | 8.34 | 2.77 | -4.40 | 4.51 | 2.75(i) | 2.31(i) | 2.11(d) | 3.34(d) | 2.39(i) | 2.39(i) |
HSE06 | 8.37 | 2.61 | -3.98 | 4.09 | 3.30(i) | 2.02(i) | 2.02(i) | 3.65(i) | 2.17(i) | 2.17(i) |
Experiment |  8.39111Reference [59, 60, 61]. |  |  |  |  |  |  |  |  |  |

##  II Computational details

The calculations were performed using the projector augmented wave (PAW)
method [62, 32] implemented in the Vienna ab initio simulation package (VASP)
[31, 32], and employing PAW pseudopotentials, specially designed for $GW$
calculations. For the exchange correlation functional, we have used PBE [29],
SCAN [52], and HSE06 [53]. For PBE and SCAN an additional on-site Hubbard
Coulomb repulsion parameter $U_{\rm{eff}}$= $U-J$ is applied to the Co and Fe
$3d$ states within the Dudarev et al. [63] approach. The electron
configurations of Co, Fe and O are 3$d^{8}$4$s^{1}$, 3$d^{7}$4$s^{1}$ and
2$s^{2}$2$p^{4}$, respectively. The conventional cubic spinel unit cell of CFO
with $Fd\bar{3}m$ space group contains eight spinel formula units. We have
used the primitive rhombohedral unit cell including two spinel formula units
with 14 atoms (Fig.1 b) to reduce the computational cost. The plane-wave
cutoff energy is set to 500 eV. For the integration over the Brillouin zone,
we use a $\Gamma$-centered 5 $\times$ 5 $\times$ 5 $\mathbf{k}$ mesh. Both the
volume and the internal parameters were optimized with the residual forces
smaller than 0.001 eV/Å-1. The band structures are interpolated using the
WANNIER90 code [64] along the high symmetry point path adopted from AFLOW [24]
and FINDSYM [65]. Our calculations with and without spin-orbit coupling (SOC),
showed that Co acquires a significant orbital moment of $0.16$$\mu_{\rm B}$,
but the band structure is only weakly modified (see Fig. S1 in supplementary
information), therefore, we proceed with the results without SOC.

Single-particle excitations were described in terms of electron and hole QPs
by adopting the $GW$ approximation. Within the first-order perturbation, the
DFT wave functions $\psi_{n}$ are used and the QP energy of a state $n$,
$\varepsilon_{n}^{QP}$ is defined as:

$\small{\varepsilon_{n}^{QP}=\varepsilon_{n}^{DFT}+\langle\psi_{n}|\Sigma(\varepsilon_{n})-V_{xc}|\psi_{n}\rangle}$
(1)

where $\varepsilon_{n}^{DFT}$ and $\varepsilon_{n}^{QP}$ are the DFT single-
particle and quasiparticle energies, respectively. $\Sigma(\varepsilon_{n})$
is the self-energy operator which is obtained by both Green’s function $G$ and
the screened Coulomb potential $W$ calculated using DFT single-particle
energies and wavefunctions. Convergence with respect to the number of bands
and the cutoff energy of the response function in the $G_{0}W_{0}$
calculations were ensured by employing 792 bands and a cutoff energy of 333 eV
(see Fig. S2 in the supplementary information) [66].

To take into account excitonic effects, we solve the Bethe-Salpeter equation
[67]:

$\small{(E_{c\mathbf{k}}^{QP}-E_{v\mathbf{k}}^{QP})A^{\lambda}_{vc\mathbf{k}}+\sigma_{v^{\prime}c^{\prime}\mathbf{k^{\prime}}}^{QP}\langle
vc\mathbf{k}|K^{eh}|v^{\prime}c^{\prime}\mathbf{k^{\prime}}\rangle
A^{\lambda}_{v^{\prime}c^{\prime}\mathbf{k^{\prime}}}=\Omega^{\lambda}A^{\lambda}_{vc\mathbf{k}}}.$
(2)

Within the Tamm-Dancoff approximation, vertical transitions from the valence
to the conduction band $(E_{c\mathbf{k}}^{QP}-E_{v\mathbf{k}}^{QP})$ are
considered. $|v^{\prime}c^{\prime}\mathbf{k^{\prime}}\rangle$ are the
corresponding electron-hole pair configurations, $A^{\lambda}_{vc\mathbf{k}}$
are the expansion coefficients on the electron-hole basis functions,
$\Omega^{\lambda}$ are the exciton eigenenergies, and $K^{eh}$ is the kernel
that takes into account the electron-hole interaction. An accurate description
of the optical spectrum in $G_{0}W_{0}+$BSE, especially the calculation of
Re[$\epsilon$($\omega$)] from the Kramers-Kronig relation requires a large
number of $\mathbf{k}$-points and empty states. For a large system like CFO,
this enhances substantially the computational time and memory demand. In this
work, the BSE calculations were performed with 24 (28) occupied (unoccupied)
bands on a $\Gamma$-centered 5 $\times$ 5 $\times$ 5 $\mathbf{k}$ mesh to
evaluate the electron-hole excitation energies in the range of 0$-$6 eV (see
Fig. S3 in the supplementary information). The dielectric function is
evaluated by using 100 (imaginary) frequency and imaginary time grid points. A
Lorentzian broadening of 0.3 eV is applied to all the calculated optical and
absorption coefficient spectra to mimic the excitation lifetime.

Figure 2: Total and projected density of states (TDOS/PDOS) of CoFe2O4
calculated with PBE+$U$: a) $U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV, b)
$U_{\rm{Co}}$=$U_{\rm{Fe}}$= 4 eV and c) $U_{\rm{Co}}$= 5 eV, $U_{\rm{Fe}}$= 4
eV); SCAN+$U$ d) $U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV,
e)$U_{\rm{Co}}$=$U_{\rm{Fe}}$= 4 eV) and f) HSE06 functionals. Bold numbers
indicate the calculated band gaps ($E_{\rm{gap}}$) in eV.

##  III  Results and discussion

###  III.1 Structural and electronic properties

####  III.1.1 Ground state structural electronic properties

We start our analysis with the structural properties of CFO obtained with
different starting exchange-correlation functionals, namely, PBE, SCAN
including a Hubbard $U$ term on the Co and Fe 3$d$ states and HSE06. The
lattice constants are presented in Table 1: the PBE+$U$ value is 8.39 Å
($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV), 8.40 Å ($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 4
eV), and 8.39 Å ($U_{\rm{Co}}$ = 5 eV and $U_{\rm{Fe}}$ = 4 eV) almost
coinciding with the experimental value of 8.39 Å [59, 60, 61]. With SCAN+$U$
the lattice parameter is 8.33 Å ($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV) and 8.344
Å ($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 4 eV), 0.61$\%$ and 0.55$\%$ less than the
experimental value, respectively. With the hybrid functional HSE06, the
lattice constant is 8.37 Å (0.24$\%$ smaller than the experimental one).

As shown in Table 1, all three functionals render a ferrimagnetic ground
state. The Fe3+ at tetrahedral and octahedral sites are aligned antiparallel
and cancel each other. The net magnetization of CFO stems from the
ferromagnetically aligned Co2+ at octahedral sites. With PBE+$U$/SCAN+$U$,
$U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV, the calculated magnetic moments are
2.65/2.70 $\mu_{\rm{B}}$ for CoOct, 4.21/4.32 $\mu_{\rm{B}}$ for FeOct and
-4.08/-4.22 $\mu_{\rm{B}}$ for FeTet. These values increase to 2.70/2.77,
4.41/4.51, and -4.29/-4.40 $\mu_{\rm{B}}$ for CoOct, FeOct and FeTet for a
higher $U_{\rm{Co}}$=$U_{\rm{Fe}}$= 4 eV, respectively. With HSE06 the
magnetic moments are slightly lower 2.61, 4.09, and -3.98 $\mu_{\rm{B}}$ for
CoOct, FeOct and FeTet, respectively. The total magnetic moment of 3
$\mu_{\rm{B}}$ per formula unit with all three functionals is in good
agreement with the experimental value of 3.25 $\mu_{\rm{B}}$ [68].

The total-, element- and orbital-projected density of states (TDOS and PDOS)
obtained with the different exchange-correlation functionals is presented in
Fig. 2 (a-f). While the bottom of the valence band (-6.0 to -8 eV) is
dominated by Fe 3$d$ states at the tetrahedral (minority spin channel) and
octahedral sites (majority spin channel), Co 3$d$ and O 2$p$ states prevail at
the top of the valence band. However, depending on the $U$ value and the
starting functional, the valence band maximum (VBM) is in the minority spin
channel with PBE+$U$ ($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV) and HSE06, whereas it
is in the majority spin channel for PBE+$U$ ($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 4)
and SCAN+$U$ ($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 and 4 eV). The conduction band is
mostly comprised of Fe 3$d$ states at the tetrahedral (majority spin channel)
and octahedral sites (minority spin channel), the conduction band minimum
(CBM) for all functionals being in the minority spin channel.

In general, the band gap increases with $U$ for both PBE and SCAN: With
PBE+$U$, the band gap is 0.92 eV for $U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV, 1.38
eV for $U_{\rm{Co}}$=$U_{\rm{Fe}}$= 4 eV and 1.49 eV for $U_{\rm{Co}}$ to 5 eV
and $U_{\rm{Fe}}$= 4 eV. SCAN+$U$ renders the same trend but significantly
larger values of 1.69 and 2.11 eV for $U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 and 4 eV,
respectively. For the hybrid functional HSE06, a band gap of 2.02 eV is
obtained. Further insight into the nature of calculated band gaps, as well as
the position of VBM and CBM, is provided by analyzing the band structures in
the following section.

Figure 3: Band structure of CoFe2O4 calculated within (a, b, c) the
independent particle (IP) approximation with the different functionals,
PBE+$U$ ($U_{\rm{Co}}$=$U_{\rm{Fe}}=4$ eV), SCAN+$U$
($U_{\rm{Co}}$=$U_{\rm{Fe}}=3$ eV) and HSE06, respectively, (blue and red
denote majority and minority spin channels), and the $G_{0}W_{0}$
approximation for (d, e, f) the majority and (g, h, i) minority spin channels,
respectively. Maroon and cyan denote the quasiparticle (QP) band structure
plotted together with the IP band structure.

####  III.1.2  Independent Particle and quasiparticle band structure

In this section, we discuss the band structures obtained with PBE$+U$,
SCAN$+U$, and HSE06 functionals within the independent particle (IP) picture
and by including quasiparticle effects within $G_{0}W_{0}$, shown in Fig. 3
(See Fig. S5 in supplementary information for other $U$ values). The IP band
structures displayed in Fig. 3 (a-c) and Fig. S5 (in supplementary
information) show that both with PBE+$U$ ($U_{\rm{Co}}=U_{\rm{Fe}}$= 4 eV) and
SCAN+$U$ ($U_{\rm{Co}}=U_{\rm{Fe}}$= 3 and 4 eV) the VBM and CBM are located
at the $\Gamma$ point in the minority and majority spin channel, respectively.
In contrast, with HSE06 and PBE+$U$ ($U_{\rm{Co}}=U_{\rm{Fe}}$= 3 eV) both the
VBM and CBM are in the minority spin channel, the former is located along
$\Gamma$-$Y$ and the latter at $\Gamma$.

Regarding the IP band gap, with PBE+$U$, we obtain an indirect gap of 0.92 eV
($U_{\rm{Co}}=U_{\rm{Fe}}$= 3 eV) in good agreement with previous reported
value of 0.95 eV using the same $U$ values [25, 26]. However, for higher $U$
values $U_{\rm{Co}}=U_{\rm{Fe}}$= 4 eV the band gap switches to a direct and
larger one (1.38 eV). Similarly, the band gap calculated with SCAN+$U$
($U_{\rm{Co}}=U_{\rm{Fe}}$= 3 and 4 eV) is direct and significantly higher,
1.69, and 2.11 eV, respectively, as presented in Table. 1. An indirect band
gap of 2.02 eV is obtained with HSE06.

Including QP corrections substantially modifies the band structure for the
semi-local functionals. The valence band in the minority spin channel shifts
only by 0.06-0.08 eV, but shows modifications beyond a rigid band shift, for
example, change in the position and in the order and dispersion of bands at
$\sim-1$ eV, at $\Gamma$ and $W$, and along $\Gamma$-$Y$ and $R$-$X$ [cf. Fig.
3 (g, h, i)]. In the majority spin channel, the valence band moves downwards
by 0.3$-$0.4 eV. As a consequence, for both PBE+$U$ and SCAN+$U$ the VBM
switches from the majority to the minority spin channel upon inclusion of QP
corrections. On the other hand, the conduction band shifts by 0.9 - 1.1 eV
upwards. This leads to an overall increase of the band gap to 1.32 eV (PBE+$U$
($U_{\rm{Co}}=U_{\rm{Fe}}=3$ eV), 1.78 eV (PBE+$U$,
$U_{\rm{Co}}=U_{\rm{Fe}}=4$ eV), 1.95 eV (SCAN+$U$,
$U_{\rm{Co}}=U_{\rm{Fe}}=3$ eV), and 2.39 eV (SCAN+$U$,
$U_{\rm{Co}}=U_{\rm{Fe}}=4$ eV) and a change to an indirect band gap.

In the case of HSE06 (cf. Fig. 3 f and i), the QP corrections are much
smaller, the VBM/CBM shift only slightly to lower/higher energy by 0.03/0.09
eV in the minority spin channel. In the majority channel, the CBM is largely
unchanged, whereas the VBM is shifted downwards by 0.28 eV at $\Gamma$.
Overall the band gap of 2.02 eV (HSE06) is enhanced by only 5% to 2.17 eV
(HSE06+$G_{0}W_{0}$). Moreover, unlike the semi-local functionals, the HSE06
band gap is an indirect one in the minority spin channel with the VBM at
$\Gamma$-$Y$ and the CBM at $\Gamma$ [see Fig. 3 (c, i)] at both the IP and QP
levels. Overall, the HSE06 functional provides an improved description of the
ground-state properties and the $GW$ corrections are smaller compared to the
semi-local exchange-correlation functionals.

Figure 4:  Optical absorption spectrum of CoFe2O4: (a, b, c) real
(Re[$\epsilon$($\omega$)]) and (d, e, f) imaginary part
(Im[$\epsilon$($\omega$)]) of the dielectric function and (g, h, i) absorption
coefficient ($\alpha(E)$) for PBE+$U$ ($U_{\rm{Co}}$=$U_{\rm{Fe}}=4$ eV),
SCAN+$U$ ($U_{\rm{Co}}$=$U_{\rm{Fe}}=3$ eV) and HSE06 as the starting
functional, respectively, within IP approximation, $G_{0}W_{0}$ and
$G_{0}W_{0}+$BSE. Vertical black lines denote the oscillator strength for the
$G_{0}W_{0}+$BSE spectra. A Lorentzian broadening of 0.3 eV is employed for
all the calculated spectra. Experimental data are adopted from Exp. 1 [69],
Exp. 2 [16] and Exp. 3 [15].

###  III.2 Optical properties

We now turn to the optical properties of the inverse spinel CFO and discuss in
detail the real, Re[$\epsilon$($\omega$)], and imaginary part,
Im[$\epsilon$($\omega$)], of the frequency-dependent dielectric function (DF),
as well as the absorption coefficient calculated with different starting
exchange-correlation functionals, namely, PBE+$U$, SCAN+$U$, and HSE06
evaluated at the independent particle (IP) level and by including
quasiparticle ($G_{0}W_{0}$) and excitonic effects ($G_{0}W_{0}+$BSE). The
theoretical spectra are compared to the experimental results of Himcinschi et
al. [16] and Zviagin et al. [69]. These measurements were performed using
ellipsometry on epitaxial films of CFO grown by pulsed-laser deposition (PLD).
The measured real part of the DF (Re[$\epsilon$($\omega$)]) shows a peak and a
shoulder at 1.38 and 2.55 eV (Zviagin et al. [69]), and at 1.95 and 2.90 eV
(Himcinschi et al. [16]), respectively. The experimental macroscopic static
electronic dielectric constant, $\epsilon_{\infty}$ = Re [$\epsilon$($\omega$
= 0)] is 6 [69]. The measured Im[$\epsilon$($\omega$)] spectra display an
onset at around 1.5 eV, a shoulder at 2.0 eV, and two broad peaks with nearly
equal intensity at around 3.5 and 5 eV, and a drop in the intensity at 6.0 eV.
The differences between the two studies may be related to the different
substrates used: in the first case, CFO was deposited on a SrTiO3(100)
substrate at 575∘C[16], thus the CFO film (bulk lattice constant $a$ = 8.39
$\AA$) was subject to a significant compressive strain of -6.8% ($a_{\rm
SrTiO_{3}}=3.905$Å). In the second study the CFO films were grown on a
MgO(100) substrate ($a_{MgO}$ =4.21 Å) at 650 ∘C leading to a lattice mismatch
of only 0.36% [69].

####  III.2.1 Optical spectrum: IP and $G_{0}W_{0}$

The IP spectra with the different starting functionals are plotted in Fig. 4.
The analysis is performed with $U_{\rm{Co}}$=$U_{\rm{Fe}}$= 4 eV for PBE+$U$
and $U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV for SCAN+$U$ (the results for other $U$
values are given in the supplementary information, Fig. S6).

The onset of the imaginary part of the optical spectrum is at 1.38 eV
(PBE+$U$), 1.69 eV (SCAN+$U$), and 2.02 eV (HSE06) and reflects the Kohn-Sham
band gap [see Table 1]. For PBE+$U$ ($U_{\rm{Co}}$=$U_{\rm{Fe}}$=4 eV), a
shoulder is observed at 2.79 eV followed by two peaks at 3.67 and 4.74 eV.
With SCAN+$U$ ($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV), the shoulder is located at
3.21 eV, and the two peaks are at 4.25 and 5.71 eV. Both semi-local
functionals reproduce the shape of the experimental spectra with respect to
the spectral features, and the intensity of the peaks, but the peak positions
are at slightly higher energies than in the experiment. With HSE06, the two
peaks are further shifted to 5.03 and 6.64 eV, compared to the experimental
values at 3.50 and 5.0 eV.

Upon including quasiparticle corrections within the $G_{0}W_{0}$
approximation, the Im[$\epsilon$($\omega$)] spectrum is blue shifted to higher
energies with respect to the IP spectrum for all functionals, but the spectral
features from the IP picture are retained to a large extent. The magnitude of
the blue shift at the onset for the independent quasiparticle approximation
(IQPA) spectrum decreases from 0.48 eV (PBE+$U$) to 0.34 eV (SCAN+$U$) and
0.20 eV (HSE06), reflecting an improved screening effect at the starting DFT
level. The prominent shoulder at around 3.5 eV emerges only after including
quasiparticle corrections, but, interestingly, it is nearly quenched for
PBE+$U$ and SCAN+$U$.

The macroscopic static electronic dielectric constant, $\epsilon_{\infty}$ =
Re [$\epsilon$($\omega$ = 0)] in the IP picture is 6.42, 5.68, and 5.08 with
PBE+$U$ ($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 4 eV), SCAN+$U$
($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV) and HSE06, respectively, where SCAN+$U$
($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV) renders good agreement with the experiment
at 6 eV [69]. Upon including QP effects, $\epsilon_{\infty}$ decreases to
5.68, 4.99, and 4.62 with PBE+$U$, SCAN+$U$, and HSE06, respectively. The
first peak of Re[$\epsilon$($\omega$)] within IP ($G_{0}W_{0}$) is at 2.36
(2.82), 2.70 (3.20) and 3.70 (4.13) eV with PBE+$U$, SCAN+$U$, and HSE06,
respectively, these values are higher compared to the experimental value of
1.38 eV [69] and 1.95 eV [16].

####  III.2.2 Optical spectrum including excitonic effects

Taking into account electron-hole interactions by solving BSE leads to a
significant spectral weight redistribution of the Im[$\epsilon$($\omega$)]
spectrum [black solid line in Fig. 4 (d, e, f), see also Fig. S6 in
supplementary information for other $U$ values with respect to the IP and IQPA
spectra]. The onset of the spectrum is at around 1.45 eV (PBE+$U$), 1.50 eV
(SCAN+$U$), and 1.61 eV (HSE06), which is in good agreement with the
experimental onset at 1.50 eV. The shoulder at 2.20 eV (PBE+$U$), 2.41 eV
(SCAN+$U$), and 2.46 eV (HSE06) corresponds to the broad shoulder at around
2.5 eV in the experimental spectrum. This is followed by a two-peak feature at
2.85 and 3.70 eV (PBE+$U$), 3.1 and 4.0 eV (SCAN+$U$), and 3.49 and 4.62 eV
(HSE06), which becomes prominent only after including the excitonic effects
and corresponds to the first broad peak at 3.5 eV (brown solid line, Exp.1
[69]). The energetic position of the second peak in the
Im[$\epsilon$($\omega$)] is at around 4.8 eV (PBE+$U$), 5.2 eV (SCAN+$U$), and
5.5 eV (HSE06) compared to the experimental value of 5.0 eV. Overall SCAN+$U$
exhibits the best agreement with the spectrum of Zviagin et al. [69] with
respect to the onset and the position and intensity of the shoulder, as well
as the overall shape of the spectrum, in particular at higher energies,
underlining the importance of the excitonic effects.

Similarly, the Re[$\epsilon$($\omega$)] spectra are redshifted to lower
energies with respect to the IP and $G_{0}W_{0}$ spectra for all functionals
upon inclusion of excitonic effects. The macroscopic static electronic
dielectric constant is 5.08 (PBE+$U$), 5.11 (SCAN+$U$), and, 4.66 (HSE06),
which is lower than the experimental value of 6.0 [16]. Furthermore, the first
peak in the theoretical Re[$\epsilon$($\omega$)] spectrum is observed at 1.99
eV (PBE+$U$), 2.28 eV (SCAN+$U$) and 2.00 eV (HSE06) compared to the
experimental peak at 1.95 eV [16]. Only HSE06 renders a shoulder at 2.92 for
the $G_{0}W_{0}+$BSE spectrum, corresponding to the experimental shoulder at
2.90 eV [16].

We have also calculated the binding energy ($E_{\textrm{b}}$) of the first
exciton in the $G_{0}W_{0}+$BSE spectrum [70]: 0.61 eV (PBE+$U$), 0.59 eV
(SCAN+$U$), and 0.85 eV (HSE06). While to our knowledge there is no
experimental report for the exciton binding energy of CFO, the obtained values
are comparable to the ones for other related oxides such as SrTiO3 (0.25 eV
with SCAN) [49] and MgO (0.59 eV with HSE06) [51] and SrZrO3 (0.31 eV with
PBE+$U$) [58].

To summarize, the overall shape of the optical spectra after BSE does not
exhibit a strong dependence on the starting ground state exchange-correlation
functional, consistent with previous findings that inclusion of quasiparticle
and excitonic effects reduces the dependence on starting exchange-correlation
functional [49, 51]. Among the three functionals, our BSE calculation starting
with SCAN+$U$ renders the best agreement with experiment with respect to the
onset of the spectrum, and energetic position and intensity of the shoulder
and peaks. Overall, our calculations indicate the inclusion of excitonic
effects is essential for the investigation of the optical and absorption
spectrum of CoFe2O4.

####  III.2.3 Absorption coefficient spectrum

In addition to the optical spectrum in Fig. 4 (g-l), we have also plotted the
absorption coefficient $\alpha(\omega)$, calculated as [71, 45, 72]:

$\alpha(\omega)=\dfrac{4\pi}{\lambda}\kappa.$ (3)

Here $\lambda$ is the wavelength of incident radiation ($\lambda=1.24\times
10^{-6}$ eV.m) and $\kappa$ is the imaginary part of the refractive index
which is related to the real Re[$\epsilon$($\omega$)] and imaginary
Im[$\epsilon$($\omega$)] part of the dielectric function:

$Re[\epsilon(\omega)]+Im[\epsilon(\omega)]=(n+i\kappa)^{2},$ (4)

where $n$ is the real part of the refractive index. By combining Equations 3
and 4, the absorption coefficient is defined as:

$\alpha(\omega)=\dfrac{4\pi}{\lambda\sqrt{2}}\sqrt{-Re[\epsilon(\omega)]+\sqrt{Re^{2}[\epsilon(\omega)]+Im^{2}[\epsilon(\omega)]}}.$
(5)

From Fig. 4, the onset of $G_{0}W_{0}+$BSE spectrum for the three exchange
correlation functionals is 1.45 (PBE+$U$), 1.50 (SCAN+$U$) and 1.61 eV
(HSE06), in good correspondence with the measured onset at 1.55 eV (obtained
for epitaxial CFO films grown at 690∘ on MgAl2O4 ($a$ = 8.08 Å ) substrate
with 3.5% compressive strain) [15]. The shoulders in the spectra are located
at 2.9 and 4.1 eV (PBE+$U$), 2.47 and 4.3 eV (SCAN+$U$), and 2.6 and 4.7 eV
(HSE06), in agreement with the two broad experimental shoulders at around 2.6
eV and 4.5 eV. In general, both HSE06 and SCAN+$U$ render excellent agreement
with the measured absorption coefficient spectrum after taking into account
the excitonic effects.

We further analyze the oscillator strengths obtained from the BSE calculations
[see Fig. 4 (g-l)] which indicate the probability of excitation at the
corresponding energy. In general, the first excitation with a non-zero
oscillator strength is interpreted as the optical band gap (the lowest
threshold for optical transitions). From Fig. 4 we observe the first optically
allowed transition marked as 1 at the onset of the spectra at 1.45 eV
(PBE+$U$), 1.50 eV (SCAN+$U$) and 1.61 eV (HSE06). Oscillator strengths with
high intensity are found at around 2.0 eV (marked as 2 and 3 in Fig. 4) for
all the functionals, at 3.5 eV (marked as 4 for PBE+$U$, and 4 and 5 for
SCAN+$U$ and HSE06) and at around 5.0 eV (marked as 6 for SCAN+$U$ and HSE06).
These are in good agreement with the reported optical transitions from
ellipsometry measurements for a single crystal of Co1.04±0.05Fe1.96±0.05O4
[60] at 2.0, 3.4 and 4.9 eV. From magneto-optical Kerr spectroscopy
transitions were reported at 1.82, 2.21, 2.60, 3.55, and 4.0 eV [23], and at
1.78, 2.05, 2.67, 3.6, 4.3 and 4.7 eV [16]. The projected band structure in
Fig. S4 indicates that the first transition is from the highest occupied band
in the minority spin channel comprised of Co 3$d$ states hybridized with O
2$p$ states to the bottom of the CB which is dominated by 3$d$ states of
octahedral Fe. This suggests that the first allowed transition has a mixed
Mott-Hubbard and charge transfer character. The transition at around 2 eV
stems from Co $3d$ $\longrightarrow$ at the top of the valence band to
Fe${}^{\rm{Oct}}_{t_{2g}}$ at the bottom of the conduction band, as shown in
the projected DOS (Fig. 2) and band structure Fig. S4 in SI.

As mentioned already in the introduction, the reported experimental direct
optical gaps show a wide range between $\sim$ 0.55 - 4.1 eV. This broad
variation can be attributed to the effect of temperature, size, crystallinity,
degree of inversion, and shape of the samples. For the fully inverse bulk
spinel our $G_{0}W_{0}$+BSE calculations indicate an optical gap of 1.50 eV
(SCAN+$U$) and 1.61 eV (HSE06) in agreement with measured values of 1.65 [20]
and 1.58 eV [21]. Further optical transitions are at around 2.0, 3.5, and 5.0
eV in agreement with the measurements [60].

Figure 5:  Impact of cation distribution on the magnetic, electronic, and
optical properties of (Co1-xFex)Tet(CoxFe2-x)OctO4 obtained with SCAN+$U$
($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV): spin density of (a) inverse ($x$ = 1.0)
(b) half ($x$ = 0.5) and (c) normal spinel ($x$ = 0.0). The blue and red
colors represent positive and negative spin density, respectively.
Additionally, the magnetic moments of cations are given in $\mu_{\rm{B}}$.
Projected density of states (PDOS) for (d) $x$ = 0.5 and (d) $x$ = 0.0, the
calculated IP band structure for (f) $x$ = 0.5 and (g) $x$ = 0.0 (blue/red
lines: majority/minority spin channels). Optical absorption spectrum: (h) real
part Re[$\varepsilon(\omega)$], (i) imaginary part Im[$\varepsilon(\omega)$]
of the dielectric function (DF) and (j) absorption coefficient ($\alpha(E)$)
for $x$ = 1.0 (black solid line), 0.5 (dashed blue line) and 0.0 (red solid
line) within $G_{0}W_{0}$+BSE calculations. Experimental data are adopted from
Exp. 1 [69] (brown shadowed area), and Exp. 2 [15] (orange shadowed area).

###  III.3 Impact of degree of inversion

As discussed in the introduction, the distribution of cations at octahedral
and tetrahedral sites can impact the structural, electronic and optical
properties of spinels. In this section, we assess the effect of the degree of
inversion on the electronic and optical properties by considering $x$ = 0.0
(normal spinel) and 0.5 (half inverse spinel) in (Co1-xFex)Tet(CoxFe2-x)OctO4
using the SCAN+$U$ functional with $U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV.

In the normal spinel ($x$ = 0.0) all the Co cations occupy tetrahedral sites
while all Fe ions are located at the octahedral sites. For the partially
inverse spinel of $x$ = 0.5, half of the Co cations occupy the tetrahedral
sites and the remaining half octahedral sites. For all degrees of inversion
modeled here, we used the primitive rhombohedral unit cell including two
spinel formula units with 14 atoms. The fully inverse spinel ($x$ = 1.0) is
favored in energy by 0.074 and 0.006 eV/f.u. compared to the half ($x$ = 0.5)
and normal ($x$ = 0.0) spinel. The calculated lattice constants are 8.357 ($x$
= 0.5) and 8.371 Å ($x$ = 0.0) which are 0.21% and 0.38% larger than the
inverse spinel (8.33 Å). The trend is in line with the experimental reports by
Venturini et al. [73] who found 8.384 Å for $x$ = 0.0 and 8.364 Å for $x$ =
1.0), as well as previous DFT calculations with the PBEsol exchange
correlation functional and $U=4$ eV for both Fe and Co by Sharma et al.[11]
who reported that the lattice constant increases from 8.332 Å($x$ = 1.0) to
8.358 ($x$ = 0.5) and 8.384 Å($x$ = 0.0). From the calculated magnetic moment,
presented in Fig. 5 b and c, Co and Fe ions preserve the high spin
configuration at octahedral and tetrahedral sites for all degrees of inversion
in agreement with previous results [36]. Due to the different sizes of Co and
Fe magnetic moments and the antiparallel orientation at octahedral and
tetrahdral sites, the total magnetic moment in the unit cell increases to 5
and 7 $\mu_{\rm B}$ for $x=0.5$ and $x=0.0$.

The PDOS for $x$ = 0.5, presented in Fig. 5 d, shows that the top of the
valence band is dominated by O 2$p$ and CoTet 3$d$ states in the majority and
CoOct 3$d$ in the minority spin channel. The bottom of the conduction band is
comprised of FeOct 3$d$ and FeTet 3$d$ states in the minority and majority
spin channels, respectively.

For $x$ = 0.0 (Fig. 5 e), the top of the valence band is dominated by CoTet
3$d$ and O 2$p$ states in both spin channels. The bottom of the conduction
band is comprised of FeOct 3$d$ (minority) and CoTet 3$d$ states (majority
spin channel). In both cases, similar to the inverse spinel ($x$ = 1.0)
presented in Fig. 2 d, the VBM is located in the majority and the CBM in the
minority spin channel. As shown in Fig. 5 f and g, the calculated band gap is
direct at the $\Gamma$ point and is reduced to 1.45 ($x$ = 0.5) and 1.19 eV
($x$ = 0.0), compared to 1.69 eV ($x$ = 1.0). Upon inclusion of QP corrections
(cf. Fig. S8 in SI), the band gaps remain direct at $\Gamma$ point and
increase to 1.90 ($x=0.5$) and 1.74 eV ($x=0.0$), in contrast to an indirect
band gap of 1.95 eV with VBM along $\Gamma$-Y and CBM at $\Gamma$ for $x=1.0$
(cf. Fig. 3 h). The VBM lies in the majority spin channel for both IP and QP
band structures for the cases with reduced degree of inversion in contrast to
the completely inverse spinel.

The calculated real and imaginary of the DF as well as the absorption
coefficient after the inclusion of the excitonic effects ($G_{0}W_{0}$+BSE
calculation) with SCAN+$U$ ($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV) for $x$ = 0.0,
0.5 and 1.0 are presented in Fig. 5(g-i). As discussed previously in Section
III.2.2, the experimental spectrum has a shoulder at 2 eV and two broad peaks
with nearly equal intensity at around 3.5 and 5 eV [16, 69]. We note that
these studies do not provide information on the degree of inversion of the
samples. In the calculated imaginary part of the optical spectrum for $x=1.0$
the first peak is split into two peaks at 3.1 and 4.0 eV with similar
intensity as the experimental spectrum (experiment one is a broad peak around
3.5 eV). However, as presented in Fig. 5(i), the cation distribution
influences the position and intensity of the shoulder and peaks of the optical
spectrum. Comparison of the optical spectra for $x$ = 0.0, 0.5, and 1.0
indicates a similar onset around 1.50-1.60 eV. Upon moving Co2+ to the
tetrahedral sites ($x$ = 0.5 and 0.0), the intensity of the shoulder at around
2 eV decreases. For $x$ = 0.5, one broad featureless peak is observed at
around 3.15 to 5.9 eV. In the case of the normal spinel ($x$ = 0.0), the
shoulder at 2 eV has the lowest intensity with respect to $x$ = 0.5 and 1.0,
followed by two small peaks at 3.0 and 3.6 eV, and an increased intensity of
the peak at 4.7 eV. By analyzing the calculated optical spectra and oscillator
strength (Fig. S7 in SI), an optical band gap (the lowest threshold for
optical transitions) of 1.64, 1.57, and 1.50 eV is obtained for $x$ = 0.0,
0.5, and 1.0, respectively. Our findings suggest that the optical band gap
decreases with decreasing degree of inversion, but overall the differences are
small which is consistent with previous experimental and theoretical studies
for another spinel, ZnFe2O4 [10]. On the other hand, the shape of the spectrum
shows substantial differences, which allows us to distinguish between
different degrees of inversion in CFO samples.

The spectrum of the real part of DF, presented in Fig. 5 h, becomes broader
and shifts to higher energies with decreasing degree of inversion. The
spectrum shows a peak at 2.8 eV and a shoulder at $3.5$ eV ($x$ = 1.0), a
rather featureless broad peak between 1.15 - 3.4 eV ($x$ = 0.5), and two peaks
at 2.84 and 4.42 eV ($x$ = 0.0). Additionally, the macroscopic static
electronic dielectric constant is found to decrease from 5.11 ($x$ = 1.0) to
4.31 ($x=0.5$), and 4.54 ($x=0.0$).

From the analysis of the oscillator strength presented in Fig. S7 in SI, both
$x$ = 0.0 and 0.5 show transitions below 1 eV with very small non-zero
oscillator strength (see insets in Fig. S7 (g-i)). We note that small
polarons, as observed in other transition metal oxides, e.g. Co3O4 and Fe2O3
[38, 74], tend to have much more pronounced midgap transitions. Fontijn et al.
[23] proposed that in CoFe2O4 the transitions below 1 eV originate from the
presence of Co2+ cations at tetrahedral sites. This is consistent with the
calculated optical spectra and PDOS analysis since these transitions are
absent in the fully inverse spinel.

Other optical transitions with high-intensity oscillator strength are marked
in Fig. S7 at 3.0, 3.6, and 4.7 eV for $x$ = 0.0 and at 2.0 and 3.2 eV for $x$
= 0.5. The absorption coefficient spectra of $x$ = 0.0, 0.5, and 1.0 are
presented in Fig. 5 j and compared with the experimental spectrum adopted from
[15]. With decreasing degree of inversion the onset of the calculated
absorption coefficient spectra increases from 1.50 eV ($x$ = 1.0) to 1.57 eV
($x$ = 0.5) and 1.64 eV ($x$ = 0.0). Moreover, the spectra for different
degree of inversion exhibit distinct shapes that may be used as a fingerprint.
The best agreement with the experimental spectrum is obtained for the fully
inverse structure ($x$ = 1.0).

##  IV Summary

We have systematically investigated the electronic and optical properties of
CoFe2O4 using different levels of description starting with the independent
particle picture, and subsequently including quasiparticle ($G_{0}W_{0}$)
correction and excitonic ($G_{0}W_{0}+$BSE) effects. Moreover, the effect of
different starting exchange-correlation functionals (PBE+$U$, SCAN+$U$, and
HSE06) and a variation of Hubbard $U$ term on the electronic and optical
properties of CFO was explored. In addition, we investigated the effect of the
degree of inversion $x$ on the electronic and optical properties of the
(Co1-xFex)Tet(CoxFe2-x)OctO4 with SCAN+$U$.

The starting exchange-correlation functional has a significant influence on
the electronic structure at the IP level, in particular, with respect to the
size and type of band gap (direct/indirect), and the position of the VBM and
CBM in the minority/majority spin channel. While an indirect band gap of 0.92
and 2.02 eV is obtained with PBE+$U$ ($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV) and
HSE06 in the minority spin channel with VBM along $\Gamma$-$Y$ and CBM at
$\Gamma$, a direct band gap of 1.38 and 1.69 eV is obtained with PBE+$U$
($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 4 eV), SCAN+$U$ ($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3
eV) wherein the VBM/CBM is located in the majority/minority spin channel at
$\Gamma$. The VBM is predominantly comprised of O 2$p$ and Co 3$d$ states,
whereas the CBM consists of Feoct 3$d$ states. However, the deviations between
the different starting functionals reduce appreciably after including the
quasiparticle effects, similar to previous findings for SrTiO3 [49]. Including
quasiparticle effects ($G_{0}W_{0}$ ), enhances the band gap to 1.78 eV
(PBE+$U$, $U_{\rm{Co}}$= $U_{\rm{Fe}}$= 4 eV), 1.95 eV (SCAN+$U$,
$U_{\rm{Co}}$= $U_{\rm{Fe}}$= 3 eV) and 2.17 eV (HSE06) and leads to an
indirect band gap for all functionals. Moreover, modification of the band
structure beyond a rigid band shift underline the critical contribution of the
non-local character of the self-energy term in the $GW$ approximation.

Concerning the optical spectra, the imaginary part of the dielectric function
obtained with SCAN+$U$ ($U_{\rm{Co}}$= $U_{\rm{Fe}}$= 3 eV) and HSE06 shows
good agreement with the experimental optical spectra [69, 16] with respect to
the energetic position and intensity of peaks only after including excitonic
effects by solving the Bethe Salpeter equation. Also the absorption
coefficient obtained with SCAN+$U$ ($U_{\rm{Co}}$= $U_{\rm{Fe}}$= 3 eV) fits
very well the measured one [15]. From the analysis of oscillator strength, the
lowest threshold for optical transitions, the optical band gap, is at 1.45,
1.50, and 1.61 eV with PBE+$U$ ($U_{\rm{Co}}$= $U_{\rm{Fe}}$= 4 eV), SCAN+$U$
($U_{\rm{Co}}$= $U_{\rm{Fe}}$= 3 eV) and HSE06, respectively, close to the
experimental optical gap of 1.65 [20] and 1.58 eV [21]. Moreover, the
caclulated spectra show transitions at $\sim$2, 3.5, and 5 eV, in agreement
with the experimental findings [60].

Additionally, we explored the impact of cation distribution at the tetrahedral
and octahedral sites on the structural and optical properties of
(Co1-xFex)Tet(CoxFe2-x)OctO4 ($x$ = 0.0, 0.5, and 1.0) with SCAN+$U$
($U_{\rm{Co}}$=$U_{\rm{Fe}}$= 3 eV). With decreasing degree of inversion, the
lattice constant as well as the total magnetic moment per f.u. increase in
agreement with previous theoretical and experimental studies [73, 36]. While
at the IP/QP level the band gap decreases with the degree of inversion,
1.69/1.96 eV ($x$ = 1.0), 1.45/1.90 eV ($x$ = 0.5) and 1.19/1.74 eV ($x$ =
0.0), the optical gap after including excitonic effects shows a slight
increase from 1.50 eV ($x$ = 1.0) to 1.57 eV ($x$ = 0.5) and 1.64 eV ($x$ =
0.0). The presence of Co ions in the tetrahedral sites significantly modifies
the overall shape of the spectrum and leads to transitions below 1 eV with
very small non-zero oscillator strength which are not present in the fully
inverse spinel structure, consistent with previous experimental suggestions
[15].

Figure 6: (a) Inverse of dielectric function $\varepsilon^{-1}$ from
$G_{0}W_{0}$ calculation and the corresponding fit according to equation 6
with HSE06 functional, (b and c) Comparison of the real
Rm[$\epsilon$($\omega$)] and imaginary Im[$\epsilon$($\omega$)] parts of
dielectric function using $G_{0}W_{0}+$BSE and model BSE approach.
Experimental data are adopted from Exp. 1 [69] and Exp. 2 [16]

The detailed analysis of the electronic and optical properties of CFO
employing DFT calculations and state-of-the-art many-body perturbation theory
(MBPT) is useful not only for the interpretation of experimental measurements,
but is also a prerequisite for exploring the incorporation of CFO in
heterostructures and nanocomposites in view of carrier separation and
reduction of recombination rates.

##  Appendix A  Model BSE

To reduce the computational cost in $G_{0}W_{0}+$BSE calculations, we tested a
less computationally demanding approach for the static screening, the model
BSE (mBSE) scheme [54, 55, 56, 57].

In this approach, the imaginary part of the dielectric constant is replaced by
the local model function and is fitted to $G_{0}W_{0}$ calculations using
[57]:

$\varepsilon^{-1}_{\textbf{k}+\textbf{G}}=1-(1-\varepsilon^{-1}_{\infty})e^{\dfrac{-|\textbf{k}+\textbf{G}|^{2}}{4\beta^{2}}},$
(6)

where $\beta$ is the range separation parameter, G is the lattice vector, and
$\varepsilon_{\infty}$ is the ion-clamped static dielectric function. Here,
$\beta$ is obtained by fitting the screened Coulomb kernel diagonal values
from the $G_{0}W_{0}$ calculation as shown in Fig. 6 (a). A scissor operator,
$\Delta$ is applied to the Kohn-Sham eigenenergies to mimic the QP effect and
is defined as a difference between the $G_{0}W_{0}$ and IP band gap. The
electron-hole interactions are considered by solving BSE using the Kohn-Sham
wave functions.

In the mBSE calculations starting from HSE06, $\beta=$ 1.414 and
$\varepsilon_{\infty}=$ 0.221 (obtained from the fit to the $G_{0}W_{0}$
dielectric function shown in Fig. 6 (a)) are used as input. A
$\Gamma$-centered 5 $\times$ 5 $\times$ 5 $\mathbf{k}$-mesh is employed for
the calculation. To converge the electron-hole excitation energy in the range
of 0-6 eV, similar to the previous BSE calculations, 24 occupied and 28
unoccupied bands are included in the mBSE calculations. As depicted in Fig. 6
(b and c), the onset of the Im[$\epsilon$($\omega$)] of the model BSE spectrum
is at 1.60 eV and is in very good agreement with the $G_{0}W_{0}+$BSE onset at
1.61 eV. The first transition at 1.63 eV with a non-zero oscillator strength
is in agreement with 1.61 eV from $G_{0}W_{0}+$BSE calculation. The feature at
around 2.46 eV and the peak 3.49 eV are concurrent with the $G_{0}W_{0}+$BSE
spectrum. Moreover, the peak at 4.2 eV is in close correspondence with the
peak at 4.7 eV. An overall good agreement of the model BSE and
$G_{0}W_{0}+$BSE spectra is traced back to only small modifications of the
HSE06 band structure by including quasiparticle effects beyond the rigid
shift, as described in Fig. 3 (f and i). We note that starting with PBE+$U$
and SCAN+$U$, we obtain a poor agreement between the model BSE and
$G_{0}W_{0}+$BSE spectra [See Fig. S9 in SI]. This is attributed to the
changes in the electronic structure after $G_{0}W_{0}$, wherein we observed
band modifications in the band dispersion.

###### Acknowledgements.

We acknowledge support by the German Research Foundation (DFG, Deutsche
Forschungsgemeinschaft) within the Collaborative Research Center TRR247
(Project No. 388390466, Subproject B4), CRC1242 (Project No. 278162697,
Subproject No. C02) and computational time at magnitUDE of the Center of
Computer Science and Simulation (DFG grant INST 20876/209-1 FUGG,INST
20876/243-1 FUGG).

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$\mathbb{R}^{d}$. This then is a generalisation of what we think of as the
topological space $\mathbb{R}^{d}$; that is such a $d$ dimensional manifold is
_locally_ homeomorphic to $\mathbb{R}^{d}$. The key aspect here is that the
set $\mathcal{M}$ can be given any topology $\mathcal{O}_{\mathcal{M}}$ on it,
but we can still locally relate it to a space we are familiar with (the real
space); see Figure 17. Each map $\phi$ that does this is called the
_coordinate map_. An atlas is a collection of such charts, such that the
domains of each of the chart maps is a cover of $M$.

Figure 17: A topological manifold $\mathcal{M}$ with examples of its
coordinate charts. We see here that despite being a complicated topological
entity (see the hold in the middle), locally, it is equivalent to
$\mathbb{R}^{2}$. Further, note that the chart transition map
$\phi\circ\psi^{-1}:V_{\mathbb{R}^{2}}\rightarrow U_{\mathbb{R}^{2}}$ exists
because a homeomorphism is invertible, and is continuous since the composition
of continuous maps remains so.

We note here that because open sets of the manifold are homeomorphic to
$\mathbb{R}^{d}$, which is endowed with the standard topology, our manifold is
locally compact. This means that every point on the manifold has a
neighbourhood that is contained in a compact subset of X; intuitively,
compactness gives us a notion of being closed (contains limit points) and
bounded (Cederquist and Negri, 1995). In addition to this, our manifolds are
_Hausdorff_ (Lee, 2001). This means that if we take any 2 points on a
manifold, there exists at least one neighbourhood of each point such that
these subsets are non-intersecting. These properties give our manifolds nice
properties that make them easy to work with. As an example, a trivial topology
that can be given to a set $X$ is $\\{X,\emptyset\\}$. It is possible to show
that every map onto $X$ is continuous, and as such this topology doesn’t
provide us with any useful structure; the Hausdorff property ensures that
there are enough open sets to reason about interesting things.

In Figure 17, there is a map that is defined to take elements in the
intersection of two open sets from one chart to another. More precisely, given
$U,V\in\mathcal{O}_{\mathcal{M}}$ and charts on these sets
$(U,\phi),(V,\psi)\in\mathcal{A}_{\mathcal{M}}$, where
$\phi:U\rightarrow(U_{\mathbb{R}^{d}}\subset\mathbb{R}^{d})$ and
$\psi:V\rightarrow(V_{\mathbb{R}^{d}}\subset\mathbb{R}^{d})$, and if $U\cap
V\neq\emptyset$, then $\phi\circ\psi^{-1}:\psi(U\cap V)\rightarrow\phi(U\cap
V)$. Such a map is called a chart transition map. These maps are important in
defining differential properties on a manifold.

######  Definition 28 ($C^{k}$ \- Atlas)

An atlas $\mathcal{A}_{\mathcal{M}}$ of a topological manifold $\mathcal{M}$
is called a $C^{k}$-atlas if all chart transition maps are $k$ times
continuously differentiable.

In this way, if our $\mathcal{A}_{\mathcal{M}}$ atlas satisfies these
conditions, we call $\mathcal{M}$ a $C^{k}$-differentiable manifold. A smooth
manifold then is a $C^{\infty}$-manifold; that is, it is infinitely
differentiable. Such properties are defined to enable us to carry out calculus
on manifolds in an invariant way; topologies don’t provide enough restrictions
because it is possible to construct maps that are homeomorphisms, but don’t
preserve differentiability. Furthermore, requiring that smoothness is
preserved at the intersection of open sets also exposes another key aspect of
the philosophy of manifolds, differential or otherwise. A manifold attempts to
describe its properties in a coordinate independent way. That is, the manifold
exists, with its properties, without the need for chart maps; the charts are
our ways of interacting with the manifold. However, if the chart is valid, and
respects the conditions we put on it, then our choice of a chart shouldn’t
change the properties of the underlying manifold. This is because if there are
two charts in our atlas, $(U,\phi),(U,\psi)\in\mathcal{A}_{\mathcal{M}}$,
which are based on the same open set $U\in\mathcal{O}_{\mathcal{M}}$, then
doing either chart transition $\phi\circ\psi^{-1}$ or $\psi\circ\phi^{-1}$
should not alter any properties we see on either coordinate. Thus, the
condition in Definition 28 ensures that smoothness properties are preserved
between coordinates, and allows us to measure whether they can be preserved
through maps between differential manifolds. A map $f$ on $M$ is said to be
smooth, if $f\circ\phi^{-1}$ is smooth, for any
$(U,\phi)\in\mathcal{A}_{\mathcal{M}}$. A map that does preserve smoothness,
and is invertible is called a _diffeomorphism_. In this way, the chart
transition maps are therefore perhaps the most obvious examples of
diffeomorphisms.

As a slight tangent, this philosophy contributes to the beauty of differential
geometry. It allows us to define properties intrinsic to a space without
worrying about how we choose to represent them; these properties are then
independent of the coordinate system we choose. This is of course a very
desirable attribute, since we don’t want to worry about whether our theories
are the way they are because of a particular choice in coordinates that we
made.

###  A.3 Groups and Group Actions

######  Definition 29 (Group)

A set $\mathcal{G}$ with an operation
$\bullet:\mathcal{G}\times\mathcal{G}\rightarrow\mathcal{G}$ is called a group
$(\mathcal{G},\bullet)$ if:

  1. a)

$\bullet$ is associative

  2. b)

there exists $e\in\mathcal{G}$ such that $e\bullet g=g$ for any
$g\in\mathcal{G}$, and

  3. c)

$\forall g\in\mathcal{G}$ there exists $g^{-1}\in\mathcal{G}$ such that
$g\bullet g^{-1}=e$ and $g^{-1}\bullet g=e$.

A group is said to be commutative or _Abelian_ if $g\bullet h=h\bullet g$.

The key way in which we will be using a group is through its action on another
set.

######  Definition 30 (Group action (left))

The (left) action of a group $(\mathcal{G},\bullet)$ on a set $\mathcal{X}$ is
a map $\rho:\mathcal{G}\times X\rightarrow X$ that satisfies:

  1. a)

$\rho(h,\rho(g,x))=\rho(h\circ g,x)$,

  2. b)

$\rho(e,x)=x$,

for $g,h\in\mathcal{G}$, $e\in\mathcal{G}$ is the neutral element in
$\mathcal{G}$, and $x\in X$.

A right action reverses the order of the input; we brevity, we will only
consider left group actions, and therefore won’t qualify which we use. A group
action gives us a way to traverse a set in terms of a set of transformations.
That is, we can think of a group as a set of transformations, and the action
of an element of a group is applying a particular transformation an element of
another group. An example of a group is SO(3), which is group of all spherical
rotations; an element of SO(3) can act on an element of $\mathbb{R}^{3}$ by
rotating it, in the intuitive sense.

Groups play an important role in the idea of symmetries and invariances (Weyl,
2015). A particular reason for that are the two conditions that a group action
must satisfy. These conditions ensure a sort of consistency in the behaviour
of the action. Consider the following scenario. We have three points $x,y,z\in
X$. Suppose $y=g_{xy}(x)$, $z=g_{yx}(y)$ and the $z=g_{xz}(x)$. The first
condition ensures that $g_{xz}=g_{xy}\bullet g_{yz}$, that we can blindly
choose from $\mathcal{G}$ whether to move from $x$ to $z$ directly, or to do
so through an intermediate point $y$. The second condition ensures this
consistency holds for the neutral element of the group too.

######  Definition 31 (Orbit)

Given a group $\mathcal{G}$ and its action $\rho$ on a set $X$, the orbit of
$\mathcal{G}$ centered at a point $x\in X$, which is denoted by
$O_{\mathcal{G}}(x)$, is a set defined as:

$O_{\mathcal{G}}(x)=\big{\\{}y\>\big{|}\>y=\rho(g,x)\>\forall
g\in\mathcal{G}\big{\\}}.$

Since $\mathcal{G}$ includes the neutral element, $x\in O_{\mathcal{G}}(x)$.
The consistency requirements of a group action, together with the requirements
of a group imply that an orbit forms a well defined equivalence class. That
is, if $x,y\in O_{\mathcal{G}}(x)$, then
$O_{\mathcal{G}}(x)=O_{\mathcal{G}}(y)$. In other words, there is an
equivalence relation, where $x\sim y$ if $\exists g\in\mathcal{G}$ such that
$x=g(y)_{\mathcal{G}}$, which is of course used in the definition of an orbit.
With an equivalence relation comes a quotient space $X/\sim$; this is the
space of equivalence classes. The quotient space defined by orbits of a
$\mathcal{G}$ is denoted by $X/\mathcal{G}$ and is called the _orbit space_.

A group action is called _free_ if its isotropy groups are trivial, and only
contain the neural element. This implies that for a free group action, if
$x,y\in O_{\mathcal{G}}(x)$, then $O_{\mathcal{G}}(x)=O_{\mathcal{G}}(y)$.
This means that the orbits of a free action are well defined equivalence
classes.

An important class of groups that we consider are Lie Groups (Nakahara, 2003;
Lee, 2001; Hall, 2015). Lie groups are simply groups that are also smooth
manifolds.

###  A.4 Pseudogroups

A pseudogroup is a generalisation of a group, often written for local
diffeomorphisms, where appropriate. They play an important role in partial
symmetries (Lawson, 1998).

######  Definition 32 (Pseudogroups)

(Lisle and Reid, 1998) If $M$ is a smooth manifold, and $\mathcal{G}$ is a
collection of local diffeomorphisms on open subsets of $M$ into $M$, then
$\mathcal{G}$ is a pseudogroup if:

  1. a)

$\mathcal{G}$ is closed under restrictions: if $\tau:U\rightarrow M$, for $U$
an open set of $M$, is in $\mathcal{G}$, then $\tau|_{V}\in\mathcal{G}$ for
any $V\subseteq U$ is also in $\mathcal{G}$,

  2. b)

if $U\subseteq M$ is an open set, and $U=\bigcup_{s}U_{s}$, and
$\tau:U\rightarrow M$, with $\tau|_{U_{s}}\in\mathcal{G}$, then
$\tau\in\mathcal{G}$,

  3. c)

$\mathcal{G}$ is closed under composition: if
$\tau_{1},\tau_{2}\in\mathcal{G}$, then $\tau_{1}\circ\tau_{2}\in\mathcal{G}$,
whenever the composition is defined (i.e over intersections of the appropriate
domain and codomain),

  4. d)

$\mathcal{G}$ contains an identity diffeomorphism over $M$,

  5. e)

$\mathcal{G}$ is closed under inverse: if $\tau\in\mathcal{G}$, then
$\tau^{-1}\in\mathcal{G}$, with a domain $\tau(U)$, where $U$ is the domain of
$\tau$.

We can see that the latter 3 conditions look similar to what we would expect
from a group of differmorphisms. A pseudogroup allows for such structure when
the diffeomorphisms aren’t defined globally.

##  Appendix B Proofs

###  B.1 Induced topology on the Task space

Proof [Theorem 9] To complete this proof, we need to show the following:

  1. 1.

$\mathcal{T}\in\mathcal{O}_{\mathcal{T}}$ and
$\emptyset\in\mathcal{O}_{\mathcal{T}}$: True by definition.

  2. 2.

$(U_{1}\cup U_{2})\in\mathcal{O}_{\mathcal{T}}$ for
$U_{1},U_{2}\in\mathcal{O}_{\mathcal{T}}$: For a $t\in U_{1}\cup U_{2}$, there
exists $\mathcal{B}_{t,1}(\epsilon_{1})\subseteq U_{1}$ or
$\mathcal{B}_{t,2}(\epsilon_{2})\subseteq U_{2}$ by construction of $U_{1}$
and $U_{2}$. Thus, the condition is satisfied.

  3. 3.

$\bigcap_{i=1}^{n}{U_{i}}\in\mathcal{O}_{\mathcal{T}}$ for
$U_{i}\in\mathcal{O}_{\mathcal{T}}$: For a $t\in U_{1}\cap U_{2}$, there
exists $\mathcal{B}_{t}(\epsilon_{i})\subseteq U_{i}$ by construction of
$U_{i}$. Suppose
$\epsilon^{*}=\min(\epsilon_{1},\epsilon_{2},...,\epsilon_{n})$. Then,
$\mathcal{B}_{t}(\epsilon^{*})\subseteq\bigcap_{i=1}^{n}{U_{i}}$. This follows
from the assumption of continuity of the loss function w.r.t. to the model
space and $[0,\infty]$.

###  B.2 Relatedness from $\mathcal{F}$

Proof [Theorem 17] A leaf ${L}$ is an immersed, connected, smooth submanifold
of a manifold on which $\mathcal{F}$ is defined (Lee, 2001; Stefan, 1974).
Thus, we need to show that, on such ${L}$, there exists a countable
(pseudo)group of transformations. Assume that ${L}$ has a dimension of $d$.

To begin with, we will identify some properties of the topology on ${L}$.
Assume that ${L}$ has subset topology derived from the foliation (Camacho and
Neto, 2013). From Lee (2001, Theorem 1.10), we know that ${L}$ has a countable
topological basis $\mathcal{B}$ of coordinate balls. Take the domains of the
charts of $\mathcal{F}$ restricted to ${L}$ (denoted by $\mathcal{F}|_{L}$);
these are open subsets of ${L}$, and form an open cover. We can create a
refinement of this cover, where for each domain $U\in\mathcal{F}|_{L}$, we
find $B_{U}\subseteq\mathcal{B}$ where $\bigcup_{\mathcal{B}^{i}\in
B_{U}}\mathcal{B}^{i}=U$. By restricting the chart maps to the appropriate
$\mathcal{B}^{i}$, we now have an altas on ${L}$ that is made up of coordinate
balls. We will denote this as
$\mathcal{A}_{\mathcal{B}}=\\{(\mathcal{B}^{i},\phi_{i})\\}$, where $\phi_{i}$
is the restriction of the appropriate chart map; we have simplified the
notation here by ignoring its construction from the original chart induced by
the foliation. Further, if we write
$\mathcal{B}^{i}\in\mathcal{A}_{\mathcal{B}}$, we mean the domain of a chart
in $\mathcal{A}_{\mathcal{B}}$.

We also know that ${L}$ is connected. Thus, given any two points on
$p,q\in{L}$, we can find a sequence
$\mathcal{B}_{1},...,\mathcal{B}_{n}\in\mathcal{B}$ such that
$p\in\mathcal{B}_{1}$ and $q\in\mathcal{B}_{n}$, and
$\mathcal{B}^{i}\cap\mathcal{B}_{i-1}\neq\emptyset$. One can think of this as
a sequence of connected coordinate balls that connect $p$ and $q$.

Next, let us look at the Euclidean space $\mathbb{R}^{d}$, equipped with the
standard topology. Let us equip $\mathbb{R}^{d}$ with the standard frame. That
is, the standard frame $V=(V_{1},...,V_{d})$ is a collection of vector fields
$V_{i}$, where the $j$-th component of the vector $V_{i}(x)$ is given by
$V^{j}_{i}(x)=\delta^{j}_{i}$ for $x\in\mathbb{R}^{d}$. Note that since $V$ is
a global frame, we have that $(V_{1}(x),...,V_{d}(x))$ is a basis for the
tangent space at $x\in\mathbb{R}^{d}$.

We know that $V$ is a global frame, and that each $V_{i}$ is a complete vector
field. Thus the flow
$\theta_{i}:\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ exists.
Then, $\theta_{i}(t):\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$, for some
$t\in\mathbb{R}$ is a diffeomorphism. We define the following map
$\tau_{(x_{1},...,x_{d})}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ as,

$\tau_{(x_{1},...,x_{d})}(x)=\theta_{d}(x_{n})\circ...\circ\theta_{1}(x_{1})(x).$
(9)

where $x_{i}\in\mathbb{R}$. For all $(x_{1},...,x_{d})\in\mathbb{R}^{d}$, the
transformations $\tau_{(x_{1},...,x_{d})}$ form an additive group. It’s
inverse is given by,

$\tau_{(x_{1},...,x_{d})}^{-1}(x)=\tau_{(-x_{1},...,-x_{d})}(x)=\theta_{d}(-x_{n})\circ...\circ\theta_{1}(-x_{1})(x).$
(10)

Furthermore, this group is transitive in $\mathbb{R}^{d}$, since we can find a
transformation from any point $x\in\mathbb{R}^{d}$ to any other point
$y\in\mathbb{R}^{d}$.

Suppose $x=(x_{1},...,x_{d})$ and $y=(y_{1},...,y_{d}$). The following is then
known to be true.

$y=\theta_{n}(y_{n}-x_{n})\circ...\circ\theta_{1}(y_{1}-x_{1})(x),$ (11)

Let us denote this as a transformation $\tau_{xy}$. Note that there is an
inverse
$\tau_{xy}^{-1}=\tau_{yx}=\theta_{n}(x_{n}-y_{n})\circ...\circ\theta_{1}(x_{1}-y_{1})$,
where,

$x=\theta_{n}(x_{n}-y_{n})\circ...\circ\theta_{1}(x_{1}-y_{1})(y).$ (12)

Thus, given the coordinates of any two points, we can construct a
diffeomorphism between them using the standard frame.

Let us go back to our leaf ${L}$. Each
$\mathcal{B}^{i}\in\mathcal{A}_{\mathcal{B}}$ is homeomorphic to a standard
ball in $\mathbb{R}^{d}$. That is for each $\mathcal{B}^{i}$, there is a
$\mathcal{B}_{\mathbb{R}^{d}}^{r}$, for some $r\in\mathbb{R}^{+}$, that it is
homeomorphic to, and this homeomorphism is given by the chart map. Each such
coordinate ball is itself homeomorphic to $\mathbb{R}^{d}$ via a map such as,

$h_{i}(x_{i})=\frac{x_{i}}{r-||x||}=y_{i},$ (13)

for an $x\in\mathcal{B}_{\mathbb{R}^{d}}^{r}$ and $y\in\mathbb{R}^{d}$. Note
that $h_{i}$ exists for each of the coordinates. The inverse of this map is
given by,

$h_{i}^{-1}(y_{i})=\frac{ry_{i}}{1+||y||}=x_{i}.$ (14)

Note that these maps are smooth. Then, we can construct a diffeomorphism
$\pi_{(x_{1},...,x_{d})}^{i}:\mathcal{B}^{i}\rightarrow\mathcal{B}^{i}$, for
$x_{1},...,x_{d}\in\mathbb{R}$ as,

$\pi_{(x_{1},...,x_{d})}^{i}(p)=\phi_{i}^{-1}\circ
h_{i}^{-1}\circ\tau_{(x_{1},...,x_{d})}\circ h_{i}\circ\phi_{i}(p).$ (15)

The inverse of this map is,

${\pi_{(x_{1},...,x_{d})}^{-1}}^{i}(p)=\phi_{i}^{-1}\circ
h_{i}^{-1}\circ\tau_{(x_{1},...,x_{d})}^{-1}\circ h_{i}\circ\phi_{i}(p).$ (16)

Then, for any $p,q\in\mathcal{B}^{i}$, we can construct a map $\pi^{i}_{pq}$,

$\pi^{i}_{pq}(p)=\phi_{i}^{-1}\circ h_{i}^{-1}\circ\tau_{xy}\circ
h_{i}\circ\phi_{i}(p)=q,$ (17)

where $x=h_{i}\circ\phi_{i}(p)$ and $y=h_{i}\circ\phi_{i}(q)$. The inverse
${\pi^{-1}_{pq}}^{i}=\pi^{i}_{qp}$ of this map from $q\in\mathcal{B}^{i}$ to
$p\in\mathcal{B}^{i}$ is given by,

$\pi^{i}_{qp}(q)=\phi_{i}^{-1}\circ h_{i}^{-1}\circ\tau_{yx}\circ
h_{i}\circ\phi_{i}(p)=p.$ (18)

$\pi_{(x_{1},...,x_{d})}^{i}$ is then a group of diffeomorphisms that acts
transitively on $\mathcal{B}^{i}$. We have effectively used the homeomorphism
defined in Equation 13 to move the transformations on $\mathbb{R}^{d}$ given
in Equation 9 to transformation in $\mathcal{B}^{i}$. Let us denote by
$\Pi^{i}$ the set of all transformations of the form
$\pi_{(x_{1},...,x_{d})}^{i}$.

An open subset of $\mathcal{B}^{i}$ is denoted as $U^{i}$. We will also denote
a countable (possibly finite) sequence of elements from $\mathbb{R}^{d}$ as
$\mathcal{S}_{\mathbb{R}^{d}}=\\{x^{1},x^{2},...\\}$, and a countable sequence
of coordinate balls
$\mathcal{S}_{U^{i},\mathcal{B}}=\\{U^{i},\mathcal{B}_{2},...\\}$, where
$\mathcal{B}_{j}\in\mathcal{A}_{\mathcal{B}}$. Now, we define a set of
transformations $\Pi$, where each $\pi\in\Pi$ is a map
$\pi^{\mathcal{S}_{U^{i},\mathcal{B}}}_{\mathcal{S}_{\mathbb{R}}^{d}}:U^{i}\rightarrow{L}$,
given by,

$\pi^{\mathcal{S}_{U^{i},\mathcal{B}}}_{\mathcal{S}_{\mathbb{R}}^{d}}=...\circ\pi^{\mathcal{B}_{j}}_{x_{j}}\circ...\circ\pi^{\mathcal{B}_{2}}_{x_{2}}\circ\pi^{U_{i}}_{x_{1}}(p),$
(19)

such that the image and domain of each subsequent transformation is valid.
That is, the intersection of image and domain
$\mathrm{im}(\pi^{\mathcal{B}_{j}}_{x_{j}})\cap\mathrm{dom}(\pi^{\mathcal{B}_{j+1}}_{x_{j+1}})\neq\emptyset$.
Thus $\Pi$ contains all transformations acting on the leaf ${L}$ that can be
written as countable compositions of maps from any $\Pi^{i}$, as long as the
ranges and domains make sense.

We now show that $\Pi$ satisfies each condition in Definition 32.

  1. a)

Satisfied by definition, since $\mathrm{dom}(\pi^{i})=\mathcal{B}_{q}$,
$\pi^{i}|_{U}$ for $U\subset\mathcal{B}^{i}$ exists,

  2. b)

follows from (1),

  3. c)

since all compositions are included, this follows from definition,

  4. d)

by settings $x_{i}=(0,...,0)$ for all $x_{i}$ in
$\mathcal{S}_{\mathbb{R}}^{d}$, we obtain the identity maps,

  5. e)

follows from Equation 15.

Finally, we must show that $\Pi$ is transitive. We do this be showing
constructively that between any two points on ${L}$, we can find a map between
them in $\Pi$. Recall that due to the connectedness of ${L}$, we can find a
sequence $\\{\mathcal{B}_{n}\\}_{pq}$ of coordinate balls that connect two
points $p,q\in\mathcal{B}^{i}$. Take the intersection
$\mathcal{B}^{i}\cap\mathcal{B}^{i}-1$. We will denote by $p_{i,i-1}$ to be
any point in this intersection. Then we can define a transformation $\pi_{pq}$
between points $p,q\in{L}$ as,

$\pi_{p,...,p_{i,i-1},...,q}(p)=\pi_{p_{n,n-1},q}\circ...\circ\pi_{p,p_{2,1}}(p)=q.$
(20)

It’s inverse is given by,

$\pi_{q,...,p_{i,i-1},...,p}(q)=\pi_{p_{2,1},p}\circ...\circ\pi_{q,p_{n,n-1}}(q)=p.$
(21)

###  B.3 MAML

Proof [Theorem 20] Since the loss function is given by
$L((t_{1},t_{2}),(m_{1},m_{2}))=\sum_{i=1}^{2}(m_{i}-t_{i})^{2}$, assuming
that the coordinate system is centered at $m_{0}$, the vector field on
$\mathcal{M}$ is given by:

$\mathrm{M}((m_{1},m_{2}))=(\dot{m_{1}},\dot{m_{2}})=2m_{1}\frac{\partial}{\partial
m_{1}}+2m_{2}\frac{\partial}{\partial m_{2}}.$ (22)

This is a system of differential equations which can be solved to give us a
flow $\theta(k)=(m_{1}(k),m_{2}(k))$; the initial value condition is
$m_{i}(0)=0$, since we have centered the coordinate system at $m_{0}=(0,0)$.

The flow is given by:

$m_{i}(k)=t_{i}e^{-2tk}+t_{i}.$ (23)

Since $L(t,m_{t}^{*})=0$, we have that for the $m(k)$ at which
$\>\big{|}\>L(t,m(k))-L(t,m_{t}^{*})\>\big{|}\>=\epsilon$,
$L(t,m(k))=\epsilon$. Thus,

$\sum_{i=1}^{2}((t_{i}e^{-2tk}+t_{i})-t_{i})^{2}=\epsilon.$ (24)

This can be rearranged to produce

$\sum_{i=1}^{2}t_{i}^{2}=\frac{\epsilon}{e^{-4k}}.$ (25)

Proof [Corollary 21] Equation 24 can be solved for a $k$, which gives us:

$k_{\epsilon}=4\ln\bigg{(}\frac{\epsilon}{t_{1}^{2}+t_{2}^{2}}\bigg{)}.$ (26)

This can then be plugged into Equation 23 to give us:

$m_{i}=-t_{i}\bigg{(}\frac{\epsilon}{t_{1}^{2}+t_{2}^{2}}\bigg{)}+t_{i}.$ (27)