# Simplified approach to the magnetic blue shift of Mott gaps

Mohsen Hafez-Torbati<EMAIL_ADDRESS>Condensed Matter Theory,
Department of Physics, TU Dortmund University, 44221 Dortmund, Germany
Frithjof B. Anders<EMAIL_ADDRESS>Condensed Matter Theory,
Department of Physics, TU Dortmund University, 44221 Dortmund, Germany  Götz
S. Uhrig<EMAIL_ADDRESS>Condensed Matter Theory, Department of
Physics, TU Dortmund University, 44221 Dortmund, Germany

###### Abstract

The antiferromagnetic ordering in Mott insulators upon lowering the
temperature is accompanied by a transfer of the single-particle spectral
weight to lower energies and a shift of the Mott gap to higher energies
(magnetic blue shift, MBS). The MBS is governed by the double exchange and the
exchange mechanisms. Both mechanisms enhance the MBS upon increasing the
number of orbitals. By performing a polynomial fit to numerical dynamical
mean-field theory data we provide an expansion for the MBS in terms of hopping
and exchange coupling of a prototype Hubbard-Kondo-Heisenberg model and
discuss how the results can be generalized for application to realistic Mott
or charge-transfer insulator materials. This allows estimating the MBS of the
charge gap in real materials in an extremely simple way avoiding extensive
theoretical calculations. The approach is exemplarily applied to
$\alpha$-MnTe, NiO, and BiFeO3 and an MBS of about $130$ meV, $360$ meV, and
$157$ meV is found, respectively. The values are compared with the previous
theoretical calculations and the available experimental data. Our ready-to-use
formula for the MBS simplifies the future studies searching for materials with
a strong coupling between the antiferromagnetic ordering and the charge
excitations, which is paramount to realize a coupled spin-charge coherent
dynamics at a femtosecond time scale.

##  I Introduction

Spintronics utilizes the charge and the spin degrees of freedom of electrons
for data storage and information processing. While it originally relies on
ferromagnetic bits, a rapidly growing interest for developing a spintronic
technology based on antiferromagnetic materials has been witnessed in the last
decade. In contrast to their ferromagnetic counterparts, antiferromagnetic
materials combine multiple unique properties which make them ideal candidates
for the construction of the next generation spintronic devices.
Antiferromagnets are resilient to disturbing magnetic fields which supports
long-term data retention, create no stray field allowing high-density memory
integration, and display a spin dynamics up to 1000 times faster than
ferromagnets providing access to terahertz writing speed regime. But the
faster dynamics due to substantially larger exchange interactions also
represents an obstacle for efficient probing and manipulating the magnetic
states [1, 2, 3].

In recent years, the interaction of electromagnetic radiation with
antiferromagnetic states has been demonstrated as a powerful tool to detect
and to control the magnetic order in antiferromagnetic materials [3]. A
coherent dynamics of the local magnetization in antiferromagnetic insulators
is induced and manipulated by ultrashort laser pulses [4, 5, 6] reaching
frequencies as high as 20 THz [7, 8, 9]. However, this concerns the magnetic
properties of the material only. An important step for the further development
of this research area is the utilization of spin and charge coupling. Aiming
at an efficient integration of magnonics, spintronics, and conventional
electronics the conversion of magnetic signals to charge signals and vice-
versa [10, 11] is crucial. This makes ultrafast spin-charge coupling a
cornerstone of future information processing. An essential first step in this
direction is to identify materials displaying a strong coupling between the
spin and the charge degrees of freedom which does not rely on the typically
weak spin-orbit coupling. One notes that the strength of the spin-charge
coupling determines the characteristic time scale at which the conversion of
spin signals into charge responses can take place and govern the spin-charge
oscillation dynamics.

Mott insulators, or the closely related charge-transfer insulators, are among
the promising materials for antiferromagnetic spintronics [2]. In Mott
insulators in transition metal compounds strong repulsive interaction between
electrons in the $3d$ shell is reponsible for the energy gap opening in the
single-particle spectral function (density of states). This gap defines the
energy necessary to create an electron and a hole independent from each other
in the system and is known as the charge gap or the Mott gap. Depending on the
number of unpaired electrons in the $3d$ shell one may realize Mott insulators
with different number of relevant orbitals. Most of the Mott insulators
undergo a transition from a paramagnetic state to an antiferromagnetic state
upon reducing the temperature below a critical value called Néel temperature
$T_{\rm N}$. The magnetic shift of the Mott gap is given by

$\Gamma_{\rm MG}(T)=G_{\rm MI}(T)-G_{\rm PI}(T),$ (1)

where $G_{\rm MI}(T)$ stands for the gap in the magnetic insulator (MI), and
$G_{\rm PI}(T)$ stands for the gap in the paramagnetic insulator (PI) at the
temperature $T$. A finite value of $\Gamma_{\rm MG}(T)$ represents a direct
coupling between the spin and the charge degrees of freedom. For $T<T_{\rm N}$
the PI is not the stable phase due to the spin ordering but $G_{\rm PI}(T)$
can still be analyzed by extrapolating the results to below $T_{\rm N}$ in
experiment or by enforcing a paramagnetic solution in theory. A magnetic blue
shift (MBS) corresponding to $\Gamma_{\rm MG}(T)>0$ is observed in several
previous experimental [12, 13, 14, 15, 16, 17, 18] and theoretical
investigations [18, 19, 20, 21, 22, 23, 24] of antiferromagnetic insulators.
We unveiled in Ref. [24] that the MBS originates from the double exchange and
exchange mechanisms. A strong MBS enables to go beyond the local magnetization
dynamics [7, 8, 9] and induce an ultrafast coherent manipulation of the
transport properties. This highlights the importance of a systematic analysis
of the influential mechanisms involved in the MBS of the Mott gap (1).

This paper is devoted to a systematic study of the effect of antiferromagnetic
ordering on the single-particle spectral function in Mott insulators. We
specifically address how such a spin-charge coupling changes upon changing the
number of orbitals. Our investigation relies on a dynamical mean-field theory
(DMFT) analysis of a generic three-dimensional Hubbard-Kondo-Heisenberg (HKH)
model. This model combines a single-band Hubbard model with localized spins
$S$ as in a Kondo lattice model. The latter also represent electrons in
orbitals so that we achieve an approximate description of a multi-orbital
model with $2S+1$ orbitals in total. The antiferromagnetic ordering is
accompanied by a transfer of spectral weight to lower energies and a MBS of
the Mott gap. The double exchange and exchange mechanisms both enhance the MBS
upon increasing the number of orbitals.

For our prototype model we provide an expansion for the MBS in terms of
hopping and exchange coupling by performing a polynomial fit to numerical DMFT
data. We discuss how the expansion can be generalized for application to real
materials. This allows estimating the MBS of the charge gap in Mott as well as
in charge-transfer insulator materials in an extremely simple way which avoids
extensive theoretical calculations. We exemplarily apply the approach to
$\alpha$-MnTe, NiO, and BiFeO3 as promising candidates for antiferromagnetic
spintronics. For $\alpha$-MnTe we find an MBS of about $130$ meV which is in a
very good agreement with the previous theoretical calculations and
experimental data. For NiO and BiFeO3 we obtain an MBS of about $360$ and
$157$ meV, respectively, which are also compared with the available
experimental data. Our ready-to-use formula for the MBS simplifies future
investigations searching for materials with strong spin-charge coupling which
would then set the stage to realize a coupled spin-charge coherent dynamics at
the unprecedented high frequencies in the THz region.

The paper is organized as follows. In Section II we discuss the model and the
method. The results are presented in Section III. Section IV is devoted to the
application of our results to real materials. The paper is concluded in
Section V.

##  II Model and method

###  II.1 Hubbard-Kondo-Heisenberg model

A full description of the charge and the spin degrees of freedom in strongly
correlated multi-orbital systems requires an analysis of a multi-orbital
Hubbard model whose reliable investigation is a grand challenge for current
theoretical research. Some simplifications can be applied in the case that the
system is deep in the Mott regime. In Mott insulators the charge excitations
are high in energy and the low-energy properties of the system are governed by
spin excitations. A common and successful strategy to address the low-energy
excitations is by mapping the multi-orbital Hubbard model to the Heisenberg
model

$H=J\sum_{\langle i,j\rangle}\vec{\mathcal{S}}_{i}\cdot\vec{\mathcal{S}}_{j}$
(2)

with the maximum local quantum spin number $\mathcal{S}$ in accord with Hund’s
first rule. The notation $\langle i,j\rangle$ limits the exchange spin-spin
interaction to nearest neighbor sites. However, such a mapping completely
removes the charge degrees of freedom and prevents any access to the charge
excitations which are the main focus of this paper.

Figure 1: Description of a three-orbital Mott insulator as a Hubbard-Kondo-
Heisenberg model (3) with a localized spin $S=1$. The itinerant electron is
described by the Hubbard model with nearest-neighbor hopping $t$ and onsite
repulsion $U>0$. The spin of the itinerant electron is coupled by the Hund
coupling $J_{\rm H}$ to the localized spin. There are nearest-neighbor
exchange interactions $J$ between electron and localized spins resulting from
the exchange processes.

Hence, we have to go beyond the Heisenberg model (2) and address the charge
gap in multi-orbital Mott insulators within a tractable model. We follow the
idea initialy proposed in Ref. [17] and applied successfully to $\alpha$-MnTe
in Ref. [24]. It is discussed here in details for the sake of completeness.
The idea relies on considering one orbital as itinerant and the other orbitals
as localized. The electron in the itinerant orbital feels the existance of the
localized electrons via an effective spin-spin interaction, see Fig. 1. We
stress that the itinerant orbital is a representative for all orbitals. It
does not mean we make a distinction between the orbitals. We will discuss
below that this approach takes into account on-site charge fluctuations $\pm
1$ around the half filling, but no further ones as, e.g., $\pm 2$. The system
is described by a HKH model given by

$H=H_{\rm Hu}+H_{\rm Ko}+H_{\rm He}$ (3)

with

$\displaystyle H_{\rm Hu}=$ $\displaystyle-t\sum_{\langle
i,j\rangle}\sum_{\sigma}(c^{{\dagger}}_{i,\sigma}c^{{\vphantom{\dagger}}}_{j,\sigma}+{\rm
H.c.})\\!+\\!U\sum_{i}n^{{\vphantom{\dagger}}}_{i,\downarrow}n^{{\vphantom{\dagger}}}_{i,\uparrow}\
,$ (4a) $\displaystyle H_{\rm Ko}=$
$\displaystyle-2J_{\text{H}}\sum_{i}\vec{S}_{i}\cdot\vec{s}_{i}\ ,$ (4b)
$\displaystyle H_{\rm He}=$ $\displaystyle+J\sum_{\langle
i,j\rangle}(\vec{S}_{i}\cdot\vec{s}_{j}+\vec{S}_{j}\cdot\vec{s}_{i}+\vec{S}_{i}\cdot\vec{S}_{j})\
,$ (4c)

where $c^{{\dagger}}_{i,\sigma}$ and $c^{{\vphantom{\dagger}}}_{i,\sigma}$ are
the usual fermionic creation and annihilation operators and
$n^{{\vphantom{\dagger}}}_{i,\sigma}=c^{{\dagger}}_{i,\sigma}c^{{\vphantom{\dagger}}}_{i,\sigma}$
is the occupation number operator at the lattice site $i$ with spin $\sigma$.
The Hubbard model (4a) with the nearest-neighbor hopping $t$ and the onsite
interaction $U$ between electrons with opposite spins
$\sigma=\uparrow,\downarrow$ describes the itinerant electrons, the Kondo term
(4b) describes the ferromagnetic Hund coupling $J_{\rm H}>0$ between the
electron spin $\vec{s}_{i}$ and the localized spin $\vec{S}_{i}$ at each
lattice site $i$, and the Heisenberg term (4c) describes the virtual exchange
interactions between the itinerant and the localized spins at neighboring
sites. The Hamiltonian is schematically depicted in Fig. 1. The quantum spin
number of the localized spin $S$ depends on the number of orbitals. The spin
$S$ represents $2S$ orbitals so that including the itinerant orbital $2S+1$
orbitals are described in total. Hence, we have $S=0$ for the single orbital
case and $S=2$ for the five-orbital case, which is the maximum value in
transition metal compounds. In order to assess the influence of the number of
orbitals we vary $S$. For the case $S=0$ there are already data available in
the literature. Hence, we mainly focus on the representative values $S=1$ and
$S=2$ leaving other values to interpolation, although we present data in
Section III.3 also for the $S=1/2$ and the $S=3/2$ cases.

Throughout this paper we fix the exchange interaction $J$ in (4c) to
$J=4t^{2}/\Delta$ where $\Delta=U+2SJ_{\rm H}$ is the bare charge gap, i.e.,
the charge gap in the absence of any hopping. This is to guarantee that the
low-energy properties of Hamiltonian (3) for $U,J_{\rm H}\gg t$ match those of
the Heisenberg model (2) with the spin quantum number $\mathcal{S}=S+1/2$ and
the exchange interaction $J=4t^{2}/\Delta$. The exchange interaction
$J=4t^{2}/\Delta$ is independent from the number of orbitals; it describes the
exchange coupling between a pair of electrons on linked sites. We assume equal
intra- and inter-orbital hopping elements and the number of contributing
electrons is taken into account by the size of the local spin $S$ consistent
with the low-energy Heisenberg model Eq. (2). The effective interaction (4c)
is limited to the leading contribution as the system is considered to be deep
in the Mott regime.

The Hamiltonian (3) goes beyond the low-energy Heisenberg model (2) by
providing an appropriate description of the multi-orbital Mott insulators in
the one hole and the one double-occupancy subspaces, counted relative to half-
filling of the orbitals. The states which contain more than one hole and one
double occupancy at the same lattice site are neglected in the Hamiltonian
(3). However, in the limit of large $U$ and $J_{\rm H}$ these states are high
in energy and are not expected to contribute to the low-lying charge
excitations. In addition, we assume that the Hund coupling is large enough to
ensure that the spins of the local electrons are aligned parallely. In Section
III.1 we compare the low-energy properties of the HKH model (3) and the
corresponding Heisenberg model (2) and discuss the effect of the neglected
spin states on the thermal fluctuations.

We point out that in general multi-orbital systems the crystal field splitting
modifies the bare charge gap $\Delta=U+2SJ_{\rm H}$. However, this
contribution is negligible compared to the intra-orbital interaction $U$ and
the Hund coupling $J_{\rm H}$. In addition, we only aim at an estimate of the
magnetic blues shift deep in the Mott regime at half filling.

It is already an established and efficient method to determine the exchange
interactions in Mott insulators by fitting theoretical magnon [25, 26, 27] or
triplon [28, 29, 30, 31] dispersions computed for Heisenberg models to
experimental data, mostly inelastic neutron scattering data. The Hubbard
interaction and the Hund coupling can be estimated based on atomic physics or
density-functional theory analysis, which in combination with the exchange
interactions allow us to determine the hopping via $J=4t^{2}/\Delta$. Hence,
the HKH model (3) provide a well-justified way to estimate the charge gap in
multi-orbital Mott insulators. We will estimate hopping elements from the
exchange values $J$. The idea is to use the minimum of experimental input for
the intended estimate. Of course, density functional theory input could also
be used. But this would require additional demanding calculations.

One should note that different bare charge gaps $\Delta=U+2SJ_{\rm H}$ (and
exchange interactions $J=4t^{2}/\Delta$) will be obtained if the same Hubbard
interaction $U$ and the same Hund coupling $J_{\rm H}$ are used for different
spin lengths $S$. This prevents a direct comparison of the results obtained
for different spin lengths since it is the hopping $t$ and the bare charge
$\Delta$ (or $t$ and $J=4t^{2}/\Delta$) which determine the MBS. For this
reason we compare results obtained for $J_{\rm H}=0.3U/S$ and a $U$
independent from $S$. In this way, the bare charge gap $\Delta=1.6U$ depends
only on $U$ and for constant $U$ the exchange interaction stays constant. The
results for the MBS will reflect the influence of the spin length only.
However, in Section III.3 we will consider also other combinations of the Hund
coupling $J_{\rm H}$ and the Hubbard interaction $U$ in order to show that the
MBS in the Mott regime does not depend on $U$ and $J_{\rm H}$ individually,
but only on their combination through the bare charge gap $\Delta$.

###  II.2 Dynamical mean-field theory

We study the charge gap in both the MI and the PI phases of the HKH model on
the generic three-dimensional cubic lattice using the DMFT technique. The
chosen theoretical approach of DMFT is an established method for strongly
correlated systems with large coordination numbers which fully takes into
account the local quantum fluctuations [32]. The self-energy is approximated
as local but it can depend on frequency in contrast to static mean-field
theories. We emphasize that a frequency-dependent self-energy is essential
[33] to describe the paramagnetic Mott insulator and to address the MBS in Eq.
(1). The DMFT is already employed to address Kondo lattice systems [34, 35,
36]. We applied the method specifically to the HKH model in Ref. [24]. Here,
we review the procedure for the sake of completeness.

We assume collinear antiferromagnetic Néel order with the spin polarization in
the $z$ direction. In the limit of large coordination number justifying the
DMFT the effect of the intersite exchange interactions (4c) can be captured by
a mean-field approximation [37] as

$H_{\rm He}=-\sum_{i}h_{i}^{\rm iti}s_{i}^{z}-\sum_{i}h_{i}^{\rm
loc}S_{i}^{z}$ (5)

with the effective magnetic fields $h_{i}^{\rm iti}$ acting on the itinerant
spin and $h_{i}^{\rm loc}$ acting on the localized spin given by

$\displaystyle h_{i}^{\rm iti}$ $\displaystyle=6J\langle S_{i}^{z}\rangle\ ,$
(6a) $\displaystyle h_{i}^{\rm loc}$ $\displaystyle=6J\langle
S_{i}^{z}+s_{i}^{z}\rangle\ .$ (6b)

The approximation (5) reduces the Hamiltonian (3) to a model involving only
local interactions in addition to electron hopping. We expect the mean-field
treatment (5) of the exchange interaction (4c) to be more accurate for larger
localized spins, where the fluctuations become less relevant.

Although the DMFT formulated in momentum space is suitable for the current
problem we opt for the real-space DMFT [38, 39, 40] due to its easy
adaptability to study the spatial dependence of the MBS in thin films in the
future as well as due to the efficient and flexible implementation we have for
the method at hand [41] which has been already applied to multiple problems
[42, 43, 44, 24]. The main disadvantage of the real-space DMFT is the
finiteness of the system. We consider periodic $L\times L\times L$ clusters
with mainly $L=10$. But, we check that the results even close to the Néel
temperature remain the same with the results obtain for $L=20$. We stress that
although we use the real-space DMFT we fully exploit the lattice symmetry to
compute the lattice Green’s function and to address the local impurity problem
[41]. This is the reason we can easily reach large system sizes.

The DMFT requires the solution of a self-consistency problem. Its iteration
loops start with an initial guess for the local polarizations $\langle
S_{i}^{z}\rangle$ and $\langle s_{i}^{z}\rangle$ and the local self-energy
$\Sigma_{i,\sigma}(i\omega_{n})$ at a single representative lattice site $i$,
which allows us to determine the effective magnetic fields in Eq. (5) and the
self-energy on the whole lattice because the Néel phase is invariant under the
combination of the spin-up and spin-down sublattices swap and the spin-flip
transformation. The lattice Green’s function is calculated using the lattice
Dyson equation

${\boldsymbol{G}}(i\omega_{n})=\left(i\omega_{n}\mathds{1}-{\boldsymbol{H}}_{0}-{\boldsymbol{\Sigma}}(i\omega_{n})\right)^{-1},$
(7)

where $\mathds{1}$ is the identity matrix,
${\boldsymbol{\Sigma}}(i\omega_{n})$ denotes the self-energy matrix with only
non-zero diagonal elements
$\Sigma_{i,\sigma}(i\omega_{n})=[{\boldsymbol{\Sigma}}(i\omega_{n})]_{i,\sigma;i,\sigma}$,
and ${\boldsymbol{H}}_{0}$ is the matrix representation of the operator

$H_{0}=-\sum_{i}(h_{i}^{\rm iti}s_{i}^{z}+\mu n_{i})-t\sum_{\langle
i,j\rangle}\sum_{\sigma}(c^{{\dagger}}_{i,\sigma}c^{{\vphantom{\dagger}}}_{j,\sigma}+{\rm
H.c.})$ (8)

in the one-particle subspace $\\{i,\sigma\\}$. The chemical potential
$\mu=U/2$ is used to satisfy the half-filling condition. One notes that the
lattice Green’s function (7) is constructed for itinerant electrons.

To obtain the local Green’s function
$\mathcal{G}_{i,\sigma}(i\omega_{n})=[{\boldsymbol{G}}(i\omega_{n})]_{i,\sigma;i,\sigma}$
we find the two columns of the inverse matrix in Eq. (7) corresponding to the
representative site $i$ with up spin $\sigma=\uparrow$ and down spin
$\sigma=\downarrow$. The full inversion in Eq. (7) is not needed because each
column of the inverse matrix can be computed independent from the others. The
inverse dynamical Weiss field
$\mathcal{G}^{(0)}_{i,\sigma}(i\omega_{n})^{-1}$, which is the local
propagator in the auxiliary impurity problem without interaction, is obtained
from the local Dyson equation

$\mathcal{G}^{(0)}_{i,\sigma}(i\omega_{n})^{-1}=\mathcal{G}_{i,\sigma}(i\omega_{n})^{-1}+\Sigma_{i,\sigma}(i\omega_{n})^{\vphantom{-1}}\
.$ (9)

The local impurity problem at the representative site $i$ is described by an
effective Anderson-Kondo model

$\displaystyle H_{i}^{{\vphantom{\dagger}}}=-\mu
n^{{\vphantom{\dagger}}}_{i}-h_{i}^{\rm
iti}s_{i}^{z}+Un^{{\vphantom{\dagger}}}_{i,\downarrow}n^{{\vphantom{\dagger}}}_{i,\uparrow}-h_{i}^{\rm
loc}S_{i}^{z}-2J^{\vphantom{\dagger}}_{\\!H}\vec{S}_{i}\cdot\vec{s}_{i}$ (10)
$\displaystyle+$
$\displaystyle\sum_{\ell=1}^{n_{b}}\sum_{\sigma}\epsilon^{i}_{\ell}a^{{\dagger}}_{\ell,\sigma}a^{{\vphantom{\dagger}}}_{\ell,\sigma}+\sum_{\ell=1}^{n_{b}}\sum_{\sigma}\left(a^{{\dagger}}_{\ell,\sigma}V^{i}_{\ell,\sigma}c^{{\vphantom{\dagger}}}_{i,\sigma}+{\rm
H.c.}\right),$

where $a^{{\dagger}}_{\ell,\sigma}$ and
$a^{{\vphantom{\dagger}}}_{\ell,\sigma}$ are the fermionic creation and
annihilation operators at the bath site $\ell$ with the spin
$\sigma=\uparrow,\downarrow$. The bath sites approximate the effect of the
surrounding sites in the lattice. The bath parameters $\epsilon^{i}_{\ell}$
and $V^{i}_{\ell,\sigma}$ are determined by fitting the inverse dynamical
Weiss field (9) to the function [45]

$\tilde{\mathcal{G}}^{(0)}_{i,\sigma}(i\omega_{n})^{-1}=i\omega_{n}+\mu+{\rm
sgn}(\sigma)\frac{h_{i}^{\rm
iti}}{2}-\sum_{\ell=1}^{n_{b}}\frac{|V^{i}_{\ell,\sigma}|^{2}}{i\omega_{n}-\epsilon^{i}_{\ell}}\
,$ (11)

where ${\rm sgn}(\sigma)=+1$ for $\sigma=\uparrow$ and $-1$ for
$\sigma=\downarrow$. We stress that $\vec{s}_{i}$ and $\vec{S}_{i}$ in Eq.
(10) are fully quantum mechanical spin operators. The local problem (10) is
solved using exact diagonalization (ED) with a finite number of bath sites
$n_{b}$. The local polarizations $\langle S_{i}^{z}\rangle$ and $\langle
s_{i}^{z}\rangle$ and the local interacting Green’s function
$\tilde{\mathcal{G}}_{i,\sigma}(i\omega_{n})$, using the Lehmann
representation, are computed. The self-energy is updated in each recursive
iteration via

$\Sigma_{i,\sigma}(i\omega_{n})^{\vphantom{-1}}=\tilde{\mathcal{G}}^{(0)}_{i,\sigma}(i\omega_{n})^{-1}-\tilde{\mathcal{G}}_{i,\sigma}(i\omega_{n})^{-1}\
,$ (12)

and is used together with the local polarizations for the next iteration. The
iterations are continued until the convergence within a given tolerance is
reached.

We perform the DMFT iterations mainly with 200 positive Matsubara frequencies,
$\omega_{n}=(2n+1)\pi T$ with $n=0,1,\cdots,199$. However, for the
paramagnetic phase at low temperatures we increased the number of positive
frequencies even up to 1000. This is because in the paramagnetic Mott
insulator the self-energy diverges at low temperatures and more Matsubara
frequencies are needed to achieve accurate results, i.e., to achieve results
which no longer depend on the number of Matsubara frequencies used.

The spin-averaged single-particle spectral function $A(\omega)$ is obtained
from the real-frequency Green’s function of the effective Anderson-Kondo model
(10) as

$A(\omega)=-\frac{1}{2}\frac{1}{\pi}\sum_{\sigma}{\rm
Im}\left[\tilde{\mathcal{G}}_{i,\sigma}(\omega+i\eta)\right]$ (13)

where $\eta=0.05t$ is the broadening factor. The spectral function $A(\omega)$
does not depend on the lattice site $i$ even in the antiferromagnetic phase
since we average over both spin components. The ED solver with a finite number
of bath sites approximates a continuous spectral structure by a set of
discrete sharp peaks. Although the fine details of the spectral function are
not produced by the ED solver, the method is considered accurate for the
charge gap and is used for the single-orbital Hubbard model to benchmark the
results obtained by quantum Monte Carlo solver [22] which suffer from the
requirement of analytic continution to access the real-frequency dynamics. In
the case of Anderson-Kondo model the quantum Monte Carlo solver furthermore
suffers from the notorious fermionic sign problem [34, 46] and is restricted
to the Ising part of the Hund interaction. Therefore, we opt for the ED
solver.

##  III Results

###  III.1 Néel temperature

Before proceeding to the charge excitations, we first examine the low-energy
properties of the HKH model (3) by computing the Néel temperature $T_{\rm N}$
and comparing it with the Néel temperature of the corresponding Heisenberg
model (2).

The DMFT takes into account the local quantum fluctuations but neglects the
quantum fluctuations which are non-local. The local spin and charge quantum
fluctuations in the HKH model (3) are eliminated in the Heisenberg model (2).
The Heisenberg model involves only non-local spin quantum fluctuations. Thus,
we expect a DMFT analysis of the low-energy features of the HKH model (3) to
be equivalent to a mean-field analysis of the corresponding Heisenberg model
(2). This had already been observed in the single-orbital Hubbard model [47].

The local mean-field treatment of the Heisenberg model is the same as the
mean-field treatment of the corresponding Ising model because we consider
collinear Néel order. The mean-field equation for the local spin polarization
$m$ of the spin-$\mathcal{S}$ Ising model [48] is given by

$m=\mathcal{S}-\frac{\sum_{n=0}^{2\mathcal{S}}n\exp(-ZJmn/T)}{\sum_{n=0}^{2\mathcal{S}}\exp(-ZJmn/T)}$
(14)

where $Z$ is the coordination number and $n$ takes integer values. The Néel
temperature of the spin-$\mathcal{S}$ Ising model in the mean-field
approximation [48] reads

$\displaystyle T_{\rm
N}=\frac{Z}{3}J\mathcal{S}(\mathcal{S}+1)=2J(S+1/2)(S+3/2)$ (15)

where the coordination number is given by $Z=6$ for the generic cubic lattice
and we substituted $\mathcal{S}=S+1/2$.

Figure 2: The main plot shows the local spin polarization $m=|\langle
S_{i}^{z}+s_{i}^{z}\rangle|$ vs temperature $T$ in the Hubbard-Kondo-
Heisenberg (HKH) model (3) for the localized spins $S=1$ and $S=2$. The DMFT
results of the HKH model (symbols) are compared with the mean-field theory
(MFT) results of the corresponding low-energy Heisenberg model (lines). The
exchange interaction is fixed to $J=4t^{2}/\Delta=0.05t$ with the bare charge
gap $\Delta=U+2SJ_{\rm H}=1.6U$. The inset compares the DMFT Néel temperature
$T_{\rm N}$ of the HKH model for $S=1$ and $S=2$ with the MFT Néel temperature
given in Eq. (15) of the corresponding low-energy Heisenberg model, see the
main text. The results remain the same for the number of bath sites $n_{b}=4$
and $n_{b}=6$.

The main plot in Fig. 2 displays the DMFT results (symbols) for the local spin
polarization $m=|\langle S_{i}^{z}+s_{i}^{z}\rangle|$ of the HKH model for the
localized spins $S=1$ and $S=2$ vs temperature $T$. We fixed $J_{\rm
H}=0.3U/S$ and $U=50t$ which corresponds to the bare charge gap $\Delta=80t$
and the exchange interaction $J=4t^{2}/\Delta=0.05t$. We find no difference in
the results obtained for the number of bath sites $n_{b}=4$ and $n_{b}=6$ as
in both cases a perfect description of the dynamical Weiss field is achieved.

The mean-field theory results of the corresponding low-energy Heisenberg model
obtained from the solution of Eq. (14) are included in the main plot of Fig. 2
(lines) for comparison. The main plot in Fig. 2 indicates perfect agreement
between the DMFT solution of the HKH model and the mean-field theory solution
of the corresponding low-energy Heisenberg model for the local spin
polarization corroborating our expectations and arguments on the various kinds
of fluctuations.

As the exchange interaction $J$ is increased, the high-energy excitations
present in the HKH model (3) and neglected in the Heisenberg model (2) start
to contribute to the thermal fluctuations more and more. This explains the
observed deviation of the DMFT transition temperature from Eq. (15) in the
inset of Fig. 2. The precise location of $T_{\rm N}$ is extracted by
performing a square root fit as expected for mean-field theories to the local
magnetization data close to the transition temperature. The figure indicates a
deviation of $T_{\rm N}$ from Eq. (15) for $J>0.16t$. This deviation is larger
for larger localized spin $S$.

The origin of this deviation is not only the charge excitations but also the
spin excitations with the latter playing even a more important role. In the
Heisenberg model (2) the size of the local spin is always $\mathcal{S}=S+1/2$.
The HKH model (3), however, involves local spin states with the total spin
$S+1/2$ and $S-1/2$. The local spin excitation is given by $(2S+1)J_{\rm
H}=0.3\Delta(2S+1)/1.6S<\Delta$ since we fixed $J_{\rm H}=0.3U/S$. The ratio
of the local spin excitation to the Néel temperature (15) at $J=0.2t$ equals
to $\approx 2.5$ for $S=2$ and $7.5$ for $S=1$. This explains the larger
deviation from the mean-field Heisenberg results for $S=2$ compared to $S=1$
which we observe in the inset of Fig. 2. In addition, we point out that due to
the intersite couplings the actual excitations of the system are dispersive
and smaller than the local excitations.

The HKH model (3) takes into account the spin excitations of order of $J_{\rm
H}$ present in multi-orbital Mott systems only partially. As one can see, for
example, for the three-orbital case sketched in the left panel of Fig. 1 the
Hilbert space of the singly occupied orbitals is given by

$\displaystyle\mathcal{H}=\frac{1}{2}\otimes\frac{1}{2}\otimes\frac{1}{2}$
$\displaystyle=$ $\displaystyle(1\oplus 0)\otimes\frac{1}{2}\ ,$ (16)

while the HKH model in the right panel of Fig. 1 involves only the states
$1\otimes\frac{1}{2}$, and the state $0\otimes\frac{1}{2}$ is neglected. In
the case of more orbitals there are more excited spin states which are
neglected by describing a multi-orbital system by the HKH model. These
neglected excited states have an energy gap of order of $(2S+1)J_{\rm H}$ and
as long as $(2S+1)J_{\rm H}\gg T$ holds, they are indeed irrelevant and do not
contribute substantially to the thermal fluctuations.

Figure 3: The spectral function $A(\omega)$ vs frequency in the magnetic
insulator (MI) phase (a) and in the paramagnetic insulator (PI) phase (b) for
the localized spin $S=2$. The Hubbard interaction is given by $U=32t$ and the
Hund coupling by $J_{\rm H}=0.3U/S$ implying the bare charge gap
$\Delta=1.6U$. The results are for the number of bath sites $n_{b}=6$. The
spectral functions for the different temperatures $T$ are shifted vertically
for clarity. The Néel temperature is given by $T_{\rm N}\approx 1.37t$. The
black arrows indicate the peaks from which the Mott gap in the MI phase
$G_{\rm MI}(T)$ and in the PI phase $G_{\rm PI}(T)$ are read off.

The inset of Fig. 2 in fact signals that for $J>0.16t$ the high-energy
excitations start to contribute to thermal fluctuations and neither the
Heisenberg model (2) nor the HKH model (3) provide a fully appropriate
description of multi-orbital Mott systems at temperatures $T\approx T_{\rm N}$
for magnetic exchange couplings of this size. But, for $J<0.16t$ the thermal
fluctuations are solely due to the low-energy excitations included in both the
Heisenberg model and the HKH model. In the following, we limit our disscusion
to $J<0.16t$ corresponding to $\Delta>25t$ where the HKH model is perfectly
justified and examine how the low-energy excitations, which are fully taken
into account, influence the Mott gap.

Although for the low-energy properties such as the Néel temperature a simple
mean-field treatment of the Heisenberg model (2) leads to precisely the same
results as the DMFT of the HKH model (3) we emphasize that for accessing the
charge gap it is essential to go beyond the Heisenberg model and the static
mean-field approximation and address the HKH model with the DMFT.

###  III.2 Spectral redistribution and magnetic blue shift

In this section we examine how the formation of the magnetic ordering below
$T_{\rm N}$ affects the charge excitations in multi-orbital Mott insulators.
We present data for the spectral function in Eq. (13) for the cases $S=1$ and
$S=2$ and compare the results with the results obtained for the single-orbital
Hubbard model in previous studies. This allows us to identify the influence of
the number of orbitals on the coupling between the charge gap and the magnetic
ordering.

Figure 4: The same as Fig. 3 but for the localized spin $S=1$ with the Néel
temperature $T_{\rm N}\approx 0.59t$.

Fig. 3 displays the spectral function $A(\omega)$ vs frequency $\omega$ for
the HKH model (3) at different temperatures for the localized spin $S=2$, and
Fig. 4 depicts the same for the localized spin $S=1$. The Hubbard interaction
is given by $U=32t$ and the Hund coupling by $J_{\rm H}=0.3U/S$ leading to the
bare charge gap $\Delta=U+2SJ_{\rm H}=1.6U$. The results in the MI phase (a)
and in the PI phase (b) are shown separately. Below the Néel temperature
$T_{\rm N}$, the MI is the stable phase but the metastable PI phase can also
be studied by enforcing a paramagnetic solution of the DMFT equations. We keep
$J_{\rm H}S={\rm const.}$ to acquire a bare charge gap independent from $S$.
This allows us to study the effect which is solely due to the size of the
localized spin on the MBS. This procedure implies to use different
$J_{\text{H}}$ for different spin sizes.

It is evident from both figures that the spectral function remains essentially
unchanged in the PI phase upon reducing the temperature from the Néel
temperature $T_{\rm N}$ down to zero. Including non-local spin fluctuations
beyond DMFT may approach the spectral functions in the PI phase closer to the
one in the MI phase. However, these contributions are expected to be
negligible in systems with large coordination numbers where the DMFT is
justified. In contrast to the PI phase, there is a shift of the electron
($\omega>0$) and the hole ($\omega<0$) peaks denoted by arrows away from the
Fermi energy $\omega=0$ in the MI phase. This leads to an MBS of the Mott gap.
We find no change in the spectral function in the MI phase below $T=0.2t$ for
$S=2$ and below $T=0.1t$ for $S=1$ meaning that these temperatures can be
viewed as equivalent to zero temperature. Although the results are obtained
for the finite number of bath sites $n_{b}=6$ in the ED impurity solver, we
observe a very similar behavior also for $n_{b}=4$ shown in Appendix A. The
shift of the electron and hole spectral peaks towards higher energies due to
antiferromagnetic ordering is already realized in the single-orbital Hubbard
model [21, 22, 23]. Our findings in Figs. 3 and 4 extend these studies to the
multi-orbital case.

Due to the finite number of bath sites the spectral function is not smooth but
consists of separate sharp peaks whose weight distribution approximate the one
of the continuous spectral function. Nevertheless, one can still see from
Figs. 3 and 4 how the spectral function in the MI phase changes upon
increasing the temperature from $0$ to $T_{\rm N}$ and takes a similar form as
the spectral function in the PI phase. For continuity, one expects the
spectral function in the MI and in the PI to be exactly the same as $T\to
T_{\text{N}}$. The main difference is that at $T\approx T_{\rm N}$ the
electron and the hole spectral contributions in the MI phase are shifted
further towards the Fermi energy in contrast to the PI phase. This slight
inconsistency originates from the finite number of bath sites used in the ED
solver as well as from the generally accepted fact that achieving accurate
results near a transition point is demanding even for static quantities, let
alone dynamical functions. This suggests that the spectral function in the PI
phase should be considered more accurate for temperatures close to $T_{\rm
N}$.

One can see from Figs. 3 and 4 that for $T\to 0$ the low-energy electron and
hole peaks, indicated by arrows, carry a spectral weight which is larger in
the MI phase than in the PI phase. Such a difference between the spectral
weight distribution in the MI and in the PI phases was first pointed out for
the single-orbital Hubbard model in Ref. [21]. It was found qualitatively
consistent with photoemission spectroscopy measurements on Cr-doped V2O3 [21].
In Ref. [24] we demonstrated that such a magnetic redistribution of the
spectral weight would result in an increase in the free energy unless the
effect is compensated by an MBS of the Mott gap. One has to keep in mind that
a decrease in the free energy is the prerequisite for the MI phase to
stabilize. The MBS of the Mott gap (blue arrows) and the magnetic
redistribution of spectral weight (red arrows) are summarized schematically in
Fig. 5. The fact that the magnetic redistribution of the spectral weight and
the MBS of the Mott gap occur in multi-orbital systems, see Figs. 3 and 4,
very similar to the single-orbital case [24] corrobarates the view that they
are generic spectral consequences of antiferromagnetic ordering in Mott
insulators. This view is further supported by the connection to the decrease
of the free energy [24].

Figure 5: Schematic representation of the spectral function $A(\omega)$ in the
paramagnetic insulator (PI) and in the magnetic insulator (MI) at the
temperature $T\to 0$. The magnetic blue shift of the Mott gap is denoted by
continuous blue arrows and the magnetic redistribution of spectral weight by
dashed red arrows.

So far, we discussed the effects common to the representative values $S=0$,
$S=1$, and $S=2$. Next, we turn to the differences between the results for
these cases which indicate the general trends for varying $S$, i.e., the
number of orbitals $2S+1$. The study of these differences is essential in
order to identify Mott insulators representing a maximum coupling between the
magnetic order and the electronic structure.

As one can see from Figs. 3(b) and 4(b) the electron and the hole peaks
defining the gap $G_{\rm PI}(T)$ show no strong dependence on the size of the
localized spin $S$ although for the $S=2$ case the gap seems a bit larger than
for the $S=1$ case. However, in the MI phase the gap $G_{\rm MI}(T)$ as $T\to
0$ is clearly larger for the $S=2$ case in Fig. 3(a) in contrast to the $S=1$
case in Fig. 4(a).

To investigate the gap in more details and to reveal the dependence of the MBS
in Eq. (1) on $S$, we plot the gap in the PI phase $G_{\rm PI}(T)$ and in the
MI phase $G_{\rm MI}(T)$ as function of $T$ in Fig. 6 for $S=2$ (a) and for
$S=1$ (b). The local spin polarization $m$, right axis, is also shown. The
figures contain the results for the number of bath sites $n_{b}=6$ and
$n_{b}=4$. The gap is extracted from the positions of the peaks closest to the
Fermi energy specified by arrows for $n_{b}=6$ in Figs. 3 and 4 and for
$n_{b}=4$ in Figs. 12 and 13 in Appendix A.

Fig. 6 shows that the gap in the PI phase remains essentially unchanged as
function of temperature while in the MI phase the gap increases and the spin
polarization grows upon $T\to 0$ for both $n_{b}=6$ and $n_{b}=4$. The fact
that the gaps in the MI phase and in the PI phase do not continuously join at
the Néel temperature originates from the finite number of bath sites in the ED
solver as we discussed above for the spectral function.

Figure 6: The Mott gap in the magnetic insulator (MI) phase $G_{\rm MI}(T)$
and in the paramagnetic insulator (PI) phase $G_{\rm PI}(T)$ plotted vs
temperature $T$ for the localized spin $S=2$ (a) and $S=1$ (b). We set the
Hubbard interaction to $U=32t$ and the Hund coupling to $J_{\rm H}=0.3U/S$.
The bare charge gap is given by $\Delta=U+2SJ_{\rm H}=1.6U$. The figure also
includes the local spin polarization $m=|\langle S_{i}^{z}+s_{i}^{z}\rangle|$,
see right axis for its scale. The results for the number of bath sites
$n_{b}=6$ and $n_{b}=4$ are compared.

For temperatures near zero in Fig. 6 we find a nice agreement between the
results for $n_{b}=4$ and $n_{b}=6$ which permits an accurate estimate of the
gap and an investigation of the effect of the size of the localized spin on
the MBS. The PI gap $G_{\rm PI}(T)$ for the localized spin $S=1$ in Fig. 6(b)
is around $45t$ while for the localized spin $S=2$ in Fig. 6(a) it is around
$45.5t$. The MI gap $G_{\rm MI}(T)$ at $T\to 0$ is around $49.1t$ for $S=1$ in
Fig. 6(b) and it is around $50.7t$ for $S=2$ in Fig. 6(a). These gap values
correspond to an MBS $\Gamma_{\rm MG}(0)\approx 4.1t$ for $S=1$ and an MBS
$\Gamma_{\rm MG}(0)\approx 5.2t$ for $S=2$ which are significantly larger than
the MBS $\Gamma_{\rm MG}(0)\approx 1.1t$ we found in Ref. [24] for the single-
orbital case ($S=0$) with the same bare charge gap $\Delta\approx 50t$. Hence,
we arrive at the summarizing conclusion that increasing the size of the
localized spin increases the Mott gap in both the PI phase and the MI phase.
But, the increase in the MI phase is larger leading eventually to a larger MBS
for larger localized spin or, equivalently, larger number of involved
orbitals.

###  III.3 Hopping and exchange contributions to
magnetic blue shift

The double exchange mechanism is well-known since 1950s to be responsible for
the ferromagnetic ordering in doped perovskite manganites. The strong
ferrromagnetic Hund coupling (Hund’s first rule) forces the spin of the
itinerant electron to be aligned with the localized spin at each lattice site.
Hence, the electron motion is strongly hindered if the adjacent local spins
are antiparallel while parallel orientation allows the electron to move [49,
50, 51, 52, 37].

In Ref. [24] we revealed that in ferromagnetic Kondo lattice systems the
double exchange mechanism reduces the effective hopping upon transition from a
paramagnetic insulator to an antiferromagnetic insulator, which narrows the
spectral bandwidth and results in an MBS which is enhanced relative to its
value in a pure Hubbard model. For the Hubbard-Kondo model consisting of the
terms (4a) and (4b),

$H_{\rm HK}=H_{\rm Hu}+H_{\rm Ko},$ (17)

with a localized spin $S=2$ and a bare charge gap $\Delta\approx 50t$, we
identified an MBS $\Gamma_{\rm MG}(0)\approx 3.2t$ larger than $\Gamma_{\rm
MG}(0)\approx 1.1t$ in the single-orbital Hubbard model (4a) alone.
Nevertheless, the MBS $\Gamma_{\rm MG}(0)\approx 3.2t$ is still considerably
smaller than $\Gamma_{\rm MG}(0)\approx 5.2t$ as we found for the full HKH
model (3) with the same localized spin $S=2$. Since the double exchange
mechanism plays the same role in the HK model as in the HKH model we attribute
this additional enhancement to the exchange interaction (4c).

To unveil the role of the exchange interaction and of the double exchange
mechanism in the MBS in the HKH model we plot the hole contribution
($\omega<0$) of the spectral function in Fig. 7 at $T=0$ for different values
of $U$. The bare charge gap is given by $\Delta=U+2SJ_{\rm H}=1.6U$. The
spectral functions in the PI and MI phase for localized spins $S=1$ and $S=2$
are shown for comparison. The results in the MI phase are shifted vertically
for clarity. We display only the hole contribution for a better visibility of
the differences. The electron contribution is the mirror image of the hole
contribution with respect to the Fermi energy $\omega=0$ due to the electron-
hole symmetry of the studied model. This symmetry of the HKH model is always
perfectly realized in our results as one can see, for instance, from Figs. 3
and 4.

Figure 7: The spectral function $A(\omega)$ vs frequency $\omega$ at $T=0$ for
various values of the Hubbard $U$. The bare charge gap is given by
$\Delta=U+2SJ_{\rm H}=1.6U$ which corresponds to the Hund coupling $J_{\rm
H}=0.3U$ for the localized spin $S=1$ and $J_{\rm H}=0.15U$ for $S=2$. The
results are for the number of bath sites $n_{b}=6$. The frequency range is
restricted to the range of the hole contribution of the spectral function. The
electron contribution is the mirror image of the hole contribution with
respect to the Fermi energy $\omega=0$. At each panel the results for the
magnetic insulator (MI) phase and for the paramagnetic insulator (PI) phase
with different localized spins $S=2$ and $S=1$ are shown for comparison. The
results in the MI phase are shifted vertically for clarity. The arrows are
used in panel (b) to illustrate the magnetic blue shift, in panel (c) to show
the shift of the spectral function in the MI phase relative to the PI phase,
and in panel (d) to specify the spectral bandwidth in the MI phase, see the
main text.

The difference between the peaks closest to the Fermi energy (the rightmost
peaks) in the MI phase and in the PI phase in Fig. 7 equals half of the MBS,
$\Gamma_{\rm MG}(0)/2$. This difference is illustrated by double-head arrows
in panel (b) of Fig. 7 for the different localized spins $S$. The message of
Fig. 7 is very clear: Increasing the bare charge gap $\Delta$ from Fig. 7(a)
to Fig. 7(d) decreases the MBS. Concerning the dependence on $S$ the larger
$S$ entails a larger MBS.

One can identify two main origins behind the MBS, which become more and more
distinct as $\Delta$ becomes larger in Fig. 7. One contribution is a shift of
the spectral function in the MI phase towards higher energies (away from the
Fermi energy) relative to the spectral function in the PI phase. This can be
seen especially from Fig. 7(c) and Fig. 7(d) where the middle peak of the
spectral function in the MI phase is shifted with respect to the middle peak
of the spectral function in the PI phase. This shift is indicated in Fig. 7(c)
by arrows and is larger for the larger localized spin and it decreases upon
increasing the bare charge gap $\Delta$. A similar, but smaller shift of the
spectral function can be seen in our results for the single-orbital Hubbard
model and for the Hubbard-Kondo model in Ref. [24]. This shift stems from the
effective antiferromagnetic exchange interaction which results from the second
order lowering in energy by the virtual hopping of a spin-$\uparrow$ electron
onto a site with a spin-$\downarrow$ electron or vice versa. For parallel
spins this lowering does not occur due to Pauli’s principle [52, 53]. The
exchange interaction can be described in the MI phase by a mean-field
effective magnetic field with the strength

$h_{i}^{{\vphantom{\dagger}}}=6J\langle S^{z}_{i}+s^{z}_{i}\rangle$ (18)

at the lattice site $i$. One notes that the term $6J\langle s^{z}_{i}\rangle$,
absent in (6a), is due to the itinerant Hamiltonian (4a). The effective
magnetic field (18) induces a shift of the electron and the hole spectral
contributions away from the Fermi energy in the MI phase. The relation (18)
explains the larger shift of the spectral function we observe for the larger
localized spin, and the smaller shift we observe for larger $\Delta$ because
it corresponds to a smaller $J=4t^{2}/\Delta$. We refer to this contribution
to the MBS as the exchange contribution.

Figure 8: The magnetic blue shift $\Gamma_{\rm MG}(T)$ given by Eq. (1) at
$T=0$ vs the inverse bare charge gap $\Delta=U+2SJ_{\rm H}$ in the Hubbard-
Kondo-Heisenberg (HKH) model (3) with the localized spin $S=\frac{1}{2},1,2$.
The hopping parameter $t$ is used as the unit of energy on both axes. The
results for the Hubbard-Kondo (HK) model with $S=2$ and for the Hubbard model
from Ref. [24] are included for comparison. The results are obtained for
$J_{\rm H}=0.3U/S$ varying $U$, except for the filled circles (HKH model with
$S=2$) and the filled squares (HKH model with $S=1$) which are for $J_{\rm
H}=0.225U$ independent from $S$.

The other contribution to the MBS originates from a narrower spectral
bandwidth in the MI phase in contrast to the spectral bandwidth in the PI
phase due to the double exchange mechanism. This can best be seen in panel (d)
in Fig. 7. In this panel we exemplarily depict the spectral bandwidth in the
MI phase for $S=1$ and $S=2$ by double-head arrows. In Ref. [24] we discussed
how the double exchange mechanism shrinks the spectral bandwidth upon
transition from the PI to the MI phase leading to a contribution proportional
to the hopping to the MBS. We refer to this contribution as the double
exchange contribution or the hopping contribution.

The additional aspect one can read off from Fig. 7 is that how the hopping
contribution depends on the size of the localized spin $S$. Upon increasing
the size of the localized spin from $S=1$ to $S=2$ the spectral bandwidth in
both the PI phase and the MI phase decreases. However, this decrease is larger
in the MI phase than in the PI phase which results in a larger hopping
contribution to the MBS for larger localized spin.

In order to extract quantitative values of the hopping and the exchange
contributions to the MBS for the localized spins $S=\frac{1}{2},1,2$ in the
HKH model (3) we plot the MBS $\Gamma_{\rm MG}(0)$ vs the inverse bare charge
gap $\Delta$ in Fig. 8. The hopping parameter $t$ is used as the unit of
energy on both axes. A quadratic fit shown by a continuous line nicely
describes the data leading to the following expansion for the MBS in the HKH
model in powers of $J/t=4t/\Delta$,

$\displaystyle\frac{\Gamma_{\rm MG}(0)}{t}$ $\displaystyle=$ $\displaystyle
C_{0}+C_{1}\frac{4t}{\Delta}+C_{2}\frac{16t^{2}}{\Delta^{2}}+\cdots$ (19)
$\displaystyle=$ $\displaystyle
C_{0}+C_{1}\frac{J}{t}+C_{2}\frac{J^{2}}{t^{2}}+\cdots\ .$

One can see from Eq. (19) that there is a hopping contribution $\Gamma_{\rm
MG}^{(t)}(0)=C_{0}t$ and an exchange contribution $\Gamma_{\rm
MG}^{(J)}(0)=C_{1}J$ to the MBS. The hopping contribution is proportional to
the offset and the exchange contribution is proportional to the slope in Fig.
8 in the limit $t/\Delta\to 0$. The expansion coefficients in Eq. (19) are
provided in Table 1 for different spin $S$. Note that Table 1 provides the
expansion coefficients also for the $S=3/2$ case. The data for $S=3/2$ are not
included in Fig. 8 in order not to overload the figure. In Fig. 8, we also
included our results for the Hubbard model (4a) and the HK model (17) with the
localized spin $S=2$ from Ref. [24] and their linear fit for comparison. The
results in Fig. 8 are obtained for $J_{\rm H}=0.3U/S$ varying $U$, except for
the filled circles (HKH model with $S=2$) and the filled squares (HKH model
with $S=1$) which are computed for a fixed $J_{\rm H}=0.225U$ independent from
$S$. The fit is performed for the data with $J_{\rm H}=0.3U/S$ but nicely
describes also the data obtained for $J_{\rm H}=0.225U$. This confirms that
the MBS depends only on the bare charge gap $\Delta=U+2SJ_{\rm H}$ and not on
$J_{\rm H}$ and $U$ individually.

| $S=0$ | $S=1/2$ | $S=1$ | $S=3/2$ | $S=2$
---|---|---|---|---|---
$C_{0}$ | $0$ | $1.16$ | $1.84$ | $2.30$ | $2.64$
$C_{1}$ | $14.4$ | $28.3$ | $33.0$ | $38.5$ | $42.4$
$C_{2}$ | $0$ | $-33.5$ | $-50.4$ | $-73.6$ | $-86.8$
Table 1: The expansion coefficients for the MBS of the Mott gap in Eq. (19).
The results are for the HKH model in Eq. (3) with the localized spin $S$. One
notes that the localized spin $S$ in the HKH model corresponds to a Mott
insulator with the number of orbitals $2S+1$.

One can see from Table 1 that the hopping $C_{0}$ and the exchange $C_{1}$
coefficients both increase upon increasing the size of the localized spin in
the HKH model (3). One notes that there is also a negative quadratic
contribution $C_{2}$ which becomes larger for larger localized spins and
suppresses the MBS as $t/\Delta$ increases. The exchange coefficient
$C_{1}=28.3$ for $S=1/2$ is almost just twice the exchange coefficient
$C_{1}=14.4$ for $S=0$. This is nicely consistent with our explanation of the
origin of the exchange contribution of the MBS based on the mean-field
effective magnetic field in Eq. (18) which suggests a proportionality of
$C_{1}$ to $S+1/2$. For larger localized spins, however, the exchange
coefficient starts to deviate from the value $(2S+1)\times 14.4$. We associate
such a deviation to higher order terms beyond the mean-field effective
magnetic field (18) which become more and more important for larger localized
spins.

An equal hopping contribution in the HKH model (3) and in the HK model (17)
having the same localized spin $S$ is expected since in the limit $t/\Delta\to
0$ the Heisenberg term (4c) vanishes (since we have fixed $J=4t^{2}/\Delta$ in
Eq. (4c)). Indeed, Fig. 8 shows a nice agreement between the hopping
contribution in the HKH model ($C_{0}\approx 2.64$) and in the HK model
($C_{0}\approx 2.74$) for the localized spin $S=2$. This provides a
corroborating consistency test for our results. As $t/\Delta$ is increased
from zero in Fig. 8, the MBS in the HKH model with $S=2$ increases much faster
than the MBS in the HK model due to the additional exchange interaction (4c).

##  IV Application

The expansion for the MBS of the Mott gap in Eq. (19) is obtained for the
simple cubic lattice with the hopping and the exchange interactions limited to
nearest-neighbor sites. In real materials, however, one needs to deal with
various kinds of lattice structures and long-range terms beyond nearest
neighbor are commonly involved. The aim of this section is to generalize Eq.
(19) such that it can be employed to estimate the MBS for a larger variety of
antiferromagnetic compounds.

As we discussed in the previous section, one contribution to the MBS
originates from the exchange interaction and can be understood based on the
effective mean-field magnetic field in Eq. (18). This suggests to substitute
the exchange $J$ in Eq. (19), which holds for $Z=6$ neighbors, by an effective
value $\tilde{J}$

$ZJ\longrightarrow Z\tilde{J}=\sum_{i}\sigma_{i}Z_{i}J_{i}$ (20)

for a system with exchange interactions represented by $J_{i}$ for the $i$th
neighbor and the corresponding coordination number $Z_{i}$. The sign factor
$\sigma_{i}=\pm 1$ depends on the spin orientation on the lattice sites linked
by $J_{i}$. It takes the positive value $+1$ if the sites linked by $J_{i}$
have antiparallel spin ordering and the negative value $-1$ if the sites
linked by $J_{i}$ have parallel spin ordering. This is because an
antiferromagnetic exchange interaction $J_{i}>0$ linking sites with
antiparallel spin ordering or a ferromagnetic exchange interaction $J_{i}<0$
linking sites with parallel spin ordering enhance the effective mean-field
magnetic field while an antiferromagnetic exchange interaction linking sites
with parallel spin ordering or a ferromagnetic exchange interaction linking
sites with antiparallel spin ordering suppress the effective mean-field
magnetic field. This accumulates the energetically favored spin alignment in
the mean-field theory.

The expansion in Eq. (19) is obtained for an ideal antiferromagnetic system,
i.e., the exchange interaction is always antiferromagnetic linking only sites
with antiparallel spin ordering. Accordingly, we expect the generalization to
be valid for a system which contains no or weak ferromagnetic exchange
interactions in contrast to the antiferromagnetic ones. In addition, the
antiferromagnetic exchange interactions linking sites with parallel spin
ordering should be weak.

As we discussed in the previous section and also in Ref. [24], the hopping
contribution to the MBS stems from the narrowing of the spectral bandwidth by
the double exchange mechanism. The double exchange mechanism reduces the
effective hopping between sites with antiparallel spin ordering upon
transition from the PI to the MI phase. If the sites have parallel spin
ordering the double exchange mechanism enhances the effective hopping [24]
which is expected to induce a decrease in the MBS. This suggests the
substitution of the hopping $t$ in Eq. (19) by an effective hopping
$\tilde{t}$ according to

$Zt\longrightarrow Z\tilde{t}=\sum_{i}\sigma_{i}Z_{i}t_{i}$ (21)

where $t_{i}$ is the hopping to the $i$th neighbor given by
$t_{i}=\sqrt{\Delta J_{i}}/2$ for the antiferromagnetic exchange interaction
$J_{i}>0$. The bare charge gap is computed from the Hubbard $U$ and the Hund
coupling $J_{\rm H}$ as $\Delta=U+2SJ_{\rm H}$. We set $t_{i}=0$ for the
ferromagnetic exchange interaction $J_{i}<0$ which is supposed to be weak. The
sign factor $\sigma_{i}=\pm 1$ has the same definition as above, i.e., it is
$+1$ if $t_{i}$ is between sites with antiparallel spins and $-1$ if $t_{i}$
is between sites with parallel spins.

The exchange interactions can be extracted adequately by fitting the inelastic
neutron scattering data of low-energy excitations to the magnon dispersion
obtained by means of spin wave theory for instance. The approach has already
be applied to a variety of magnetic materials and the exchange interactions
are known. The atomic physics or the density functional theory calculations
yield an estimate of the Hubbard interaction $U$ and the Hund coupling $J_{\rm
H}$ which determine the bare charge gap $\Delta=U+2SJ_{\rm H}$. Hence, the
relation (19) with the generalizations (20) and (21) reads

$\Gamma_{\rm
MG}(0)=C_{0}\tilde{t}+C_{1}\tilde{J}+C_{2}\frac{\tilde{J}^{2}}{\tilde{t}}$
(22)

and provides a convenient way of evaluating the MBS of the Mott gap in
antiferromagnetic materials.

In the following we apply the method to the room temperature antiferromagnets
$\alpha$-MnTe, NiO, and BiFeO3 which are considered as promising candidates
[2] for application in antiferromagnetic spintronics. These systems are better
described as charge-transfer insulators [54] rather than Mott insulators. The
charge gap, known as the charge-transfer gap, is defined between the occupied
$p$ band and the empty upper Hubbard band. The MBS of the charge-transfer gap
can be approximated as half the MBS of the Mott gap [24],

$\Gamma_{\rm CTG}(T)\approx\frac{1}{2}\Gamma_{\rm MG}(T)\ .$ (23)

This relies on the assumption that the effect of the magnetic ordering on the
$p$ band is negligible as it is only indirectly affected by the magnetic
ordering, and that the shifts of the upper and the lower Hubbard bands to
higher energies are equal which is typical for half-filled Mott insulators.
For a more detail discussion we refer the reader to Ref. [24].

We point out that we neglect vertex corrections in our calculations of the
one-particle propagator and its spectral density. They enhance the formation
of excitons which influence the charge gap as measured in experiment. But in
DMFT they do not enter in conductivity at all [55] and are expected to be
small anyway for local interactions in large dimensions. In lower dimensions,
this can be different [56, 57].

###  IV.1 $\alpha$-MnTe

We start with the MBS in $\alpha$-MnTe for which we performed an extensive
analysis in Ref. [24] and for which there is also experimental data available
[16, 17]. This permits to examine how well the generalization in Eqs. (20) and
(21) is grounded before proceeding to the other compounds. The
antiferromagnetic order in $\alpha$-MnTe develops below the Néel temperature
$T_{\rm N}\approx 310$ K with the magnetic Mn+2 ions forming triangular
ferromagnetic layers which are stacked antiferromagnetically. The magnetic
order and the neighboring sites up to the fourth neighbor are specified in
Fig. 9. The exchange interactions involving up to third neighbor are computed
by fitting the magnon dispersion to the inelastic neutron scattering data
[26]. In Ref. [58] the exchange interactions are slightly modified adding also
a fourth neighbor term in order to attain accurate results for the Néel
temperature. We use the exchange couplings of Ref. [58] given by $J_{1}=3.072$
meV, $J_{2}=0.0272$ meV, $J_{3}=0.4$ meV, and $J_{4}=0.16$ meV (according to
our definition of the exchange interaction in Eq. (2)) which are all
antiferromagnetic allowing to assign a hopping term to them. One notes that
the dominent terms $J_{1}$ and $J_{3}$ are between sites with antiparallel
spin order. The terms $J_{2}$ and $J_{4}$ linking parallel spins are weak. We
consider a Hubbard interaction $U=5.5$ eV and a Hund coupling $J_{\rm H}=0.8$
eV based on the atomic physics [17] and the density functional theory [58].
The half filled $3d$ shell of Mn+2 ions corresponds to the localized spin
$S=2$ in the HKH model (3). These values result in the hopping parameters
$t_{1}\approx 81.74$ meV, $t_{2}\approx 7.69$ meV, $t_{3}\approx 29.49$ meV,
and $t_{4}\approx 18.65$ meV.

Figure 9: Antiferromagnetic order in $\alpha$-MnTe. There are ferromagnetic
triangular layers which are stacked antiferromagnetically. The 1st, 2nd, 3rd,
and 4th neighbors are specified.

Substituting the hopping and the exchange parameters in Eqs. (21) and (20)
with the coordination numbers $Z_{1}=2$, $Z_{2}=6$, $Z_{3}=12$, and $Z_{4}=2$
and the spin orientation signs $\sigma_{1}=\sigma_{3}=1$ and
$\sigma_{2}=\sigma_{4}=-1$ according to Fig. 9, we find the effective hopping
$\tilde{t}=72.34$ meV and the effective exchange $\tilde{J}=1.74$ meV. This
leads to the MBS of the charge-transfer gap

$\displaystyle\Gamma_{\rm CTG}(0)$
$\displaystyle=\frac{C_{0}}{2}\tilde{t}+\frac{C_{1}}{2}\tilde{J}+\frac{C_{2}}{2}\frac{\tilde{J}^{2}}{\tilde{t}}$
$\displaystyle\approx(95+37-2)\ {\rm meV}=130\ {\rm meV}\ ,$ (24)

where the expansion coefficients for $S=2$ in Table 1 are used. In Ref. [24]
we performed an extensive explicit analysis of the HKH model involving hopping
and exchange interactions up to the fourth neighbor with the above parameter
values for $\alpha$-MnTe. Our results provided a very nice description of the
experimental data [16, 17] for the MBS in $\alpha$-MnTe. We obtained an MBS of
the charge-transfer gap $\Gamma_{\rm CTG}(0)\approx 120$ meV. This indicates
that Eq. (IV.1) has reproduced the result of the explicit analysis with only
$8\%$ error.

We check further the results in Eq. (IV.1) by comparing the hopping and the
exchange contributions with the results of the explicit calculations in Ref.
[24] individually. In Ref. [24] we carried out a polynomial fit similar to Eq.
(19) and found that the MBS $120$ meV is consisted of a hopping contribution
of about $76$ meV and an exchange contribution of about $43$ meV. Comparing
with the results in Eq. (IV.1) we identify an error of $25\%$ for the hopping
contribution and an error of about $14\%$ for the exchange contribution, which
still depicts overall a nice agreement. We stress that the results in Eq.
(IV.1) are achieved by just some simple substitutions while the results in
Ref. [24] are obtained through extensive theoretical calculations.

We would like to mention that if the Hubbard interaction changes from $U=4$ eV
to $U=7$ eV and the Hund coupling from $J_{\rm H}=0.7$ eV to $J_{\rm H}=1.0$
eV the MBS in Eq. (IV.1) changes from $\Gamma_{\rm CTG}(0)\approx 120$ meV to
$\Gamma_{\rm CTG}(0)\approx 140$ meV. This change is solely due to the hopping
contribution as the exchange interactions are kept fixed. This shows that
there is no significant dependence of the MBS on the Hubbard interaction and
the Hund coupling. One only needs to have a rough estimate of these
parameters; the exchange interactions are the essential ones.

###  IV.2 NiO

We proceed to the antiferromagnetic insulator NiO which has the Néel
temperature $T_{\rm N}\approx 530$ K. The magnetic Ni+2 ions form
ferromagnetic $(111)$ planes stacked antiferromagnetically on a face-centered
cubic lattice [59, 60, 61]. The magnetic structure of NiO is shown in Fig. 10.
Each magnetic ion is surrounded by twelve first neighbors having parallel
alignment with six of them and antiparallel alignment with the other six. The
alignment with the second neighbors is always antiparallel. The onset of the
magnetic ordering is found to be accompanied by a small rhombohedral
distortion which continues to zero temperature [62, 63]. The lattice
distortion is also expected to induce a band gap blue shift [16, 17] which is
described by an empirical Varshni function [64].

Measurement of the magnon dispersion by inelastic neutron scattering suggest
first neighbor ferromagnetic exchange interactions $J_{1,{\rm p}}=-1.39$ meV
and $J_{1,{\rm ap}}=-1.35$ meV and a second neighbor antiferromagnetic
exchange interaction $J_{2}=19.01$ meV [65]. The term $J_{1,{\rm p}}$ links
sites with parallel spin ordering and the term $J_{1,{\rm ap}}$ links sites
with antiparallel spin ordering, see Fig. 10. The small difference between
$J_{1,{\rm p}}$ and $J_{1,{\rm ap}}$ originates from the lattice distortion.
The antiferromagnetic interaction $J_{2}$ is much larger than the
ferromagnetic interactions $J_{1,{\rm p}}$ and $J_{1,{\rm ap}}$. This suggests
our approach to be applicable to NiO.

The Hubbard interaction and the Hund coupling in NiO were estimated from the
local density approximation combined with the DMFT as $U=8$ eV and $J_{\rm
H}=1$ eV [66]. The electron configuration $3d^{8}$ of the Ni+2 ions implies
that two orbitals need to be taken into account corresponding to the HKH model
(3) with the localized spin $S=1/2$. We set the first neighbor hopping
$t_{1}=0$ due to the ferromagnetic exchange interaction and find the second
neighbor hopping $t_{2}=\sqrt{\Delta J_{2}}/2\approx 206.8$ meV. The effective
exchange parameter in Eq. (20) and the effective hopping parameter in Eq. (21)
can easily be obtained as $\tilde{J}=J_{2}+J_{1,{\rm ap}}-J_{1,{\rm p}}=19.05$
meV and $\tilde{t}=t_{2}\approx 206.8$ meV. Using the expansion coefficients
provided in Table 1 for $S=1/2$, we find the MBS of the charge-transfer gap in
NiO

$\displaystyle\Gamma_{\rm CTG}(0)$
$\displaystyle=\frac{C_{0}}{2}\tilde{t}+\frac{C_{1}}{2}\tilde{J}+\frac{C_{2}}{2}\frac{\tilde{J}^{2}}{\tilde{t}}$
$\displaystyle\approx(120+269-29)\ {\rm meV}=360\ {\rm meV}\ ,$ (25)

which is almost three times larger than the MBS we found for $\alpha$-MnTe in
Eq. (IV.1). This large MBS in NiO, despite the small localized spin, stems
from the large effective exchange interaction $\tilde{J}=19.05$ meV in
contrast to the effective exchange interaction $\tilde{J}=1.74$ meV in
$\alpha$-MnTe.

Figure 10: Magnetic structure of NiO. The second neighbor exchange interaction
($J_{2}$) is always between sites with antiparallel spin orientation. The
first neighbor exchange interaction can link sites with parallel ($J_{1,{\rm
p}}$) or antiparallel ($J_{1,{\rm ap}}$) spin ordering.

The optical absorption spectra of NiO has been the subject of extensive
research in the past decades [67, 68, 69, 70, 71, 72] and a band gap of about
$3.5$-$4.3$ eV is suggested depending on the method used to exctract the gap
from the experimental data [73]. The temperature dependence of the band gap
measured from the Néel temperature to the room temperature indicates an
increases of about $350$ meV [74]. If we assume based on our numerical results
[24] on the temperature dependences that below room temperature there is no
significant further magnetic contribution to the blue shift of the band gap as
the magnetization tends to saturate, the Varshni function fitted to the
experimental data [75] captures the band gap blue shift due to the lattice
distortion in NiO. Subtracting the lattice distortion contribution from the
blue shift of the band gap $350$ meV one finds an MBS of about $300$ meV. This
is in an overall nice agreement with our estimate $360$ meV in Eq. (IV.2).

###  IV.3 BiFeO3

We now turn to the multiferroic semiconductor BiFeO3 which has a band gap of
about $2.7$ eV at the room temperature. The transition metal ions Fe+3 with a
half filled $3d$ shell develop a Néel antiferromagnetic order on a pseudo-
cubic lattice structure below the Néel temperature $T_{\rm N}\approx 640$ K,
see Fig. 11. Each Fe+3 magnetic moment is surrounded by six first neighbor
Fe+3 with an antiparallel magnetic moment alignment and twelve second neighbor
Fe+3 with a parallel magnetic moment alignment [76, 77, 78].

Figure 11: Néel antiferromagnetic order on a simple cubic lattice describing
the magnetic structure in BiFeO3. The first neighbor exchange interaction
$J_{1}$ links sites with antiparallel spin ordering and the second neighbor
exchange interaction $J_{2}$ links sites with parallel spin ordering.

The exchange interactions in BiFeO3 are determined by the inelastic neutron
scattering as $J_{1}=4.38$ meV for the first neighbor and the much weaker
$J_{2}=0.15$ meV for the second neighbor [78]. A Hubbard interaction $U=5$ eV
and a Hund coupling $J_{\rm H}=1$ eV sound plausible from the different
density functional theory analyses [79, 80, 81]. These values result in the
first neighbor hopping $t_{1}\approx 99.3$ meV and the second neighbor hopping
$t_{2}\approx 18.4$ meV. The effective hopping and the effective exchange
interaction can be found from Eqs. (21) and (20) as $\tilde{t}\approx 62.5$
meV and $\tilde{J}=4.08$ meV. Using the expansion coefficients for $S=2$ in
Table 1 one obtains

$\displaystyle\Gamma_{\rm CTG}(0)$
$\displaystyle=\frac{C_{0}}{2}\tilde{t}+\frac{C_{1}}{2}\tilde{J}+\frac{C_{2}}{2}\frac{\tilde{J}^{2}}{\tilde{t}}$
$\displaystyle\approx(83+86-12)\ {\rm meV}=157\ {\rm meV}$ (26)

for the MBS of the charge-transfer gap in BiFeO3.

Despite extensive experimental research there are conflicting conclusions on
the direct or the indirect nature of the band gap in BiFeO3 and its
temperature evolution. A direct band gap for BiFeO3 is found in Refs. [82, 83,
84, 85, 86] which indicates a blue shift of about 100 meV upon reducing the
temperature from $T_{\rm N}\approx 640$ K to zero [86]. A much stronger change
in the band gap is observed in Ref. [87]. The change is about $500$ meV
between the Néel temperature and the room temperature [87]. A detailed Raman
scattering study of the electronic band structure of BiFeO3 associates the
strong band gap blue shift observed in Ref. [87] to the indirect nature of the
band gap [88]. Band structure calculations based on the density functional
theory also indicate an indirect band gap in BiFeO3 [87, 80, 79]. More
detailed investigations are needed to determine the temperature evolution of
the band gap in BiFeO3 and its microscopic origin, specifically including the
effect of the antiferromagnetic ordering on the band gap blue shift. This will
allow a comparison of the experimental data with our theoretical result in Eq.
(IV.3) and can unveil the direct or the indirect nature of the band gap in
this technologically attractive material.

##  V conclusion

We perform a systematic study of the effect of antiferromagnetic ordering on
the single-particle spectral function in Mott insulators involving various
number of orbitals. Our analysis relies on the DMFT of a HKH model. The model
goes far beyond the low-energy Heisenberg model and permits investigation of
charge excitations. The antiferromagnetic ordering is accompanied by a
transfer of the spectral weight to lower energies and an MBS of the Mott gap.
The MBS increases upon increasing the number of orbitals due to the double
exchange and exchange mechanisms.

We provide an expansion for the MBS of the Mott gap in terms of the hopping
and the exchange coupling in our prototype model. We show how such an
expansion can be generalized for application to realistic Mott or charge-
transfer insulator materials. This allows estimating the MBS in real materials
in an extremely simple manner avoiding extensive theoretical calculations. The
approach is exemplarily applied to $\alpha$-MnTe, NiO, and BiFeO3 as
attractive compounds for antiferromagnetic spintronics application [2]. We
find an MBS of about $130$ meV for $\alpha$-MnTe which is in an overall very
good agreement with the previous theoretical calculations and experimental
data. For NiO and BiFeO3 we obtain an MBS of about $360$ and $157$ meV,
respectively. Our results for NiO also match the available experimental data.
For BiFeO3 there are currently controversial results in the literature on the
direct or the indirect nature of the band gap and its temperature dependence.
A systematic study of the MBS in BiFeO3 and its comparison with our
theoretical prediction can unveil the direct or the indirect nature of the
band gap in this multiferroic compound.

We emphasize that the formula (22) is intended to establish a quick, ready-to-
use estimate for the MBS. Our results paves the path for identifying materials
with a strong spin-charge coupling which is a prerequisite to go beyond the
pure magnetic state in Refs. [7, 8, 9] and to realize a coupled spin-charge
coherent dynamics at a femtosecond time scale.

###### Acknowledgements.

This study was funded by the German Research Foundation (DFG) in the
International Collaborative Research Centre TRR 160 (Project B8).

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##  Appendix A Spectral function for $n_{b}=4$

Figure 12: The spectral function $A(\omega)$ of the Hubbard-Kondo-Heisenberg
model (3) plotted vs frequency $\omega$ in the magnetic insulator (MI) phase
(a) and in the paramagnetic insulator (PI) phase (b) for the localized spin
$S=2$ and the number of bath sites $n_{b}=4$ in the ED impurity solver. The
Hubbard interaction is fixed to $U=32t$ and the Hund coupling to $J_{\rm
H}=0.3U/S$, which result in the bare charge gap $\Delta=1.6U$. The spectral
functions for the different temperatures $T$ are shifted vertically for
clarity. The Néel temperature is given by $T_{\rm N}\approx 1.37t$. The arrows
specify the peaks from which the Mott gap in the MI phase $G_{\rm MI}(T)$ and
in the PI phase $G_{\rm PI}(T)$ are read off. Figure 13: The same as Fig. 12
but for the localized spin $S=1$ with the Néel temperature $T_{\rm N}\approx
0.59t$.

We plot the spectral function $A(\omega)$ vs frequency at different
temperatures $T$ in the MI phase (a) and in the PI phase (b) in Fig. 12 for
the localized spin $S=2$ and in Fig. 13 for the localized spin $S=1$. The
results are shown for the bare charge gap $\Delta=U+2SJ_{\rm H}=1.6U$ for
$U=32t$ and the number of bath sites $n_{b}=4$. Similar to the results for the
number of bath sites $n_{b}=6$ in Figs. 3 and 4 discussed in the main text, we
find that the spectral function in the PI phase remains unchanged upon
reducing the temperature, but in the MI phase there is a shift of the Mott gap
specified by arrows to higher energies as the temperature is reduced from the
Néel temperature $T_{\rm N}$ to zero. We also find a similar magnetic
redistribution of spectral weight as observed in the main text for $n_{b}=6$,
i.e., there is a larger spectral weight near the Fermi energy in the MI phase
at $T\to 0$ in contrast to the PI phase.