# Dynamical Mean-Field Theory on the Real-Frequency Axis: p-d Hybridizations
and Atomic Physics in SrMnO3

Daniel Bauernfeind<EMAIL_ADDRESS>Institute of Theoretical and
Computational Physics
Graz University of Technology, 8010 Graz, Austria  Robert Triebl  Institute of
Theoretical and Computational Physics
Graz University of Technology, 8010 Graz, Austria  Manuel Zingl  Institute of
Theoretical and Computational Physics
Graz University of Technology, 8010 Graz, Austria  Center for Computational
Quantum Physics, Flatiron Institute
New York, New York 10010, USA  Markus Aichhorn  Institute of Theoretical and
Computational Physics
Graz University of Technology, 8010 Graz, Austria  Hans Gerd Evertz  Institute
of Theoretical and Computational Physics
Graz University of Technology, 8010 Graz, Austria  Kavli Institute for
Theoretical Physics
University of California, Santa Barbara, CA 93106, USA

###### Abstract

We investigate the electronic structure of SrMnO3 with Density Functional
Theory (DFT) plus Dynamical Mean-Field Theory (DMFT). Within this scheme the
selection of the correlated subspace and the construction of the corresponding
Wannier functions is a crucial step. Due to the crystal field splitting of the
Mn-$3d$ orbitals and their separation from the O-2$p$ bands, SrMnO3 is a
material where on first sight a 3-band $d$-only model should be sufficient.
However, in the present work we demonstrate that the resulting spectrum is
considerably influenced by the number of correlated orbitals and the number of
bands included in the Wannier function construction. For example, in a
$d$-$dp$ model we observe a splitting of the $t_{2g}$ lower Hubbard band into
a more complex spectral structure, not observable in $d$-only models. To
illustrate these high-frequency differences we employ the recently developed
Fork Tensor Product State (FTPS) impurity solver, as it provides the necessary
spectral resolution on the real-frequency axis. We find that the spectral
structure of a 5-band $d$-$dp$ model is in good agreement with PES and XAS
experiments. Our results demonstrate that the FTPS solver is capable of
performing full 5-band DMFT calculations directly on the real-frequency axis.

##  I Introduction

The combination of density functional theory (DFT) and dynamical mean-field
theory (DMFT) has become the work-horse method for the modeling of strongly-
correlated materials Anisimov  _et al._ (1997); Lechermann  _et al._ (2006);
Kotliar  _et al._ (2006). For DMFT, a (multi-orbital) Hubbard model is
constructed in a selected correlated subspace, which usually describes the
valence electrons of the transition metal orbitals in a material. An adequate
basis for these localized orbitals are projective Wannier functions Anisimov
_et al._ (2005); Aichhorn  _et al._ (2009). In contrast to the Bloch wave
functions, these functions are localized in real space, and therefore provide
a natural basis to include local interactions as they resemble atomic orbitals
and decay with increasing distance from the nuclei. However, the selection of
the correlated subspace itself and the Wannier function construction are not
uniquely defined.
In the present work, we use SrMnO3 to analyze the differences of some common
models. This perovskite is an insulator 111Although most published work
suggest that the compound is insulating, the experimental magnitude of the gap
ranges from approximately $1.0\text{\,}\mathrm{eV}$ to
$2.0\text{\,}\mathrm{eV}$, see citations in the main text. with a nominal
filling of three electrons in the Mn $3d$ shell. There are various works
concerning its electronic structure, both on the experimental Lee and Iguchi
(1995); Abbate  _et al._ (1992); Chainani  _et al._ (1993); Kang  _et al._
(2008); Saitoh  _et al._ (1995); Kim  _et al._ (2010) as well as on the
theoretical side Dang  _et al._ (2014); Chen  _et al._ (2014); Søndenå  _et
al._ (2006); Mravlje  _et al._ (2012). For the construction of the correlated
subspace, we explicitly identify the following meaningful cases: The first is
a three orbital model for the $t_{2g}$ states only. For the second choice,
usually denoted as $d$-$dp$ model, the transition metal $3d$-states and the
oxygen $2p$-states are considered in the Wannier function construction, but
the Hubbard interaction is only applied to the $3d$-states. The correlated
subspace is then affected by the lower lying oxygen bands due to
hybridizations. In both cases, the full $3d$ manifold can be retained by
including the $e_{g}$ orbitals in genuine 5 orbital models.
To assess the consequences of the different low-energy models, a good
resolution of the spectral function on the real-frequency axis is beneficial.
Due to its exactness up to statistical noise, Continuous Time Quantum Monte
Carlo (CTQMC) is often used as a DMFT impurity solver Werner and Millis
(2006); Gull  _et al._ (2011); Werner  _et al._ (2006). However, when using a
CTQMC impurity solver, an analytic continuation is necessary, which results in
spectral functions with a severely limited resolution at higher frequencies
Bauernfeind  _et al._ (2017). This can make it difficult to judge the
influence of the choices made for the correlated subspace. In the present
paper, we therefore employ the real-frequency Fork Tensor Product States
(FTPS) solver Bauernfeind  _et al._ (2017). This recently developed zero
temperature impurity solver was previously applied to SrVO3, making it
possible to reveal an atomic multiplet structure in the upper Hubbard band
Bauernfeind  _et al._ (2017). This observation of a distinct multiplet
structure in a real-material calculation is an important affirmation of the
atom-centered view promoted by DMFT.
The present work also serves as a deeper investigation of the capabilities of
the FTPS solver. We show that the FTPS solver can be applied to $d$-$dp$
models, leading to new insight into the interplay of the atomic physics of the
transition metal impurity and hybridization effects with the oxygen atoms as a
natural extension to the atom-centered view. Furthermore, the physics of
SrMnO3 is different from SrVO3, since the manganate is an insulator, and thus
it constitutes a new challenge for the FTPS solver. While we presented a proof
of concept for FTPS on a simple 5-band model before Bauernfeind  _et al._
(2017), we now perform full 5-band real-frequency DFT+DMFT calculations for
both $d$-only and $d$-$dp$ models.
We find that the choices made for the correlated subspace strongly affect the
resulting spectral function and its physical interpretation. Additionally, we
show that the interplay of atomic and hybridization physics can already be
found in very simple toy models.
This paper is structured as follows. In section II we discuss the methods
employed, namely DFT, the different models obtained from different Wannier
constructions, DMFT, and the impurity solvers used. Section III focuses on the
results of the DMFT calculations and the underlying physics of these different
models. This knowledge will then be used in Sec. IV to compare the spectral
function to experiments by Kim _et al._ Kim  _et al._ (2010).

##  II METHOD

###  II.1 DFT and WANNIER BASIS

We start with the DFT density of states (DOS) from a non-spin-polarized DFT
calculation for SrMnO3 in the cubic phase (shown in Fig. 1). The calculation
was performed with Wien2k Blaha  _et al._ (2001), using 969 $k$-points in the
irreducible Brillouin zone and a lattice parameter of
$a=$3.768\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$. Around the Fermi energy
$E_{F}$, SrMnO3 has the characteristic steeple-like shaped DOS, stemming from
the Mn-$t_{2g}$ bands with a bit of O-$p_{x/y}$ contribution. Below
$-2.0\text{\,}\mathrm{eV}$, the DOS is mainly determined by oxygen bands which
also exhibit manganese hybridizations. With the exception of some additional
weight below $-5.0\text{\,}\mathrm{eV}$, the Mn-$e_{g}$ states lie mainly in
the energy range from $0.0\text{\,}\mathrm{eV}$ to $5.0\text{\,}\mathrm{eV}$.
In this work we use projective Wannier functions, where an energy interval has
to be chosen as a projection window Anisimov  _et al._ (2005); Aichhorn  _et
al._ (2009). The bands around $E_{F}$ have mainly $t_{2g}$ character,
suggesting a selection of only a narrow energy window for the Wannier function
construction ($-2.0\text{\,}\mathrm{eV}$ to $0.82\text{\,}\mathrm{eV}$). We
call this set of projective Wannier functions the 3-band $d$-only model.
However, the $t_{2g}$ orbitals also show a considerable hybridization with the
O-$2p$ states below $-2\text{\,}\mathrm{eV}$, and hence, one might want to
enlarge the projective energy window by setting its lower boundary to
$-10\text{\,}\mathrm{eV}$. We refer to this model as the 3-band $d$-$dp$
model.
At the same time, we realize that also the $e_{g}$ orbitals are not entirely
separated from the $t_{2g}$ orbitals in energy and that they have even some
weight around $E_{F}$ (see middle graph of Fig. 1). These states lie directly
above $E_{F}$ and therefore their influence on the resulting spectrum needs to
be checked. One should then use a window capturing 5 bands, the $e_{g}$ and
$t_{2g}$, as a correlated subspace (from $-2\text{\,}\mathrm{eV}$ to
$5\text{\,}\mathrm{eV}$). This is a 5-band $d$-only model. Note that empty
orbitals do not pose a problem for the FTPS solver. Like before, we can again
enlarge the energy window to include the oxygen hybridization
($-10\text{\,}\mathrm{eV}$ to $5\text{\,}\mathrm{eV}$). We denote this model
as the 5-band $d$-$dp$ model.
In total, we end up with 4 different choices. The settings for these 4 models
are summarized in Tab. 1. All of them are justified, have different
descriptive power, and have been employed in various DFT+DMFT calculations for
SrMnO3 Chen  _et al._ (2014); Dang  _et al._ (2014); Mravlje  _et al._ (2012).

Figure 1: _Top:_ Total DFT-DOS for SrMnO3. _Middle:_ partial Mn-3$d$ DOS. _Bottom:_ partial O-2$p$ DOS for SrMnO3. Below approximately $-1.5\text{\,}\mathrm{eV}$, the band structure consists of oxygen bands that have mostly $p$-character but also have some $e_{g}$ and $t_{2g}$ weight due to hybridizations. The $t_{2g}$ bands are located around the Fermi energy from $-1.5\text{\,}\mathrm{eV}$ to about $0.5\text{\,}\mathrm{eV}$, which have small $p$-character. Directly above the Fermi energy and partly overlapping with the $t_{2g}$ bands we find the $e_{g}$ bands that have small $p$-contributions as well.  Table 1: Summary of models with their projective energy windows and the parameters used in the FTPS solver: number of bath sites $N_{B}$, Fourier transform broadening $\eta_{FT}$, truncated weight $t_{w}$ and maximal bond dimension allowed for the links between impurities as well as for the links between an impurity and the first bath tensor. We keep at most this number of states and increase $t_{w}$ if needed. The number in brackets is the maximal bond dimension during ground state search, while the first number is used for the time evolution. The bath links were not restricted to any maximal bond dimension. The FTPS time evolution is performed up to $t_{max}$, given in eV-1.  Model | Window ($eV$) | Comments | $\mathbf{N_{B}}$ | $\bm{\eta_{FT}}$ | $\mathbf{t_{w}}$ | Bond dim. | $\bm{t_{max}}$
---|---|---|---|---|---|---|---
3-band $d$-only |  $-2.0$ \- $0.82$ | only major $t_{2g}$ weight around $E_{F}$ | 79 | 0.08 | $5\cdot 10^{-9}$ | - | 14.0
5-band $d$-only |  $-2.0$ \- $5.0$ | include $e_{g}$, neglect hybridizations | 49 | 0.15 | $1\cdot 10^{-8}$ | 200 (150) | 12.0
3-band $d$-$dp$-model |  $-10.0$ \- $5.0$ | include hybridized $t_{2g}$ weight on oxygen bands | 59 | 0.1 | $1\cdot 10^{-8}$ | 450 (150) | 14.0
5-band $d$-$dp$-model |  $-10.0$ \- $5.0$ |  $t_{2g}$ and $e_{g}$ bands with hybridizations | 49 | 0.2 | $1\cdot 10^{-8}$ | 200 (150) | 7.0

###  II.2 DMFT

Once the correlated subspace is defined, we use DMFT Georges  _et al._ (1996);
Metzner and Vollhardt (1989); Lechermann  _et al._ (2006); Kotliar  _et al._
(2006) to solve the resulting multi-band Hubbard model. As interaction term we
choose the 5/3-band Kanamori Hamiltonian Kanamori (1963)222Even for the five-
band calculations we choose the Kanamori Hamiltonian over the Slater
Hamiltonian Slater (1960), because the large number of interaction terms in
the latter make a treatment with the FTPS solver more involved.. Within DMFT,
the lattice problem is mapped self consistently onto an Anderson impurity
model (AIM) with the Hamiltonian

$\displaystyle H$ $\displaystyle=H_{\text{loc}}+H_{\text{bath}}$ (1)
$\displaystyle H_{\text{loc}}$
$\displaystyle=\sum_{m\sigma}\epsilon_{m0}n_{m0\sigma}+H_{\text{DD}}+H_{\text{SF-
PH}}$ $\displaystyle H_{\text{DD}}$
$\displaystyle=U\sum_{m}n_{m0\uparrow}n_{m0\downarrow}$
$\displaystyle+(U-2J)\sum_{m^{\prime}>m,\sigma}n_{m0\sigma}n_{m^{\prime}0\bar{\sigma}}$
$\displaystyle+(U-3J)\sum_{m^{\prime}>m,\sigma}n_{m0\sigma}n_{m^{\prime}0\sigma}$
$\displaystyle H_{\text{SF-PH}}$
$\displaystyle=J\sum_{m^{\prime}>m}\left(c_{m0\uparrow}^{{\dagger}}c_{m0\downarrow}c_{m^{\prime}0\uparrow}c_{m^{\prime}0\downarrow}^{{\dagger}}+\text{h.c.}\right)$
$\displaystyle-J\sum_{m^{\prime}>m}\left(c_{m0\uparrow}^{{\dagger}}c_{m0\downarrow}^{{\dagger}}c_{m^{\prime}0\uparrow}c_{m^{\prime}0\downarrow}+\text{h.c.}\right)$
$\displaystyle H_{\text{bath}}$
$\displaystyle=\sum_{ml\sigma}\epsilon_{ml}n_{ml\sigma}+V_{ml}\left(c_{m0\sigma}^{{\dagger}}c_{ml\sigma}+\text{h.c.}\right).$

Here, $c_{ml\sigma}^{{\dagger}}$ ($c_{ml\sigma}$) creates (annihilates) an
electron in orbital $m$, with spin $\sigma$ at site $l$ (site zero is the
impurity). $n_{ml\sigma}$ are the corresponding particle number operators.
$\epsilon_{m0}$ is the orbital dependent on-site energy of the impurity and
$\epsilon_{ml}$ as well as $V_{ml}$ are the bath on-site energies and the
bath-impurity hybridizations, respectively.
The interaction part of Hamiltonian (1), $H_{\text{DD}}+H_{\text{SF-PH}}$, is
parametrized by a repulsive interaction $U$ and the Hund’s coupling $J$. For
each of the models presented in Tab. 1, we choose these parameters _ad hoc_ in
order to obtain qualitatively reasonable results. In addition, for the full
5-band $d$-$dp$ model we also estimate them quantitatively via a comparison to
an experiment.
Within DFT+DMFT, a so-called double counting (DC) correction is necessary,
because part of the electronic correlations are already accounted for by DFT.
For general cases, exact expressions for the DC are not known, although there
exist several approximations Haule (2015); Park  _et al._ (2014); Karolak  _et
al._ (2010); Haule  _et al._ (2010). In the present work we use the fully-
localized-limit (FLL) DC (Eq.(45) in Ref. Held, 2007). When needed, we adjust
it to account for deviations from the true, unknown DC. Note that in the
$d$-only models, the DC is a trivial energy shift that can be absorbed into
the chemical potential Karolak  _et al._ (2010), which is already adjusted to
obtain the correct number of electrons in the Brillouin zone. This step, as
well as all other interfacing between DFT and DMFT, is performed using the
TRIQS/DFTTools package (v1.4) Parcollet  _et al._ (2015); Aichhorn  _et al._
(2016, 2009, 2011).

###  II.3 CTQMC + MaxEnt

We compare some of our results to CTQMC data at an inverse temperature of
$\beta=$40\text{\,}\mathrm{eV}^{-1}$$ obtained with the TRIQS/CTHYB solver
(v1.4) Seth  _et al._ (2016); Werner and Millis (2006). We calculate real-
frequency spectra with an analytic continuation using the freely available
$\Omega$-MaxEnt implementation of the Maximum Entropy (MaxEnt) method Bergeron
and Tremblay (2016). However, the analytic continuation fails to reproduce
high-energy structure in the spectral function, as we have shown in Ref.
Bauernfeind  _et al._ (2017) on the example of SrVO3. This is especially true
when the imaginary-time Green’s function is subject to statistical noise,
which is inherent to Monte Carlo methods.
There are in general two quantities for which one can perform the analytic
continuation. First, one can directly calculate the real-frequency impurity
Green’s function from its imaginary time counterpart (as done in Fig. 7).
Second, one can perform the continuation on the level of the impurity self-
energy Mravlje and Georges (2016) and then calculate the local Green’s
function of the lattice model (as done in Fig. 8). In the latter case, the DFT
band-structure enters on the real-frequency axis, which increases the
resolution of the spectral function.

###  II.4 FTPS

For all models studied we employ FTPS Bauernfeind  _et al._ (2017). This
recently developed impurity solver uses a tensor network geometry which is
especially suited for AIMs. The first step of this temperature $T=0$ method is
to find the absolute ground state including all particle number sectors with
DMRG White (1992). Then the interacting impurity Green’s function is
calculated by real-time evolution. Since entanglement growth during time
evolution prohibits access to arbitrary long times Schollwöck (2011), we
calculate the Green’s function up to some finite time (see Tab. 1) and predict
the time series using the linear prediction method White and Affleck (2008);
Bauernfeind  _et al._ (2017) up to times $\mathcal{O}(100eV^{-1})$. The linear
prediction could potentially produce artifacts in the spectrum, and therefore
we always make sure that every spectral feature discussed in this work is
already present in the finite-time Green’s function without linear prediction.
The main approximations that influence the result of the FTPS solver are the
broadening $\eta_{FT}$ used in the Fourier transform 333We Fourier transform
with a kernel $e^{i\omega t-\eta_{FT}|t|}$ , and the truncation of the tensor
network Bauernfeind  _et al._ (2017). The former corresponds to a convolution
with a Lorentzian in frequency space making its influence predictable, while
the truncation can be controlled by including more states. This control over
the approximations allows us to analyze spectral functions in greater detail
than what would be possible with CTQMC+MaxEnt. The parameter values for our
FTPS calculations are listed in Tab. 1.
Note that we choose $\eta_{FT}$ larger than in our previous work Bauernfeind
_et al._ (2017). The reason for doing this is two-fold: First, some of the
calculations we show in this work have a large bandwidth, which lowers the
energy resolution if we keep the number of bath sites fixed. Second, FTPS uses
a discretized bath to represent the continuous non-interacting lattice Green’s
function $G_{0}^{\text{cont}}$. When calculating the self energy
$\Sigma=G_{0}^{-1}-G^{-1}$, we can either use the discretized version of
$G_{0}^{\text{discr}}$ or the continuous one, $G_{0}^{\text{cont}}$. In this
work we choose $G_{0}^{\text{discr}}$, which is formally the correct choice.
This then requires to use a larger broadening to obtain causal self-energies
that do not show finite discretization effects from inverting
$G_{0}^{\text{discr}}$. However, when calculating the final impurity spectral
function shown in all figures, we employ a very small broadening of
$\eta_{FT}=$0.01\text{\,}\mathrm{eV}$$ in order to obtain optimal resolution.
The real-frequency approach of FTPS allows to resolve spectral features with
higher precision than CTQMC+MaxEnt. This is especially true for high energy
multiplets. On the other hand, with FTPS and real-time evolution it is
difficult to obtain perfect gaps, since the results are less precise at small
$\omega$, encoded in the long-time properties of the Green’s function which we
obtain only approximately using linear prediction White and Affleck (2008).
With FTPS we calculate the greater and lesser Green’s functions separately
Bauernfeind  _et al._ (2017). Since the greater (lesser) Green’s function has
no contribution at $\omega<0$ ($\omega>0$) we restricted the contributions of
the calculated Green’s functions in frequency space.

##  III RESULTS

Figure 2: 3-band $d$-only calculation: $t_{2g}$ correlated spectral function
for $U=$4.0\text{\,}\mathrm{eV}$$ and $J=$0.6\text{\,}\mathrm{eV}$$, as well
as $e_{g}$ DFT-DOS. The $t_{2g}$ spectrum shows a Mott insulator at half-
filling with pronounced lower and upper Hubbard bands. Figure 3: 5-band
$d$-only calculation: correlated spectral function of the $e_{g}$ and the
$t_{2g}$ orbitals for $U=$ $4.0\text{\,}\mathrm{eV}$ and $J=$
$0.6\text{\,}\mathrm{eV}$.

###  III.1 _d_ -only models

First we focus on $d$-only calculations using a projective energy window with
a lower energy boundary of $-2.0\text{\,}\mathrm{eV}$ for the Wannier-function
construction, neglecting the occupied Mn-$3d$ weight at lower energies (see
Tab. 1 and middle graph of Fig. 1). With this choice of the correlated
subspace, the occupation of the $e_{g}$ orbitals is nearly zero and the three
degenerate $t_{2g}$ orbitals are half-filled.

#### 3-band calculation

Considering only the $t_{2g}$ subspace, the resulting impurity spectral
function (Fig. 2) is gapped for the chosen interaction values. The peaks of
the lower and upper Hubbard bands are separated by $5.0\text{\,}\mathrm{eV}$
in energy, which is roughly $U+2J=$5.2\text{\,}\mathrm{eV}$$, as expected from
atomic physics Georges  _et al._ (2013).
Contrary to SrVO3, where a distinct 3-peak multiplet structure in the upper
Hubbard band is present Bauernfeind  _et al._ (2017), both SrMnO3 Hubbard
bands show only one dominant peak. The structure observed in SrVO3 was well
explained by the atomic multiplets of the interaction Hamiltonian
$H_{\text{loc}}$ in Eq. 1 for a ground state with one electron occupying the
$t_{2g}$ orbitals. The absence of such an atomic multiplet structure in this
model for SrMnO3 can be understood in a similar way: The large Coulomb
repulsion in combination with Hund’s rules (due to the density-density
interaction strengths $U$, $U-2J$ and $U-3J$) lead to a ground state
$\ket{\psi_{0}}$ which consists mostly of the states
$\ket{\uparrow,\uparrow,\uparrow}$ and
$\ket{\downarrow,\downarrow,\downarrow}$ on the impurity. Adding a particle,
when calculating the Green’s function, produces a single double occupation,
e.g.,
$c_{1,\downarrow}^{{\dagger}}\ket{\psi_{0}}=\ket{\uparrow\downarrow,\uparrow,\uparrow}$.
This state is an eigenstate of the atomic Hamiltonian, because it is trivially
an eigenstate of $H_{\text{DD}}$, and both the spin-flip and pair-hopping
terms annihilate this state. Hence, all $t_{2g}$ single-particle excitations
from the ground state have the same energy, and as a consequence, only _one_
atomic excitation energy is observed.
Although not included in the low-energy model, the uncorrelated states still
need to be taken into account for the single-particle gap of SrMnO3. On the
unoccupied side, the onset of the $e_{g}$ orbitals leads to a reduction of the
single-particle gap to about half the size of the $t_{2g}$ gap (see Fig. 2).
On the occupied side, depending on $U$ and $J$, either the lower Hubbard band
or the O-bands (at about $-1.5\text{\,}\mathrm{eV}$) determine the gap size,
and thus also the type of the insulating state (Mott or charge transfer
insulator Zaanen  _et al._ (1985)). For SrMnO3 to be clearly classified as
Mott insulator, $U+2J<$3.0\text{\,}\mathrm{eV}$$ would be required. However,
it is questionable if the $d$-only picture is correct, as in this case the
lower Hubbard band is not influenced by the $t_{2g}$/O-2$p$ hybridizations
between $-6.0\text{\,}\mathrm{eV}$ and $-2.0\text{\,}\mathrm{eV}$ (see Fig 1).
We will discuss the effect of these hybridizations in detail in Sec. III.2 and
Sec. III.3.

#### 5-band calculation

Next, we add the $e_{g}$ orbitals to the correlated subspace, which now
comprises the full Mn-$3d$ manifold. The resulting impurity spectral functions
of the $e_{g}$ and $t_{2g}$ orbitals are shown in Fig. 3. The $t_{2g}$
spectral weight does not change much compared to the 3-band calculation. This
is to be expected, because the $e_{g}$ orbitals remain nearly empty during the
calculation of the $t_{2g}$ Green’s function.
The $e_{g}$ spectral function, on the other hand, becomes much broader in
comparison to the DFT-DOS, showing spectral weight above
$4.5\text{\,}\mathrm{eV}$. The unoccupied part of the spectrum is encoded in
the greater Green’s function, i.e., adding a particle in an $e_{g}$ orbital to
the ground state. If we again assume
$\ket{\psi_{0}}\propto\ket{\uparrow,\uparrow,\uparrow}+\ket{\downarrow,\downarrow,\downarrow}$
as the $t_{2g}$ ground state, we can add a particle to the $e_{g}$ orbitals
either in a high-spin or low-spin configuration:

$\displaystyle
c_{e_{g}\uparrow}^{\dagger}\ket{\psi_{0}}\propto\underbrace{\ket{\uparrow,\uparrow,\uparrow}}_{t_{2g}}\otimes\underbrace{\ket{\uparrow,0}}_{e_{g}}+\ket{\downarrow,\downarrow,\downarrow}\otimes\ket{\uparrow,0}.$
(2)

Using the Kanamori Hamiltonian, the high-spin configuration (first term in Eq.
2) generates a single atomic excitation energy, while the low-spin
configuration (second term in Eq. 2) leads to two energies (due to the spin-
flip terms). According to this atomistic picture, the splitting of the $e_{g}$
peaks is proportional to Hund’s coupling $J$ (see Fig. 3). Their position
relative to the upper $t_{2g}$ Hubbard band is influenced by the crystal field
splitting and $J$. From this clear atomic-like structure we see that even
empty orbitals need to be included in the correlated subspace because of
correlation effects with other occupied orbitals.

###  III.2 3-band _d_ -_dp_ model

Figure 4:  3-band _d-dp_ model: spectral functions for different $J$ (top) and
different $U$ (bottom). All interaction parameters are given in eV. Upon
increasing both parameters the gap increases. Changing $J$ shifts the peak at
around $-6.0\text{\,}\mathrm{eV}$, while changing $U$ only shifts the one at
$-8.0\text{\,}\mathrm{eV}$.

In the energy region where the lower Hubbard band is located, we also find
$t_{2g}$ weight stemming from the Mn-$3d$/O-$2p$ hybridization (see middle
plot of Fig. 1). This suggests that those states should be included in the
construction of the projective Wannier functions, i.e., a $d$-$dp$ model. In
the following we will use the term High Energy Spectral Weight (HESW) to
denote the Wannier function weight on the oxygen bands (located below
$-1.5\text{\,}\mathrm{eV}$). The first and most obvious consequence of a
larger projective energy window is an increased bandwidth of the Wannier DOS.
To obtain a similar insulating behavior as in the $d$-only model we increase
$U$ and $J$. Secondly, now also the DC correction has a non-trivial effect,
since it shifts the correlated $t_{2g}$ states relative to the oxygen bands.
The $t_{2g}$ weight on the oxygen bands is rather small, which means that the
effect of the DC correction on the HESW is equally low. Thirdly, in the 3-band
$d$-$dp$ model the impurity occupation grows (the exact value depending on $U$
and $J$), changing the character of the ground state to a mix of states with
mainly three and four particles on the impurity, while in the 3-band $d$-only
calculation the occupation of the impurity was three electrons. Due to the
increased complexity of the ground state, we expect a richer dependence of the
spectrum on the interaction parameters $U$ and $J$.

Figure 5:  3-band _d-dp_ model: effect of the DC correction. _Top:_ FLL DC,
_Bottom:_ FLL DC with an additional shift of $-0.5\text{\,}\mathrm{eV}$. $U=$
$5.0\text{\,}\mathrm{eV}$ and $J=$ $0.7\text{\,}\mathrm{eV}$ are used as
interaction parameters. Contrary to all other figures, in this plot we show
the spectrum of the correlated local Green’s function. The occupation of the
$t_{2g}$ orbitals changes from $0.55$ (No DC shift) to $0.54$ (DC shift
=$-0.5\text{\,}\mathrm{eV}$) indicating that the $t_{2g}$ occupation is not
much affected by the DC. The DFT-DOS (Fig. 1) already gives an occupation of
$0.54$.

In Fig. 4 we compare calculations for different values of $J$ (top) and
different values of $U$ (bottom). Overall, the spectral functions consist of a
(smaller) lower Hubbard band connected to states from the hybridized oxygen
bands and an upper steeple-like Hubbard band of similar shape as in the
$d$-only calculation. By comparing the two peaks at $-6.0\text{\,}\mathrm{eV}$
and $-8.0\text{\,}\mathrm{eV}$, we observe that they behave differently when
changing $U$ or $J$. While the former is only affected by $J$, the latter is
not, but shifts with $U$. The resolution of the structure in the lower-
Hubbard-band/HESW complex demonstrates the capabilities of the FTPS solver.
The $t_{2g}$ gap grows when increasing either $U$ or $J$, which is a typical
sign of Mott physics at half filling Georges  _et al._ (2013). Nevertheless,
in the $d$-$dp$ model the gap size increases slowly: when increasing $U$ by
$1.0\text{\,}\mathrm{eV}$, the gap only grows by about half of that.
Considering also the uncorrelated $e_{g}$ orbitals, we observe that the
single-particle gap is not much affected by the interaction values studied. An
artificial lowering of the DC correction by $-0.5\text{\,}\mathrm{eV}$, which
corresponds to a relative shift in energy between the correlated subspace and
the uncorrelated states, also increases the $t_{2g}$ gap (Fig. 5). This growth
of the gap is mostly due to a shift of the $t_{2g}$ upper Hubbard band, since
the chemical potential is pinned by the $e_{g}$ bands 444We found that the
$e_{g}$ bands have a very small occupation already in the DFT calculation.
This does not change during the DMFT calculations and therefore, the chemical
potential is always pinned at the onset of the $e_{g}$ spectral function.. The
first excitation below $E_{F}$ has a mix of $t_{2g}$ and O-$p$ character. This
indicates that in this model, SrMnO3 is not a pure Mott insulator, but a
mixture between Mott- and charge transfer insulator. This classification is
consistent with previous results Chainani  _et al._ (1993); Saitoh  _et al._
(1995); Abbate  _et al._ (1992); Dang  _et al._ (2014).

Figure 6: Spectrum of a one-band AIM with one interacting site coupled to a
single non-interacting site (Eq. 3). The spectrum is calculated with the
absolute ground state over all particle number sectors. The on-site energy
$\epsilon_{1}$ is shown as gray dashed-dotted vertical line. The gray dashed
lines visualize the evolution of the location of the three peaks as a function
of the interaction strength $U$. The colored dotted peaks show the atomic
spectrum with peaks at $-U/2$. The upper Hubbard band (additional peak at
$\omega>0$) is not shown. All spectra have been broadened by
$\eta_{FT}=$0.2\text{\,}\mathrm{eV}$$.

Let us employ a simple toy model to qualitatively understand this intermediate
regime. We use a correlated site coupled to only one non-interacting site:

$\displaystyle H=$ $\displaystyle U(n_{0,\uparrow}-0.5)(n_{0,\downarrow}-0.5)$
$\displaystyle+\sum_{\sigma}V_{1}(c_{0,\sigma}^{\dagger}c_{1,\sigma}+h.c.)+\epsilon_{1}n_{1,\sigma}$
(3)

The purpose of the non-interacting site is to mimic the effect of the HESW. We
set the on-site energy to $\epsilon_{1}=$-2.0\text{\,}\mathrm{eV}$$ and use a
coupling to the impurity of $V_{1}=$1.0\text{\,}\mathrm{eV}$$. Since we want
to understand the occupied part of the spectrum, we focus on negative energies
only. In Fig. 6 we show the resulting spectral functions ($\omega<0$) for
various values of the interaction strength $U$ (full lines). The atomic
excitation spectra of this model (corresponding to $V_{1}=0$), whose peaks are
positioned exactly at $-U/2$, are indicated by dotted lines. This toy model
shows three important features:
_First:_ The peak highest in energy (above $-2.0\text{\,}\mathrm{eV}$)
corresponds to the lower Hubbard band for small values of $U$ 555If we use a
larger bath energy $\epsilon_{1}$, for example
$\epsilon_{1}=$-5.0\text{\,}\mathrm{eV}$$, the position of the first peak of
the impurity spectrum is proportional to $U$ at small $U$, showing that it is
indeed a lower Hubbard band. . We see that it does not cross the on-site
energy $\epsilon_{1}$ with increasing $U$, but approaches it asymptotically.
The bath site _repels_ this level and upon increasing $U$ its weight
decreases.
_Second:_ The peak lowest in energy shows the opposite behavior. The
uncorrelated site repels it towards lower energies and the spectral weight
increases when we increase $U$. For large $U$ this level asymptotically
approaches the atomic limit at energy $-U/2$ and eventually becomes the lower
Hubbard band. These two peaks together form what one could call a split lower
Hubbard band.
_Third:_ The excitation at the on-site energy $\epsilon_{1}$ shifts to lower
energy and splits under the influence of $U$. Upon increasing $U$, one part
develops into the lower Hubbard band discussed above, and the other approaches
$\epsilon_{1}$ from below, with diminishing weight.
The DMFT spectral functions (Fig. 4) also show roughly a 3-peak structure,
where the peaks at about $-1.5\text{\,}\mathrm{eV}$
($-8.0\text{\,}\mathrm{eV}$) could be the first (last) peak of the split lower
Hubbard band of our toy model. The region in between then corresponds to the
small, middle peak in the toy model stemming from the HESW.
The repulsion of the first peak explains why increasing $U$ (Fig. 4 lower
graph) has only a relatively weak effect on the size of the gap. On the other
hand, effectively shifting the oxygen bands with the DC correction to lower
energies (Fig. 5) corresponds to shifting the bath energy $\epsilon_{1}$. This
means that the repulsion gets weaker, which explains the growth of the gap.
Furthermore, when increasing $U$ we find that the peak highest in energy gets
smaller, while spectral weight is transferred to the lowest energy peak, which
is also shifted to lower energies (Fig. 4). Additionally, a lowering of the DC
correction leads to an opposite behavior, where the first peak below $E_{F}$
grows at the expense of the lowest one in energy. Note that the middle region
of our DMFT spectrum shows a $J$-dependence (Fig. 4 top), which cannot be
explained by a one-orbital toy model. Using a similar toy model with two
orbitals and Kanamori interaction, we indeed observe a splitting proportional
to $J$ in the spectra (not shown here). Since the effect is small we will
refrain from discussing it in more depth.
We emphasize that the close relation between the toy model and the actual
impurity Green’s function of SrMnO3 in the $d$-$dp$ model suggests that the
HESW has the effect of splitting the lower Hubbard band into two bands; their
separation increases with the hybridization strength. Therefore, including the
oxygen states in the model strongly influences the size of the gap.

###  III.3 5-band _d_ -_dp_ model

Figure 7:  Comparison of the spectral functions for the 5-band _d-dp_ model
between FTPS and CTQMC+MaxEnt using the impurity Green’s function. _Top:_
$e_{g}$ orbitals. _Middle:_ $t_{2g}$ orbitals. For both calculations we use
$U=$ $6.0\text{\,}\mathrm{eV}$ and $J=$ $0.8\text{\,}\mathrm{eV}$. _Bottom:_
Combined spectral function.  Figure 8: Comparison of the $e_{g}$ spectral
functions of the local lattice Green’s function for the 5-band _d-dp_ model
between FTPS and CTQMC+MaxEnt, when the analytic continuation is performed for
the self-energy. To obtain the plot with FTPS we used $\eta_{FT}=0.25$. Using
the band structure on the real-frequency axis enhances the resolution of the
CTQMC result at energies where the imaginary part of the self-energy is small,
and increases its agreement with FTPS in the occupied part of the spectral
function.

From the DFT-DOS in Fig. 1, we see that the $e_{g}$ orbitals are actually not
empty. They possess additional spectral weight at around
$-7.0\text{\,}\mathrm{eV}$, stemming from hybridizations with the oxygen
bands. Similarly to the previous section where we included hybridizations of
$t_{2g}$ and O-$2p$, we now also include the hybridizations of $e_{g}$ and
O-$2p$.
As mentioned at the beginning, only approximations to the DC correction are
known. For the present 5-band calculation we find that using the FLL DC does
not produce a pronounced gap. This can be traced back to the additional
hybridizations of $e_{g}$ with O-$2p$ (see discussion below). Furthermore, the
FLL formula is based on five degenerate orbitals. In the case at hand we find
an approximately half filled $t_{2g}$ impurity ($\langle
n_{t_{2g}0\sigma}\rangle\approx 0.5$) and about one electron in total on the
$e_{g}$ part of the impurity ($\langle n_{e_{g}0\sigma}\rangle\approx 0.2$).
One therefore needs to adapt the DC correction to reproduce experimental
results. In order to obtain a pronounced gap, we decrease the FLL DC energy by
$2.0\text{\,}\mathrm{eV}$. Note that it has been argued that very often the
FLL-DC is too high Haule (2015). A reduction of the DC can also be
accomplished by adjusting $U$ in the FLL formula Dang  _et al._ (2014); Park
_et al._ (2014). While we find that the $t_{2g}$ occupation is not much
affected by the DC (similar to Fig. 5), its effect on the $e_{g}$ occupation
is much stronger. Without a DC-shift, we find $\langle
n_{e_{g}0\sigma}\rangle\approx 0.30$, while for a shift by
$-2.0\text{\,}\mathrm{eV}$ the occupation is $\langle
n_{e_{g}0\sigma}\rangle\approx 0.19$ compared to $\langle
n_{e_{g}0\sigma}\rangle\approx 0.28$ in the DFT calculation.
Fig. 7 shows the spectral function of the full 5-band $d$-$dp$ calculation
with adjusted DC as well as the respective spectral function obtained by a
DMFT calculation using CTQMC and performing the analytic continuation for the
impurity Green’s function. Overall, the FTPS spectrum is in good agreement
with the CTQMC+MaxEnt result. However, FTPS provides a much better energy
resolution at high energies, which is especially apparent from the pronounced
peak structure in the $e_{g}$ spectrum.
Instead of calculating the real-frequency spectrum of the impurity Green’s
function as in Fig. 7, one can use the analytic continuation for the self-
energy $\Sigma(\omega)$, and calculate the local Green’s function of the
lattice model directly for real frequencies. This way, the dispersion of the
DFT band structure enters on the real frequency axis directly, which increases
the resolution of the CTQMC result, but is suboptimal for FTPS 666To calculate
the self energy, we use the bath parameters to calculate the non-interacting
Green’s function. To avoid finite size effects, we need to use a larger
broadening $\eta_{FT}$.. The resulting spectral function for the $e_{g}$
orbitals is shown in Fig. 8. As expected, we find that some features shown by
the FTPS solver are now also present in CTQMC, but the $J$-multiplet in the
unoccupied part of the spectrum still cannot be resolved. To calculate the
FTPS self-energy, used to obtain the spectrum in Fig. 8, a broadening of
$\eta_{FT}=0.25$ was used, which explains the difference to Fig 7.
From these comparisons we also see that the sharp, step-like shape of the
$e_{g}$ spectrum at $E_{F}$ is not an artifact of the FTPS solver. We note
that for the 5-band calculation presented in Fig. 7, FTPS (720 CPU-h) and
CTQMC (600 CPU-h) need similar computational effort for one DMFT iteration
777CTQMC used $128\cdot 10^{6}$ measurements and the calculations were
performed on the same processors: Intel Xeon E5-2650v2, 2.6 GHz with 8 cores..
The unoccupied part of the total spectrum (sum of the $e_{g}$ and $t_{2g}$
spectra shown in the bottom plot of Fig. 7) consists of a three peak structure
with alternating $e_{g}$-$t_{2g}$-$e_{g}$ character, which is much more
pronounced than in the 5-band $d$-only calculation (Fig. 3). Compared to the
3-band $d$-$dp$ model we find differences mainly in the occupied part of the
$t_{2g}$ spectral function (Fig. 10). This is especially apparent in the
lowest peak, which seems to be shifted from $-9.0\text{\,}\mathrm{eV}$ to
$-13.0\text{\,}\mathrm{eV}$. Although this high energy excitation is small,
the FTPS solver can reliably resolve it.

Figure 9: Effect of the $e_{g}$ hybridizations on the $t_{2g}$ spectrum of the
toy model (Eq. 4). Parameters (in eV): $U=10.0$, $J=U/10$, $E_{t_{2g}}=-U/2$,
$E_{e_{g}}=-U/2+1.0$, $\epsilon_{t_{2g}}=\epsilon_{e_{g}}=-5.0$ and
$V_{t_{2g}}=1.5$. The grey dashed dotted line shows the bath energy levels.
All spectra have been broadened by $\eta_{FT}=$0.2\text{\,}\mathrm{eV}$$.

The differences in the position of this peak are again similar to the behavior
of a toy model. Here we use a two-orbital AIM with a single bath site for each
orbital:

$\displaystyle H=$ $\displaystyle
H_{\text{int}}+\sum_{m\in(t_{2g},e_{g})}E_{m}n_{0,m}+$
$\displaystyle\sum_{\sigma}V_{m}(c_{0,m,\sigma}^{\dagger}c_{1,m,\sigma}+h.c.)+\epsilon_{m}n_{1,m,\sigma}.$
(4)

For the interaction $H_{int}$ we choose the Kanamori Hamiltonian. As before,
we use a single bath site for each orbital to mimic the effect of the HESW. We
are interested in the influence of the hybridizations of $e_{g}$ and O-$2p$ on
the $t_{2g}$ spectral function. In Fig. 9, we compare the spectrum without
$e_{g}$-HESW states ($V_{e_{g}}=0$) with the one obtained from
$V_{e_{g}}=2V_{t_{2g}}$888In the full 5-band calculation, the $e_{g}$ bath
spectral function is much larger than the one for the $t_{2g}$ orbitals in the
energy region of the oxygen bands, which we mimic by a factor of 2 in
$V_{e_{g}}$.. Although one would expect the $e_{g}$ hybridization to only have
a minor influence on the $t_{2g}$ spectrum, we observe a rather surprising
behavior. The additional hybridization leads to a stronger repulsion of the
lowest energy peak from the bath energy, qualitatively explaining the shift
from $-9.0\text{\,}\mathrm{eV}$ to $-13.0\text{\,}\mathrm{eV}$ in Fig. 10.
Additionally, this toy model provides an explanation for the necessary
adjustment of the DC correction in the 5-band calculation: The peak highest in
energy in Fig. 9 is repelled more strongly with the additional $e_{g}$
hybridizations, therefore the gap decreases. If we would want to obtain a
similar $t_{2g}$ gap as with $V_{e_{g}}=0$, the interaction in the toy model
would need to be increased to $U\approx$20\text{\,}\mathrm{eV}$$ (keeping
$J=U/10$). Since this is unphysical, the only other option is to shift the
bath site energies of the toy model. In the DMFT calculation this corresponds
to a shift in the DC correction, effectively shifting the HESW to lower
energies. This behavior can be observed in Fig. 10, where we compare the
spectra of the 3- and 5-band $d$-$dp$ models. The onset of the lower-Hubbard-
band/HESW complex is exactly at the same position in both spectra, although
the DC shift differs by $2.0\text{\,}\mathrm{eV}$.

Figure 10: Comparison of the $t_{2g}$ spectral functions of the 3-band
$d$-$dp$ and 5-band $d$-$dp$ calculations at $U=$6.0\text{\,}\mathrm{eV}$$ and
$J=$0.8\text{\,}\mathrm{eV}$$, taken from Figs. 4 and 7. In the 5-band
calculation we shifted the double counting by $-2.0\text{\,}\mathrm{eV}$ to
increase the gap. The influence of the number of bands is most apparent in the
high-energy features. The increased repulsion of the first peak of the lower-
Hubbard-band/oxygen complex (Fig. 9) makes a shift in the DC necessary, if the
single particle gap should remain the same. Figure 11: Photo Emission
Spectroscopy (PES) and X-ray Absorption Spectroscopy (XAS) compared to the
5-band $d$-$dp$ DMFT-FTPS results ($U=$5.0\text{\,}\mathrm{eV}$$ and
$J=$0.6\text{\,}\mathrm{eV}$$). The experimental curves are reproduced from
Ref. Kim  _et al._ , 2010, Fig. 5. We normalized the experimental curves to
$\int_{-9}^{0}A_{\text{PES}}d\omega=\int_{-9}^{0}A_{\text{FTPS}}d\omega$ and
$\int_{0}^{6}A_{\text{XAS}}d\omega=\int_{0}^{6}A_{\text{FTPS}}d\omega$. FTPS
as well as the experiments show a 3-peak structure of alternating $e_{g}$\-
$t_{2g}$\- $e_{g}$ character in the unoccupied part of the spectrum (indicated
by arrows). For the arrow labels we adopted the notation of Ref. Kim  _et al._
, 2010, where $e_{g}^{\uparrow}$ means an excitation into the $e_{g}$ spectrum
with majority spin, while $t_{2g}^{\downarrow}$ and $e_{g}^{\downarrow}$ are
excitations into the $t_{2g}$ and $e_{g}$ spectrum with minority spin (see
also Eq. 2).

##  IV Comparison to experiment

Equipped with a good understanding of the model-dependent effects on the
spectral function, we are finally in a position to compare our results to
experiments. Several studies concluded that the unoccupied part of the
spectrum consists of three peaks with alternating $e_{g}$\- $t_{2g}$\- $e_{g}$
character Kang  _et al._ (2008); Saitoh  _et al._ (1995); Kim  _et al._
(2010). As we have shown, with DMFT+FTPS we are able to resolve such a
structure when including the $e_{g}$ states as correlated orbitals in a
genuine 5-band model. Additionally, we need to choose the energy window, i.e.,
whether the HESW should be included in the construction of the projective
Wannier functions. The nature of the insulating state (Mott or charge
transfer) has been debated in the literature Chainani  _et al._ (1993); Saitoh
_et al._ (1995); Abbate  _et al._ (1992); Dang  _et al._ (2014), but it is
likely that SrMnO3 falls in an intermediate regime where a clear distinction
is difficult. In the present work we have come to the same conclusion. This
implies that the lower Hubbard band and the O-$2p$ bands are not separated in
energy, which favors the use of a $d$-$dp$ model. We therefore conclude that a
5-band $d$-$dp$ model is necessary to fully capture the low-energy physics of
SrMnO3.
Having decided on the model for the correlated subspace, we still need to
determine the interaction parameters $U$ and $J$ as well as the DC. To do so
we use PES and XAS data for the Mn-$3d$ orbitals obtained by Kim _et al._ Kim
_et al._ (2010) and compare to our total impurity spectrum
($6A_{t_{2g}}\left(\omega\right)+4A_{e_{g}}\left(\omega\right)$ from Fig. 7).
According to Ref. Kim  _et al._ , 2010, the XAS (PES) spectrum can be
considered to represent the unoccupied (occupied) Mn-$3d$ spectrum. In the
measured spectrum the chemical potential is in the middle of the gap. In all
our calculations, the chemical potential is determined by the onset of the
unoccupied $e_{g}$ spectrum. However, the absolute position in energy is not
exactly known in XAS Kang (2017). Our calculation is in good agreement with
the experiment when we use a rigid shift of the XAS spectrum by
$0.8\text{\,}\mathrm{eV}$ to lower energies. Additionally, we deduce from the
peak positions in the experiment that the interaction parameters used for the
calculations presented in Fig. 7 are too high. The separation of the two
$e_{g}$ peaks ($\sim J$) and also the relative position of the $t_{2g}$ upper
Hubbard band is different than in the experiment. Therefore, we decrease the
interaction parameters to $U=$5.0\text{\,}\mathrm{eV}$$ and
$J=$0.6\text{\,}\mathrm{eV}$$ but keep the static shift of the FLL DC by
$-2.0\text{\,}\mathrm{eV}$. Note that these parameters are similar to the ones
used in other DFT+DMFT studies on SrMnO3 Dang  _et al._ (2014); Chen  _et al._
(2014).
The resulting spectral function for the new set of parameters is compared to
the experimental spectrum in Fig. 11. Notably, the bandwidths of both, the
unoccupied and the occupied spectrum, agree very well with the experiment. The
unoccupied part of the experimental spectrum (XAS) shows that the first
$e_{g}$ peak is just a shoulder of the $t_{2g}$ upper Hubbard band, and that
the separation of the two $e_{g}$ peaks is about $3.2\text{\,}\mathrm{eV}$,
which is in agreement with our result. Since this separation is proportional
to the Hund’s coupling, we conclude that $J\approx$ $0.6\text{\,}\mathrm{eV}$
for this compound. The $t_{2g}$ upper Hubbard band at
$2.0\text{\,}\mathrm{eV}$ is still slightly too high in energy.
The experiment also shows a lower-Hubbard-band/oxygen complex with two main
peaks at about $-6.0\text{\,}\mathrm{eV}$ and $-2.0\text{\,}\mathrm{eV}$. As
discussed in the previous sections (bottom plot of Fig. 7), our results
identify the first peak at $-2.0\text{\,}\mathrm{eV}$ to have mainly $t_{2g}$
character and to correspond to the largest part of the split lower Hubbard
band, whereas the second peak at $-6.0\text{\,}\mathrm{eV}$ has both $e_{g}$
and $t_{2g}$ character and stems from the hybridizations with the oxygen
bands. We note that the region between these two peaks has larger spectral
weight in the experiment than in our calculations. Importantly, no prominent
spectral features are observed in the experiment around
$-8.0\text{\,}\mathrm{eV}$, strengthening our conclusion that the 3-band
$d$-$dp$ model is not sufficient to describe the experiment (see also Fig.
10).

##  V Conclusions

We have studied the influence of the choice of the correlated subspace, i.e.
the number of bands and the energy window, on the DFT+DMFT result for the
strongly correlated compound SrMnO3. For $d$-only models (neglecting $p$-$d$
hybridizations), we have shown that the empty $e_{g}$ orbitals should be
included in the correlated subspace because interactions with the half-filled
$t_{2g}$ bands affect the spectrum, leading to a multiplet structure and a
broadening of the $e_{g}$ DFT-DOS. Including the Mn-$3d$/O-$2p$ hybridizations
in a 3-band model for the $t_{2g}$ bands only, i.e., the 3-band $d$-$dp$
model, we found a situation similar to avoided crossing, which leads to an
interesting interplay of atomic physics (lower Hubbard band) and Mn-$d$/O-$p$
hybridizations. In SrMnO3, the lower Hubbard band hybridizes with the $t_{2g}$
Wannier-weight on the oxygen bands, giving rise to a spectrum that can be
approximated by three peaks. This result provides new perspectives on an
intermediate regime, where both Mott and charge transfer physics are found. By
performing a 5-band calculation including the $p$-$d$ hybridization, we
investigated the effect of the $e_{g}$ hybridization on the $t_{2g}$ spectrum.
The splitting due to avoided crossing is heavily increased, which strongly
affects the 3-peak structure and also decreases the gap. Equipped with a good
understanding of the different correlated subspaces and the effects of the
model parameters ($U$, $J$, DC) we were able to obtain a spectral function in
good agreement with experimental data. We conclude that the choice of a
suitable model for the correlated subspace is important, since the inclusion
of both the O-$2p$ hybridizations and the $e_{g}$ states is essential for a
correct description of the observed spectral function in SrMnO3.
Finally, we would also like to stress that we have shown that FPTS is a viable
real-time impurity solver for real material calculations with five bands.

## Acknowledgments

The authors acknowledge financial support by the Austrian Science Fund (FWF)
through SFB ViCoM F41 (P04 and P03), through project P26220, and through the
START program Y746, as well as by NAWI-Graz. This research was supported in
part by the National Science Foundation under Grant No. NSF PHY-1125915. We
thank J.-S. Kang for the permission to reproduce data and for helpful
discussions. We are grateful for stimulating discussions with J. Mravlje, F.
Maislinger and G. Kraberger. The computational resources have been provided by
the Vienna Scientific Cluster (VSC). All calculations involving tensor
networks were performed using the ITensor library ITe .

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#  Masser-Wüstholz bound for reducibility of Galois representations for
Drinfeld modules of arbitrary rank

Chien-Hua Chen Electronic address<EMAIL_ADDRESS>ORCID:
0000-0003-3267-5603 ; Corresponding author Mathematics Division,
National Center for Theoretical Sciences,
Taipei, Taiwan

###### Abstract

In this paper, we give an explicit bound on the irreducibility of
mod-$\mathfrak{l}$ Galois representation for Drinfeld modules of arbitrary
rank without complex multiplication. This is a function field analogue of
Masser-Wüstholz bound on irreducibility of mod-$\ell$ Galois representation
for elliptic curves over number field.

##  1 Introduction

In 1993, Masser and Wüstholz [MW93b] proved a famous result on existence of
isogeny, with degree bounded by an explicit formula, between two isogenous
Elliptic curves. Later on, they [MW93a] applied such an isogeny estimation to
give an explicit bound on the irreducibility of mod-$\ell$ Galois
representation associated to elliptic curves over number field without complex
multiplication (CM). This bound is then used to deduce a bound on the
surjectivity of mod-$\ell$ Galois representation for elliptic curves over
number field without CM.

As a function field analogue of the theory in elliptic curves, David and
Dennis [DD99] gave an isogeny estimation for Anderson $t$-modules. In
particular, they deduced an isogeny estimation for Drinfeld
$\mathbb{F}_{q}[T]$-modules over a global function field, see Theorem 2.13 for
more details. Thus it is natural to ask whether one can apply the same
strategy as Messer-Wüstholz to deduce a bound on irreducibility of
mod-$\mathfrak{l}$ Galois representation for rank-$r$ Drinfeld modules without
CM. However, the Messer-Wüstholz strategy can not be applied directly to the
context of Drinfeld modules. The main resean is when one computes the degree
of isogeny between Drinfeld modules, the degree is always a power of $q$,
which is not a prime number. Thus the computational trick in Lemma 3.1 of
[MW93a] does not work for Drinfeld modules.

However, the idea from Messer-Wüstholz inspired us to produce a similar
method. Combining with the estimation on heights between isogenous Drinfeld
modules given by Breuer, Pazuki, and Razafinjatovo (see Theorem 2.14), we can
deduce our main result for an explicit bound on irreducibility of
mod-$\mathfrak{l}$ Galois representation for Drinfeld modules of arbitrary
rank and without CM:

######  Theorem 1.1.

Let $q=p^{e}$ be a prime power, $A:=\mathbb{F}_{q}[T]$, and $K$ be a finite
extension of $F:=\mathbb{F}_{q}(T)$ of degree $d$. Let $\phi$ be a rank-$r$
Drinfeld $A$-module over $K$ of generic characteristic and assume that ${\rm
End}_{\bar{K}}(\phi)=A$. Let $\mathfrak{l}=(\ell)$ be a prime ideal of $A$,
consider the mod-$\mathfrak{l}$ Galois representation

$\bar{\rho}_{\phi,\mathfrak{l}}:\mathrm{Gal}(\bar{K}/K)\rightarrow{\rm
Aut}(\phi[\mathfrak{l}])\cong\mathrm{GL}_{r}(A/\mathfrak{l}).$

If $\bar{\rho}_{\phi,\mathfrak{l}}$ is reducible, then either

$\deg_{T}\ell-10(d+1)^{7}\log\deg_{T}\ell\leqslant\log
c_{2}+10(d+1)^{7}\left\\{\log
d+r+\log[h_{G}(\phi)+1+\frac{q}{q-1}-\frac{q^{r}}{q^{r}-1}]\right\\}$ (1)

or

$\deg_{T}\ell\leqslant\log c_{2}+10(d+1)^{7}[\log d\cdot h(\phi)]$ (2)

As a corollary of Theorem 1.1, we deduce a lowerbound for $\deg_{T}\ell$ such
that the mod-$\mathfrak{l}$ Galois representation
$\bar{\rho}_{\phi,\mathfrak{l}}$ is irreducible. See corollary 4.2 for
details.

For the special case “rank-$2$ Drinfeld modules over $\mathbb{F}_{q}(T)$”,
there is actually a finer estimation on irreducibility of mod-$\mathfrak{l}$
Galois representation made by Chen and Lee [CL19]. However, their strategy
uses the fact that a power of $1$-dimensional group representation is again a
group representation, see proof of Proposition 7.1 in [CL19]. In the rank-2
case, the reducibility of mod-$\mathfrak{l}$ Galois representation always
contributes a $1$-dimensional subrepresentation. But this is not true for
higher rank Drinfeld modules.

On the other hand, Chen and Lee [CL19] gave an explicit bound on surjectivity
of mod-$\mathfrak{l}$ Galois representations for rank-$2$ Drinfeld modules
over $\mathbb{F}_{q}(T)$ without CM. Such an explicit bound is still unknown
for higher rank Drinfeld modules. The main difficulty is the classification of
maximal subgroups (up to conjugacy classes) in $\mathrm{GL}_{r}$ over finite
field is much more complicated comparing to the $\mathrm{GL}_{2}$ case, where
one only need to take care of the Borel and Cartan cases.

##  2 Preliminaries

Let $A=\mathbb{F}_{q}[T]$ be the polynomial ring over finite field with
$q=p^{e}$ an odd prime power, $F=\mathbb{F}_{q}(T)$ be the fractional field of
$A$, and $K$ be a finite extension over $F$. Set $K\\{\tau\\}$ to be the
twisted polynomial ring with the multiplication rule
$\tau\alpha=\alpha^{q}\tau\text{ for any }\alpha\in K$. Throughout this paper,
“$\log$” refers to the logarithm with base $q$.

###  2.1 Drinfeld modules

We view $K$ as an $A$-field, which is a field equipped with a homomorphism
$\gamma:A\rightarrow K$. The $A$-characteristic of $K$ is defined to be the
kernel of $\gamma$.

######  Note 2.1.

Throughout this paper, we take $\gamma:A\rightarrow K$ to be the natural
embedding from $A$ to $K$. i.e. the $A$-characteristic of $K$ is always equal
to zero.

######  Definition 2.2.

A Drinfeld $A$-module of rank $r$ over $K$ of generic characteristic is a ring
homomorphism

$\phi:A\rightarrow K\\{\tau\\}={\rm End}_{\mathbb{F}_{q}}(\mathbb{G}_{a,K})$

such that

  1. 1.

$\phi(a):=\phi_{a}$ satisfies ${\rm deg}_{\tau}\phi_{a}=r\cdot{\rm deg}_{T}a$

  2. 2.

Denote $\partial:F\\{\tau\\}\rightarrow F$ by $\partial(\sum
a_{i}\tau^{i})=a_{0}$, then $\phi$ satsfies $\gamma=\partial\circ\phi$.

From the definition of Drinfeld $A$-module, we can characterize a Drinfeld
module $\phi$ by writing down

$\phi_{T}=T+g_{1}\tau+\cdots+g_{r-1}\tau^{r-1}+g_{r}\tau^{r},\textrm{ where
}g_{i}\in K\textrm{ and }g_{r}\in K^{*}.$

######  Proposition 2.3.

There is an isomorphism between the twisted polynomial ring $K\\{\tau\\}$ and
the ring of $q$-polynomials $(K<x>,+,\circ)$ where
$K<x>:=\left\\{\sum_{i=0}^{n}c_{i}x^{q^{i}}\mid c_{i}\in K\right\\}$ and the
mutiplication of $K<x>$ is defined to be composition of $q$-polynomials.

###### Proof.

Consider the map sending $\sum_{i=0}^{n}c_{i}\tau^{i}$ to
$\sum_{i=0}^{n}c_{i}x^{q^{i}}$. This defines an isomorphism between
$K\\{\tau\\}$ and $K<x>$.

∎

Fix a Drinfeld module $\phi$ over $K$. From the above proposition, the image
$\phi_{a}$ of Drinfeld module $\phi$ at $a\in A$ corresponds to a
$q$-polynomial $\phi_{a}(x)$. Hence for an ideal $\mathfrak{a}=<a>$ of $A$, we
may define the $\mathfrak{a}$-torsion of the Drinfeld module $\phi$ over $K$.

######  Definition 2.4.

The $\mathfrak{a}$-torsion of a Drinfeld module $\phi$ over $K$ is defined to
be

$\phi[\mathfrak{a}]:=\left\\{\textrm{ zeros of }\phi_{a}(x)\textrm{ in
}\bar{K}\right\\}\subset\bar{K}.$

Now we define the $A$-module structure on $\bar{K}$. For any elements $b\in A$
and $\alpha\in\bar{K}$. We define the $A$-action of $b$ on $\alpha$ via

$b\cdot\alpha:=\phi_{b}(\alpha).$

This gives $\bar{K}$ an $A$-module structure. And the $A$-module structure
inherits to $\phi[\mathfrak{a}]$. As our Drinfeld module $\phi$ over $K$ has
generic characteristic, we have the following proposition

######  Proposition 2.5.

$\phi[\mathfrak{a}]$ is a free $A/\mathfrak{a}$-module of rank $r$.

###### Proof.

See [Gos96], Proposition 4.5.3. ∎

Let $\mathfrak{l}$ be a prime ideal of $A$, then the $\mathfrak{l}$-torsion
$\phi[\mathfrak{l}]$ of the Drinfeld module $\phi$ is an $r$-dimensional
$A/\mathfrak{l}$-vector space. Applying the action of absolute Galois group
$\mathrm{Gal}(\bar{K}/K)$ on $\phi[\mathfrak{l}]$, we obtain the so-called
mod-$\mathfrak{l}$ Galois representation

$\bar{\rho}_{\phi,\mathfrak{l}}:\mathrm{Gal}(\bar{K}/K)\rightarrow{\rm
Aut}(\phi[\mathfrak{l}])\cong\mathrm{GL}_{r}(A/\mathfrak{l})$

for the Drinfeld module $\phi$ over $K$.

Now we define some heights of Drinfeld modules. We denote by $M_{K}$ the set
of all places of $K$ including places above $\infty$. For each place $\nu\in
M_{K}$, define $n_{\nu}:=[K_{\nu}:F_{\nu}]$ to be the degree of local field
extension $K_{\nu}/F_{\nu}$. And define $|\cdot|_{\nu}$ to be a normalized
valuation of $K_{\nu}$.

######  Definition 2.6.

Let $\phi$ be a rank-$r$ Drinfeld module over $K$ characterized by

$\phi_{T}=T+g_{1}\tau+\cdots+g_{r-1}\tau^{r-1}+g_{r}\tau^{r},\textrm{ where
}g_{i}\in K\textrm{ and }g_{r}\in K^{*}.$

  1. 1.

The naive height of $\phi$ is defined to be

$h(\phi):={\rm max}\\{h(g_{1}),\cdots,h(g_{r})\\},$

where $h(g_{i}):=\frac{1}{[K:F]}\sum_{\nu\in M_{K}}n_{\nu}\cdot{\rm
log}|g_{i}|_{\nu}$.

  2. 2.

The graded height of $\phi$ is defined to be

$h_{G}(\phi):=\frac{1}{[K:F]}\sum_{\nu\in M_{K}}n_{\nu}\cdot{\rm log}\ {\rm
max}\\{|g_{i}|_{\nu}^{1/(q^{i}-1)}\mid 1\leqslant i\leqslant r\\}$

######  Corollary 2.7.

One can observe from the definition of naive height and graded height that

$h(\phi)\leqslant(q^{r}-1)\cdot h_{G}(\phi).$

###  2.2 Isogenies

######  Definition 2.8.

Let $\phi$ and $\psi$ be two rank-$r$ Drinfeld $A$-modules over $K$. A
morphism $u:\phi\rightarrow\psi$ over $K$ is a twisted polynomial $u\in
K\\{\tau\\}$ such that

$u\phi_{a}=\psi_{a}u\text{ \rm for all }a\in A.$

A non-zero morphism $u:\phi\rightarrow\psi$ is called an isogeny. A morphism
$u:\phi\rightarrow\psi$ is called an isomorphism if its inverse exists.

Set ${\rm Hom}_{K}(\phi,\psi)$ to be the group of all morphisms
$u:\phi\rightarrow\psi$ over $K$. We denote ${\rm End}_{K}(\phi)={\rm
Hom}_{K}(\phi,\phi)$. For any field extension $L/K$, we define

${\rm Hom}_{L}(\phi,\psi)=\\{u\in L\\{\tau\\}\mid u\phi_{a}=\psi_{a}u\text{
\rm for all }a\in A\\}.$

For $L=\bar{K}$, we omit subscripts and write

${\rm Hom}(\phi,\psi):={\rm Hom}_{\bar{K}}(\phi,\psi)\text{ \rm and }{\rm
End}(\phi):={\rm End}_{\bar{K}}(\phi)$

######  Definition 2.9.

The composition of morphisms makes ${\rm End}_{L}(\phi)$ into a subring of
$L\\{\tau\\}$, called the endomorphism ring of $\phi$ over $L$. For any
rank-$r$ Drinfeld module $\phi$ over $K$ with ${\rm End}(\phi)=A$, we say that
$\phi$ does not have complex multiplication.

######  Definition 2.10.

Let $f:\phi\rightarrow\psi$ be an isogeny of Drinfeld modules over $K$ of rank
$r$, we define the degree of $f$ to be

$\deg f:=\\#{\rm{ker}}(f).$

######  Proposition 2.11.

Let $f:\phi\rightarrow\psi$ be an isogeny of Drinfeld modules over $K$ of rank
$r$. There exists a dual isogeny $\hat{f}:\psi\rightarrow\phi$ such that

$f\circ\hat{f}=\psi_{a}\textrm{ and }\hat{f}\circ f=\phi_{a}.$

Here $0\neq a\in A$ is an element of minimal $T$-degree such that ${\rm
ker}(f)\subset\phi[a]$.

###### Proof.

See [Gos96] Proposition 4.7.13 and Corollary 4.7.14.

∎

The following corollary is immediate by counting cardinalities.

######  Corollary 2.12.

As in the setting of Proposition 2.11, we have

$q^{r\cdot\deg_{T}(a)}=(\deg f)\cdot(\deg\hat{f}).$

Now we can state the key tools to derive our main result:

######  Theorem 2.13 ([DD99] Theorem 1.3).

Let $K$ be a finite extension over $F$ with $[K:F]:=d$. Suppose that there are
two $\bar{K}$-isogenous Drinfeld modules $\phi$ and $\psi$ defined over $K$.
Then there is an isogeny $f:\phi\rightarrow\psi$ such that

$\deg f\leqslant c_{2}\cdot(dh(\phi))^{10(r+1)^{7}}.$

Here $c_{2}=c_{2}(r,q)$ is a effectively computable constant depends only on
$d$ and $q$.

######  Theorem 2.14 ([BPR21] Theorem 3.1).

Let $f:\phi\rightarrow\psi$ be an isogeny of rank-$r$ Drinfeld modules over
$\bar{K}$ and suppose that ${\rm ker}(f)\subset\phi[N]$ for some $0\neq N\in
A$. Then we have

$|h_{G}(\psi)-h_{G}(\phi)|\leqslant\deg_{T}(N)+\left(\frac{q}{q-1}-\frac{q^{r}}{q^{r}-1}\right).$

##  3 Proof of Theorem 1.1

We are given a rank-$r$ Drinfeld module $\phi$ defined over $K$ with ${\rm
End}(\phi)=A$. And suppose the image of mod-$\mathfrak{l}$ Galois
representation ${\rm{Im}}\bar{\rho}_{\phi,\mathfrak{l}}$ acting on
$\phi[\mathfrak{l}]$ has an invariant $A/\mathfrak{l}$-subspace of dimension
$1\leqslant k\leqslant r-1$. Denote such an invariant subspace by $H$.

From Proposition 4.7.11 and Remark 4.7.12 of [Gos96], there is an isogeny

$f:\phi\rightarrow\phi/H$

with ${\rm ker}(f)=H$. Since $\phi$ and $f$ both are defined over $K$, one can
see that the Drinfeld module $\phi/H$ is a rank-$r$ Drinfeld module defined
over $K$ as well. In addition, we have

$\deg f=\\#H=q^{k\cdot\deg_{T}\mathfrak{l}}.$

Take a dual isogeny $\hat{f}:\phi/H\rightarrow\phi$ of $f$. The degree of
$\hat{f}$ can be computed using Corollary 2.12. We get

$\deg\hat{f}=q^{(r-k)\cdot\deg_{T}\mathfrak{l}}.$

Besides, we can find two isogenies between $\phi$ and $\phi/H$ with bounded
degree from Theorem 2.13:

  1. $\bullet$

$u:\phi\rightarrow\phi/H$ is such an isogeny defined over $\bar{K}$ with $\deg
u\leqslant c_{2}\cdot(dh(\phi))^{10(d+1)^{7}}$

  2. $\bullet$

$u^{\prime}:\phi/H\rightarrow\phi$ is such an isogeny defined over $\bar{K}$
with $\deg u^{\prime}\leqslant c_{2}\cdot(dh(\phi/H))^{10(d+1)^{7}}$

Since ${\rm End}(\phi)=A$, we have $u^{\prime}\circ u=\phi_{b}$ for some $b\in
A$.

Now we consider the composition of isogenies

$u^{\prime}\circ f\circ\hat{f}\circ
u:\phi\rightarrow\phi/H\rightarrow\phi\rightarrow\phi/H\rightarrow\phi.$

Since ${\rm End}(\phi)=A$, we can find $N_{1}$ and $N_{2}$ in $A$ such that

$u^{\prime}\circ f=\phi_{N_{1}},\textrm{ and }\hat{f}\circ u=\phi_{N_{2}}.$

Thus we have

$u^{\prime}\circ f\circ\hat{f}\circ u=(u^{\prime}\circ f)\circ(\hat{f}\circ
u)=\phi_{N_{1}N_{2}}.$

On the other hand, we compute in different order and get

$u^{\prime}\circ f\circ\hat{f}\circ u=u^{\prime}\circ(f\circ\hat{f})\circ
u=u^{\prime}\circ(\phi/H)_{\ell}\circ u=\phi_{\ell}\circ(u^{\prime}\circ
u)=\phi_{\ell b}.$

Thus we get the equality $\ell b=N_{1}N_{2}$. As $\ell$ is prime, we have
either case (1): $\ell|N_{1}$ or case (2): $\ell|N_{2}$.

  1. case (1):

$\ell|N_{1}$.

Then we may write $N_{1}=\ell\cdot\beta$ for some $0\neq\beta\in A$. From the
equality $u^{\prime}\circ f=\phi_{N_{1}}$, we have

$\log\deg u^{\prime}+k\cdot\deg_{T}\ell=r(\deg_{T}\ell+\deg_{T}\beta).$

Hence we get $\log\deg u^{\prime}=(r-k)\deg_{T}\ell+r\deg_{T}\beta$. Combining
with the bound $\deg u^{\prime}\leqslant
c_{2}\cdot(dh(\phi/H))^{10(d+1)^{7}}$, we obtain the inequality

$(r-k)\deg_{T}\ell\leqslant\log
c_{2}+10(d+1)^{7}\log[dh(\phi/H)]-r\deg_{T}\beta\leqslant\log
c_{2}+10(d+1)^{7}\log[dh(\phi/H)]$

Thus we have

$\deg_{T}\ell\leqslant\frac{1}{r-k}\cdot\left(\log
c_{2}+10(d+1)^{7}\log[dh(\phi/H)]\right)\leqslant\log
c_{2}+10(d+1)^{7}\log[dh(\phi/H)].$ ($\star$)

Now from Corollary 2.7 and Theorem 2.14, we have

$h(\phi/H)\leqslant(q^{r}-1)h_{G}(\phi/H)\leqslant(q^{r}-1)\cdot\left[h_{G}(\phi)+\deg_{T}\ell+(\frac{q}{q-1}-\frac{q^{r}}{q^{r}-1})\right].$

Deduce from the above inequality, we get

$\begin{array}[]{ccc}\log
h(\phi/H)&\leqslant&\log(q^{r}-1)+\log\left(h_{G}(\phi)+\deg_{T}\ell+(\frac{q}{q-1}-\frac{q^{r}}{q^{r}-1})\right)\\\
\\\
&\leqslant&r+\log\deg_{T}\ell+\log\left(h_{G}(\phi)+1+(\frac{q}{q-1}-\frac{q^{r}}{q^{r}-1})\right)\end{array}$

Combining with the inequality ($\star$), we have the desired inequality (1)

$\deg_{T}\ell-10(d+1)^{7}\log\deg_{T}\ell\leqslant\log
c_{2}+10(d+1)^{7}\left\\{\log
d+r+\log[h_{G}(\phi)+1+\frac{q}{q-1}-\frac{q^{r}}{q^{r}-1}]\right\\}$

  2. case (2):

$\ell|N_{2}$.

Then we may write $N_{2}=\ell\cdot\beta$ for some $0\neq\beta\in A$. From the
equality $\hat{f}\circ u=\phi_{N_{2}}$, we have

$(r-k)\deg_{T}\ell+\log\deg u=r(\deg_{T}\ell+\deg_{T}\beta).$

Thus we get $\log\deg u=k\deg_{T}\ell+r\deg_{T}\beta$. Together with the bound
$\deg u\leqslant c_{2}\cdot(dh(\phi))^{10(d+1)^{7}}$, we achieve that

$k\deg_{T}\ell\leqslant\log
c_{2}+10(d+1)^{7}\log[dh(\phi)]-r\deg_{T}\beta\leqslant
c_{2}\cdot(dh(\phi))^{10(d+1)^{7}}.$

Hence we have the inequality (2)

$\deg_{T}\ell\leqslant\frac{1}{k}\cdot\left(\log c_{2}+10(d+1)^{7}[\log d\cdot
h(\phi)]\right)\leqslant\log c_{2}+10(d+1)^{7}[\log d\cdot h(\phi)]$

This complete the proof of Theorem 1.1.

##  4 Lower bound on irreducibility of $\bar{\rho}_{\phi,\mathfrak{l}}$

Under the setting of Theorem 1.1, one may further solve the inequality (1) for
$\deg_{T}\ell$. Set

$\Omega_{\phi}:={\rm max}\left\\{\log c_{2}+10(d+1)^{7}\left(\log
d+r+\log[h_{G}(\phi)+1+\frac{q}{q-1}-\frac{q^{r}}{q^{r}-1}]\right),\log
c_{2}+10(d+1)^{7}[\log d\cdot h(\phi)]\right\\},$

and $N_{d}:=10(d+1)^{7}$. Theorem 1.1 implies that the mod-$\mathfrak{l}$
Galois representation is irreducible when

$\textrm{(1'):
}\frac{q^{\deg_{T}\ell}}{\deg_{T}\ell^{N_{d}}}>q^{\Omega_{\phi}}\textrm{ and
(2): }\deg_{T}\ell>\Omega_{\phi}$

Since when we fix a finite extension $K/F$ and a Drinfeld module $\phi$, the
numbers $N_{d}$ and $\Omega_{\phi}$ are fixed. Elementary Calculus can tell us
that the fraction $\frac{q^{\deg_{T}\ell}}{\deg_{T}\ell^{N_{d}}}$ tends to
infinity as $\deg_{T}\ell$ goes to infinity. Thus we can always find a real
number $C_{\phi,d}$ such that $\deg_{T}\ell>C_{\phi.d}$ implies
$\frac{q^{\deg_{T}\ell}}{\deg_{T}\ell^{N_{d}}}>q^{\Omega_{\phi}}$. Now we try
to solve $C_{\phi,d}$ explicitly:

######  Lemma 4.1.

Let $a,b,$ and $c$ be positive real numbers such that
$c^{1/b}\cdot\frac{b}{{\rm ln}a}\geqslant e$, here $e$ is the Euler’s number.
Then

$x>\frac{-b\cdot W_{-1}(\frac{-{\rm ln}a}{c^{1/b}\cdot b})}{{\rm ln}\ a}$

is a solution to the inequality

$\frac{a^{x}}{x^{b}}>c.$

Here ${\ln(\cdot)}:={\rm log}_{e}(\cdot)$ and $W_{-1}$ is the negative brach
of the real-valued Lambert $W$-function, i.e. the inverse function of the
complex valued function $f(y)=ye^{y}$.

###### Proof.

First of all, we solve the equation

$\frac{a^{x}}{x^{b}}=c.$

We have

$\begin{array}[]{lll}&&\frac{a^{x}}{x^{b}}=c\\\ \\\ &\Rightarrow&x{\rm
ln}(a)={\rm ln}(c)+b{\rm ln}(x)\\\ \\\ &\Rightarrow&x\cdot\frac{{\rm
ln}(a)}{b}={\rm ln}(c^{1/b})+{\rm ln}(x)\\\ \\\ (\textrm{set
}\tilde{x}=x\frac{{\rm ln}(a)}{b})&\Rightarrow&\tilde{x}={\rm
ln}(c^{1/b}\cdot\frac{b}{{\rm ln}(a)})+{\rm ln}(\tilde{x})\\\ \\\ (\textrm{set
}\tilde{c}=-{\rm ln}(c^{1/b}\frac{b}{{\rm
ln}(a)}))&\Rightarrow&-\tilde{x}+{\rm ln}(\tilde{x})=\tilde{c}\\\ \\\
&\Rightarrow&-\tilde{x}e^{-\tilde{x}}=-e^{\tilde{c}}\\\ \\\ (\textrm{Here we
use condition }c^{1/b}\cdot\frac{b}{{\rm ln}(a)}\geqslant
e)&\Rightarrow&-\tilde{x}=W_{-1}(-e^{\tilde{c}})\\\ \\\
&\Rightarrow&x=\frac{-bW_{-1}(\frac{-{\rm ln}(a)}{c^{1/b}b})}{{\rm
ln}(a)}\par\end{array}$

Now from elementary Calculus, we can see that the function
$g(x)=\frac{a^{x}}{x^{b}}$ is increasing whenever $x>\frac{b}{{\rm ln}(a)}$.
On the other hand, the output value of $-W_{-1}(\frac{-{\rm
ln}(a)}{c^{1/b}\cdot b})$ is at least $1$. Thus we have

$x>\frac{-b\cdot W_{-1}(\frac{-{\rm ln}(a)}{c^{1/b}\cdot b})}{{\rm ln}\
a}\geqslant\frac{b}{{\rm ln}(a)}.$

Thus we get

$\frac{a^{x}}{x^{b}}>c,\textrm{ whenever }x>\frac{-b\cdot W_{-1}(\frac{-{\rm
ln}a}{c^{1/b}\cdot b})}{{\rm ln}(a)}.$

∎

Now we take $x=\deg_{T}\ell$, $a=q$, $b=N_{d}$, and $c=q^{\Omega_{\phi}}$. One
can check that

$c^{1/b}\cdot\frac{b}{{\rm ln}(a)}\geqslant e.$

Therefore, Lemma 4.1 shows that

$C_{\phi,d}=\frac{-b\cdot W_{-1}(\frac{-{\rm ln}(a)}{c^{1/b}\cdot b})}{{\rm
ln}(a)}.$

And we can conclude the following corollary:

######  Corollary 4.2.

Let $q=p^{e}$ be a prime power, $A:=\mathbb{F}_{q}[T]$, and $K$ be a finite
extension of $F:=\mathbb{F}_{q}(T)$ of degree $d$. Let $\phi$ be a rank-$r$
Drinfeld $A$-module over $K$ of generic characteristic and assume that ${\rm
End}_{\bar{K}}(\phi)=A$. Let $\mathfrak{l}=(\ell)$ be a prime ideal of $A$,
consider the mod-$\mathfrak{l}$ Galois representation

$\bar{\rho}_{\phi,\mathfrak{l}}:\mathrm{Gal}(\bar{K}/K)\rightarrow{\rm
Aut}(\phi[\mathfrak{l}])\cong\mathrm{GL}_{r}(A/\mathfrak{l}).$

If $\deg_{T}\ell>{\rm max}\\{C_{\phi,d},\Omega_{\phi}\\}$, then
$\bar{\rho}_{\phi,\mathfrak{l}}$ is irreducible. Here

$\Omega_{\phi}:={\rm max}\left\\{\log c_{2}+10(d+1)^{7}\left(\log
d+r+\log[h_{G}(\phi)+1+\frac{q}{q-1}-\frac{q^{r}}{q^{r}-1}]\right),\log
c_{2}+10(d+1)^{7}[\log d\cdot h(\phi)]\right\\},$

and

$C_{\phi,d}=\frac{-b\cdot W_{-1}(\frac{-{\rm ln}(a)}{c^{1/b}\cdot b})}{{\rm
ln}(a)}.$

Where $a=q$, $b=10(d+1)^{7}$, and $c=q^{\Omega_{\phi}}$

## Acknowledgement

The author would like to thank Sophie Marcques for inspiring discussions, and
Wei-Hung Su for showing me how to solve the inequality in Lemma 4.1.

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