††thanks: These two authors contributed equally.††thanks: These two authors
contributed equally.

# Comparing scattering rates from Boltzmann and dynamical mean-field theory

M. Wais1 J. Kaufmann1 M. Battiato2 K. Held1 1 Institute of Solid State
Physics, TU Wien, Vienna, Austria  2 Nanyang Technological University, 21
Nanyang Link, Singapore, Singapore

(August 27, 2024)

###### Abstract

We compute scattering rates for electrons in the two-dimensional Hubbard model
for a one-orbital metal and a two-orbital band insulator by means of the
Boltzmann scattering equation (BSE) and dynamical mean-field theory (DMFT). As
an intermediate method between both, we also consider the BSE without momentum
conservation. In the weak interaction regime and for the band insulator, the
last two agree to very good accuracy. The BSE with momentum conservation, on
the other hand, shows slightly larger scattering rates, and a momentum
differentiation of these on the Fermi surface. For the Mott insulator at
strong interaction, the DMFT electron scattering rates are much larger and
defy a BSE description. Noteworthy, the scattering rates for the band
insulator are exceedingly small because—in contrast to the Mott
insulator—there is virtually no impact ionization.

##  I Introduction

The electronic structure, i.e., the electronic states and their broadening or
scattering rate, is arguably the most fundamental property of a solid.
Scattering processes not only affect equilibrium properties but are also
essential if a material is driven away from equilibrium. Experimentally, the
one-particle scattering rate for the (occupied) electronic states can be
measured by angular-resolved photoemission spectroscopy (ARPES)[1, 2]. If
vertex corrections can be neglected, there is a one-to-one correspondence
between this one-particle scattering rate and the two-particle scattering
rates for response functions such as the optical conductivity. Here, the width
of the Drude peak corresponds to the two-particle scattering rate that,
without vertex corrections, is directly related to the one-particle scattering
rates we calculate here[3, 4] 111Both in dynamical mean-field theory and
Boltzmann there are no vertex corrections to the optical conductivity. We
study them by using two methods that are widely employed in solid state
theory, albeit by different communities. Through our comparison, we hope to
contribute to a better mutual understanding of the strengths and weaknesses of
these methods, as well as of the very different electron-electron scattering
in a metal, band insulator, and Mott insulator.

Dynamical mean-field theory (DMFT) [6, 7, 8, 9] is one of the most widely used
approaches for strongly correlated materials. It is non-perturbative and maps
a correlated lattice model onto the solution of an Anderson impurity model in
a self-consistent way [7]. DMFT becomes exact in the limit of high dimensions
or a high connectivity of the lattice [6], which implies that the self-energy
and hence the scattering rate is momentum independent.

The Boltzmann scattering equation (BSE) [10, 11, 12, 13] has been originally
developed for gases [10] but is nowadays used to address a multitude of
different problems, all the way from nuclear physics to cosmology. Often the
transport part of this equation is combined with a crude approximation for the
scattering, the relaxation time approximation, to study transport properties.
However, the full Boltzmann scattering term can also be included, allowing
e.g. for a highly detailed reconstruction of the thermalization process. Among
the possible applications of the full Boltzmann scattering term is the
possibility of calculating scattering rates, making a direct comparison of
this approach with DMFT possible.

To the best of our knowledge such comparison has not been done in a systematic
way, and we attempt to fill this blank spot through this work. Specifically,
we study the equilibrium scattering rates for the single-orbital Hubbard model
in two dimensions as well as those for a two-orbital band insulator. The BSE
is expected to fail at strong interaction $U$, since it describes the dynamics
of the distribution function by a (momentum-resolved) rate equation with the
transition rates usually calculated in lowest order perturbation theory in $U$
(Fermi’s golden rule). DMFT, on the other hand, neglects (as an impurity model
is solved) the momentum dependence of scattering (an approximation known to
become correct in the limit of high dimensions). A final point is that, while
DMFT even allows for the construction of effective (local) scattering matrix
elements in form of the two-particle vertex, the Boltzmann scattering term
needs them as input and only performs the joint density of states (DOS)
integration and, eventually, the time propagation.

In this paper, we show that indeed at strong interaction $U$, i.e., in the
Mott insulating phase 222For an overview of the Mott-Hubbard transition and
the physics of the Mott insulator, see [55]., a BSE description of the
scattering rate is not possible. This is surprising given a good description
of the spectral redistribution caused by impact ionization [15]. The DMFT
scattering rate is much higher than what can be expected or understood in a
rigid band picture; it is intimately connected with the formation of the
Hubbard bands and shoulders therein. Conversely, at weak $U$ we obtain a
discrepancy as well. These discrepancies, noticeable larger scattering rates
and a momentum differentiation on the Fermi surface, can be traced back to the
momentum conservation or lack thereof: DMFT and BSE without momentum
conservation are in good agreement.

This paper is structured as follows: In Sec. II we introduce the Hubbard-type
models considered, and describe how scattering rates are calculated in DMFT
and with the Boltzmann scattering equation. In Sec. III we present results for
the weak-coupling single-orbital Hubbard model. Next, we compare scattering
rates for the two-orbital band insulator in Sec. IV and the Mott insulating
single-orbital Hubbard model in Sec. V. In Sec. VI we summarize the results.
Furthermore we provide additional derivations and results in the Appendix.

##  II Model and methods

###  II.1 Hubbard-type models

In this paper we study the single-orbital Hubbard model on a two-dimensional
square lattice, as well as a related two-orbital model which is a band
insulator. It is useful to employ second quantization, where operators
$c^{({\dagger})}_{\mathbf{k}m\sigma}$ annihilate (create) electrons at
momentum $\mathbf{k}$ and spin $\sigma$ in orbital $m$. Their Fourier-
transformed operators $c^{({\dagger})}_{im\sigma}$ do the same for a lattice
site $i$ instead of momentum $\mathbf{k}$; the products
$n^{\vphantom{{\dagger}}}_{\mathbf{k}m\sigma}=c_{\mathbf{k}m\sigma}^{\dagger}c^{\vphantom{{\dagger}}}_{\mathbf{k}m\sigma}$
and
$n^{\vphantom{{\dagger}}}_{im\sigma}=c_{im\sigma}^{\dagger}c^{\vphantom{{\dagger}}}_{im\sigma}$
are the particle number operators for momentum and site occupations,
respectively. Both Hubbard-type models can be described by the following
Hamiltonian

$H=\sum_{\mathbb{k}m\sigma}\epsilon^{\vphantom{{\dagger}}}_{m}(\mathbf{k})n^{\vphantom{{\dagger}}}_{\mathbf{k}m\sigma}+\frac{U}{2}\sum_{i}\\!\sum_{(l\sigma)\neq(m\sigma^{\prime})}\\!n_{il\sigma}n_{im\sigma^{\prime}}.$
(1)

The first term constitutes a tight-binding description of the system. It
describes the kinetic energy (“hopping”) of non-interacting electrons with
crystal momentum $\mathbf{k}$ and a dispersion relation
$\epsilon^{\vphantom{{\dagger}}}_{m}(\mathbf{k})$ that is assumed to be
diagonal in the orbital index $m$. This term is diagonal in momentum space.

The second term models the Coulomb repulsion $U$ between electrons. It is
strictly local at each lattice site $i$ and, for the sake of simplicity, we
take the interaction to be the same within one orbital and between different
orbitals. Consistently, there is no Hund’s rule coupling, i. e. $J=0$. A self-
interaction is excluded in the sum. We consider both, the prevalent single-
orbital Hubbard model, where orbital indices $m$ and $l$ are restricted to
this single orbital, and a two-orbital band insulator with interaction $U$. In
the latter case, the bandgap is encoded in the dependence of
$\epsilon^{\vphantom{{\dagger}}}_{m}(\mathbf{k})$ on $m\in\\{1,2\\}$ as
detailed below.

Due to the exponential scaling of the Fock space needed to represent an
$N$-particle wave function, it is completely impossible to compute the
dynamics of every single electron in the system. Instead one is bound to make
approximations such as the DMFT and BSE, for extracting relevant information
from statistically averaged quantities such as distributions or correlation
functions.

###  II.2 Dynamical mean field theory

Many-body quantum field theory, which also is the pillar upon which DMFT is
built, has the Green’s function as its basic one-particle quantity. The
retarded Green’s function is defined as follows (with operators in the
Heisenberg representation) [16]:

$\displaystyle
G_{R}(\mathbf{k},m,t)=-i\Theta(t)\Big{\langle}c_{\mathbf{k}m\sigma}^{\vphantom{{\dagger}}}(t)c_{\mathbf{k}m\sigma}^{\dagger}(0)\\!+\\!c^{{\dagger}}_{\mathbf{k}m\sigma}(0)c_{\mathbf{k}m\sigma}^{\vphantom{{\dagger}}}(t)\Big{\rangle}$
(2) $\displaystyle
G_{R}(\mathbf{k},m,\omega)=\int_{-\infty}^{\infty}\\!dt\;e^{i\omega
t}\;G_{R}(\mathbf{k},m,t).$ (3)

Here, $\Theta(t)=0$ for $t<0$ and $1$ for $t>0$ is the step function; and
$\langle...\rangle$ the grand canonical expectation value. One can further
define a self-energy

$\Sigma_{R}(\mathbf{k},m,\omega)=\big{[}G_{R}^{(0)}(\mathbf{k},m,\omega)\big{]}^{-1}-\big{[}G_{R}(\mathbf{k},m,\omega)\big{]}^{-1}$
(4)

as the difference between (inverse) non-interacting ($U=0$) Green’s function
$G_{R}^{(0)}(\mathbf{k},m,\omega)$ and interacting ($U$) Green’s function
$G_{R}(\mathbf{k},m,\omega)$, which contains all effects of the
interaction[16]. Here, and similarly in

$G_{R}^{(0)}(\mathbf{k},m,\omega)=\lim_{\alpha\rightarrow
0^{+}}\big{[}\omega+\mu+i\alpha-\epsilon_{m}(\mathbf{k})\big{]}^{-1},$ (5)

the orbital-diagonal dispersion relation allows us to avoid matrix-inversions
in the orbital indices; $\mu$ is the chemical potential.

In DMFT, which becomes exact in the limit of infinite dimensions [6], the
momentum dependence of the self-energy is neglected:
$\Sigma_{R}(\mathbf{k},m,\omega)\to\Sigma_{R}(m,\omega)$. Thus the one-
particle Green’s function of the Hubbard model in the DMFT approximation is

$G_{R}(\mathbf{k},m,\omega)=\big{[}\omega+\mu-{\epsilon}_{m}(\mathbf{k})-\Sigma_{R}(m,\omega)\big{]}^{-1},$
(6)

where the $i\alpha$ of Eq. (5) becomes obsolete since ${\rm
Im}\Sigma_{R}(\omega)$ is negative. For the actual calculation of this self-
energy in DMFT, done through a self-consistent solution of an Anderson
impurity model, we refer the reader to Refs. 7, 9, 17.

Let us instead turn to our actual task, i.e. calculating scattering rates or
quasiparticle life times. For the following considerations we drop the orbital
($m$) dependence, as the Green’s function and self-energy are anyhow diagonal
in $m$ due to the assumed dispersion relation. If we linearize the real part
of the self-energy and parameterize it through the quasiparticle weight $Z$,
i.e.,
$\mathrm{Re}\Sigma_{R}(\omega)\approx\mathrm{Re}\Sigma_{R}(0)+[1-Z^{-1}]\omega$
we can approximate Eq. (6) as

$G_{R}(\mathbf{k},\omega)\approx
Z\big{[}\omega-\tilde{\epsilon}(\mathbf{k})-Z\mathrm{Im}\Sigma_{R}(\omega)\big{]}^{-1},$
(7)

where the Green’s function has a quasiparticle pole at
$\omega=\tilde{\epsilon}(\mathbf{k})=Z[{\epsilon}(\mathbf{k})+\mathrm{Re}\Sigma_{R}(0)-\mu]$,
with a Lorentzian broadening of full-width–half-maximum of
$-2Z\mathrm{Im}\Sigma_{R}(\tilde{\epsilon}(\mathbf{k}))$. That is,
$\tilde{\epsilon}(\mathbf{k})$ is the quasiparticle energy and the broadening
indicates that

$\frac{1}{\tau[\omega=\tilde{\epsilon}(\mathbf{k})]}=-2Z\mathrm{Im}\Sigma_{R}(\omega=\tilde{\epsilon}(\mathbf{k})).$
(8)

is the inverse life time, also known as scattering rate.

Even more transparent is the role of the life time $\tau$ when we recapitulate
the physical meaning of the time-dependent retarded Green’s function Eq. (2).
For the special case of zero temperature the system is in the ground state
$|\mathrm{GS}\rangle$ and if the momentum $\mathbf{k}$ is not occupied in the
ground state, Eq. (2) is reduced to

$G_{R}(\mathbf{k},t)=-i\langle\mathrm{GS}|c^{\vphantom{{\dagger}}}_{\mathbf{k}}(t)c^{\dagger}_{\mathbf{k}}(0)|\mathrm{GS}\rangle\;.$
(9)

That is, at time $t=0$ a particle is added to the system which is thus in the
state $|\phi\rangle=c^{\dagger}_{\mathbf{k}}(0)|\mathrm{GS}\rangle$.
Projecting this state onto its propagated version at time $t>0$
$\langle\phi(t)|=e^{iE_{\text{GS}}t}\langle\mathrm{GS}|c^{\vphantom{{\dagger}}}_{\mathbf{k}}(t)$
yields the probability amplitude ($E_{\text{GS}}$ is the ground state energy)
that this state still exists after a time $t$ has elapsed [18]. This motivates
the interpretation of $|G_{R}(\mathbf{k},t)|^{2}$ as the probability that a
state created by addition of a particle at $t=0$ still exists at later time
$t>0$.

In Appendix A, we will show that this probability is approximately

$\big{|}G_{R}(\mathbb{k},t)\big{|}^{2}\propto
e^{2Z\mathrm{Im}\Sigma_{R}(\tilde{\epsilon}(\mathbf{k}))\,t}\equiv
e^{-t/\tau(\tilde{\epsilon}(\mathbf{k}))},$ (10)

which again leads to Eq. (8) for the life time $\tau$.

Technically, we calculate the DMFT self-energy on Matsubara frequencies [19]
by continuous-time quantum Monte Carlo [20] with symmetric improved estimators
[21] using the w2dynamics program package [22, 23]. The retarded self-energy
at real (physical) frequencies is then obtained by the maximum entropy
analytic continuation [24, 25, 26].

###  II.3 Boltzmann scattering equation

The key quantity of the BSE [10, 11, 12, 13] is the distribution function,
whose dynamics is described through the leading-order contributions of the
particle-particle interaction (for the models considered). In cases where
elementary particles interact strongly, it is recommendable to rewrite the
Hamiltonian in terms of weakly interacting quasiparticles so that the leading
order perturbation theory can be applied to the weaker effective quasiparticle
interaction.

Here, we assume that a quasielectron description is possible and that these
quasiparticles are characterized by a certain set of quantum numbers, namely
the momentum $\mathbb{k}$, spin $\sigma$ or orbital-index $n$, and a
corresponding quasiparticle dispersion relation
$\tilde{\epsilon}_{n\sigma}(\mathbb{k})$. Then the distribution function
$f_{n\sigma}(t,\mathbb{k})$ corresponds to the expectation value of the
occupation number operator of a single-particle state $n_{\mathbf{k}n\sigma}$
at time $t$. In the following the spin will be absorbed into the band index
for brevity.

The BSE in case of a spatially homogeneous system without external fields but
with a fermionic particle-particle scattering reads [11, 12, 13]

$\begin{split}&\frac{\mathrm{d}f_{n_{0}}(\mathbb{k}_{0})}{\mathrm{d}t}=\frac{1}{2}\sum_{n_{1}n_{2}n_{3}}\int\mathrm{d}^{d}k_{1}\mathrm{d}^{d}k_{2}\mathrm{d}^{d}k_{3}\Big{[}W_{n_{0}\dots
n_{3}}(\mathbb{k}_{0}\dots\mathbb{k}_{3})\\\
&\quad\times\Big{(}(1-f_{n_{0}}(\mathbb{k}_{0}))(1-f_{n_{1}}(\mathbb{k}_{1}))f_{n_{2}}(\mathbb{k}_{2})f_{n_{3}}(\mathbb{k}_{3})\\\
&\quad\quad\quad-
f_{n_{0}}(\mathbb{k}_{0})f_{n_{1}}(\mathbb{k}_{1})(1-f_{n_{2}}(\mathbb{k}_{2}))(1-f_{n_{3}}(\mathbb{k}_{3}))\Big{)}\Big{]}\end{split}$
(11)

for a $d$-dimensional system. Here, $W_{n_{0}\dots
n_{3}}(\mathbb{k}_{0}\dots\mathbb{k}_{3})$ is defined as

$\begin{split}&W_{n_{0}\dots
n_{3}}(\mathbb{k}_{0}\dots\mathbb{k}_{3})=w_{n_{0}\dots
n_{3}}(\mathbb{k}_{0}\dots\mathbb{k}_{3})\\\
&\quad\quad\times\delta(\tilde{\epsilon}_{n_{0}}(\mathbb{k}_{0})+\tilde{\epsilon}_{n_{1}}(\mathbb{k}_{1})-\tilde{\epsilon}_{n_{2}}(\mathbb{k}_{2})-\tilde{\epsilon}_{n_{3}}(\mathbb{k}_{3}))\\\
&\quad\quad\times\sum_{\mathbb{G}}\delta(\mathbb{k}_{0}+\mathbb{k}_{1}-\mathbb{k}_{2}-\mathbb{k}_{3}+\mathbb{G})\;;\end{split}$
(12)

and the scattering amplitude $w_{n_{0}\dots
n_{3}}(\mathbb{k}_{0}\dots\mathbb{k}_{3})$ can be calculated by perturbation
theory (Fermi’s Golden rule) and is $\sim U^{2}$ (explicit formulas follow in
the context of the specific models below). The two delta-distributions
$\delta(\cdot)$ ensure momentum and energy conservation at the scattering
event and the sum $\sum_{\mathbb{G}}$ runs over all reciprocal lattice vectors
$\mathbb{G}$.

In thermal equilibrium, the distribution of electrons is given by the Fermi-
Dirac distribution,
$f_{\textrm{FD}}(\tilde{\epsilon})=1/\big{(}1+\mathrm{exp}[\beta(\tilde{\epsilon})]\big{)}$
with the inverse temperature $\beta=1/T$, and the chemical potential $\mu$
already absorbed in $\tilde{\epsilon}$. The Fermi-Dirac distribution is a
fixed point of the Boltzmann equation Eq. (11) and therefore properly
represents an equilibrium system.

The scattering rate $1/\tau_{n}(\mathbb{k})$ of a test-particle that is added
in the state $(n,\mathbb{k})$ in thermal equilibrium can be calculated within
the Boltzmann framework as (for a derivation, see [27]):

$\begin{split}&\frac{1}{\tau_{n_{0}}(\mathbb{k}_{0})}=\frac{1}{2}\sum_{n_{1}n_{2}n_{3}}\int\mathrm{d}^{d}k_{1}\mathrm{d}^{d}k_{2}\mathrm{d}^{d}k_{3}\Big{[}W_{n_{0}\dots
n_{3}}(\mathbb{k}_{0}\dots\mathbb{k}_{3})\\\
&\times\Big{(}(1-f_{\textrm{FD}}(\tilde{\epsilon}_{n_{1}}(\mathbb{k}_{1})))f_{\textrm{FD}}(\tilde{\epsilon}_{n_{2}}(\mathbb{k}_{2}))f_{\textrm{FD}}(\tilde{\epsilon}_{n_{3}}(\mathbb{k}_{3}))\\\
&+f_{\textrm{FD}}(\tilde{\epsilon}_{n_{1}}(\mathbb{k}_{1}))(1-f_{\textrm{FD}}(\tilde{\epsilon}_{n_{2}}(\mathbb{k}_{2})))(1-f_{\textrm{FD}}(\tilde{\epsilon}_{n_{3}}(\mathbb{k}_{3})))\Big{)}\Big{]}.\end{split}$
(13)

The calculation of the scattering rate above is done numerically with the
method presented in Ref. 27. Notice that DMFT scattering rates are only energy
(and orbital) dependent. In the BSE we can, on the other hand, add a
quasiparticle at every momentum $\mathbb{k}$ which then necessarily has the
quasiparticle energy $\tilde{\epsilon}_{n}(\mathbb{k})$. When we later plot
the BSE scattering rates as a function of energy, there will be different
$1/\tau_{n}(\mathbb{k})$’s at the same energy $\tilde{\epsilon}$. Note that
the many-body life time broadening discussed above also allows us to add
particles away from $\tilde{\epsilon}_{n}(\mathbb{k})$ in DMFT, albeit the
spectral density of such states is strongly suppressed if the broadening is
weak.

###  II.4 BSE without momentum conservation

Prospective differences between the BSE and DMFT may emerge because of (i)
strong coupling effects beyond the perturbative treatment of the scattering in
the BSE rate equation and (ii) neglecting the momentum dependence in DMFT. The
latter not only reflects in the momentum-independent DMFT self-energy but also
in disregarding the momentum conservation at scattering events in DMFT. That
is, the DMFT self-energy is calculated from Feynman diagrams to all order in
$U$ but with the interaction only on an impurity which per construction breaks
momentum conservation.

We can apply the same approximation also to Boltzmann scattering. That is, we
remove in Eq. (12) the momentum conserving delta-distributions
$\sum_{\mathbb{G}}\delta(\mathbb{k}_{0}+\mathbb{k}_{1}-\mathbb{k}_{2}-\mathbb{k}_{3})\to\frac{1}{V_{BZ}}$,
where $V_{BZ}$ is the volume of the first Brillouin-zone, as was proposed in
Ref. 15. Eq. (13) can then be simplified to a purely energy-dependent
scattering rate $1/\tau_{n}(\epsilon)$ that is calculated as [15, 27]

$\begin{split}&\frac{1}{\tau_{n_{0}}(\epsilon_{0})}=\frac{1}{2}\sum_{n_{1}n_{2}n_{3}}\int\mathrm{d}\epsilon_{1}\mathrm{d}\epsilon_{2}\mathrm{d}\epsilon_{3}\Big{[}\tilde{w}_{n_{0}\dots
n_{3}}(\epsilon_{0}\dots\epsilon_{3})\\\
&\quad\times\delta(\epsilon_{0}+\epsilon_{1}-\epsilon_{2}-\epsilon_{3})A_{0}^{n_{1}}(\epsilon_{1})A_{0}^{n_{2}}(\epsilon_{2})A_{0}^{n_{3}}(\epsilon_{3})\\\
&\quad\quad\quad\times\Big{(}(1-f_{\textrm{FD}}(\epsilon_{1}))f_{\textrm{FD}}(\epsilon_{2})f_{\textrm{FD}}(\epsilon_{3})\\\
&\quad\quad\quad+f_{\textrm{FD}}(\epsilon_{1})(1-f_{\textrm{FD}}(\epsilon_{2}))(1-f_{\textrm{FD}}(\epsilon_{3}))\Big{)}\Big{]},\end{split}$
(14)

where $A_{0}^{n}(\epsilon)$ is the normalized DOS of band $n$ and
$\tilde{w}_{n_{0}\dots n_{3}}(\epsilon_{0}\dots\epsilon_{3})$ is the thus
modified scattering amplitude that depends on the energies only.

Notice that in Eq. (14) we have explicitly used the fact that the interaction
is itself momentum-independent (which is the case for the purely local
interaction in the Hubbard model). In the general case Eq. (14) cannot be
derived, but it can be constructed as an approximation [28, 29]. In the
following we will refer to Eq. (14) as Boltzmann without momentum conservation
(BSE without $\mathbb{k}$).

Note that the structure of Eq. (14) is way simpler than Eq. (13): it can be
computed by inverting analytically the energy-conserving delta distribution in
Eq. (14) and then using standard numerical integration techniques.

##  III One-band Hubbard model at weak coupling

Figure 1: Scattering rates $1/\tau$ normalized by the interaction squared
($U^{2}$) for the two-dimensional Hubbard model at half-filling calculated by
DMFT and BSE with and without $\mathbf{k}$ conservation. The case $\beta=20$
could not be calculated with full Boltzmann due to computational limitations.
The scattering rates shown are the same for both spins in the paramagnetic
phase.

As a first comparison, we discuss the case of the prototypical one-band
Hubbard model in two dimensions at half-filling. Depending on the strength of
the local interaction $U$ and the temperature $T=1/\beta$, such a system is
predicted by DMFT to be either metallic or Mott-insulating.

For the weak coupling case we may employ Boltzmann theory with the dispersion
relation of the non-interacting Hamiltonian, which is

$\begin{split}&\epsilon(\mathbf{k})=-2t[\cos(k_{x})+\cos(k_{y})]\end{split}$
(15)

for ${\mathbb{k}}=(k_{x},k_{y})\in[-\pi,\pi)\otimes[-\pi,\pi)$ (lattice
constant $a\equiv 1$; unit-cell volume $V_{\text{UC}}=a^{2}=1$) and the
corresponding DOS [30]

$A_{0}(\omega)=\int_{\text{BZ}}\\!\\!\frac{d^{2}{k}}{V_{\text{BZ}}}\
\,\delta(\omega\\!-\\!\epsilon(\mathbb{k}))=\frac{1}{2\pi^{2}t}\mathrm{K}\Bigg{(}\\!\\!\sqrt{1-\left(\frac{\omega}{4t}\right)^{2}}\Bigg{)}\textrm{}$
(16)

where $\mathrm{K}(\ldots)$ is the complete elliptic integral of first kind. As
hopping parameter and unit of energy we choose $t\equiv 1$ in the following.
The scattering amplitude for this system can be calculated in perturbation
theory as

$w(\mathbb{k}_{0}\dots\mathbb{k}_{3})=\frac{2\pi}{{V_{BZ}}^{2}}U^{2}\delta_{\sigma_{0}\bar{\sigma}_{1}}\delta_{\sigma_{2}\bar{\sigma}_{3}}$
(17)

with the short-hand notation $\bar{\sigma}_{i}\equiv-\sigma_{i}$ for the BSE
scattering rate Eq. (13), and

$\tilde{w}(\epsilon_{0}\dots\epsilon_{3})=2\pi
U^{2}\delta_{\sigma_{0}\bar{\sigma}_{1}}\delta_{\sigma_{2}\bar{\sigma}_{3}}$
(18)

for the case of BSE without $\mathbf{k}$ in Eq. (14), cf. Ref. 15.

In Fig. 1 we show the calculated scattering rates for different temperatures
comparing DMFT and the BSE with and without momentum conservation. The quasi-
particle renormalization is $Z\approx 1$ for these values of the interaction.
In order to compare the structure of the scattering rates for different
interaction strengths, we divide the scattering rate by $U^{2}$. The Boltzmann
scattering rates then become completely independent of $U$. In contrast, the
DMFT scattering rates depend on $U$ in a non-trivial fashion (Fig. 1 shows
$U=1$ and $U=2$) since it is a non-perturbative approach. Nonetheless in the
limit $U\rightarrow 0$, the DMFT normalized scattering rates must be
$U$-independent.

Comparing the DMFT scattering rates for both interaction strengths, one
notices that the thus normalized scattering rates lie almost on top of each
other for the inverse temperatures $\beta=1.0$, $\beta=2.5$ and $\beta=20$,
while they slightly deviate for $\beta=0.5$, $\beta=1.5$, $\beta=2.0$. Since
there is a rather large uncertainty from the maximum entropy analytical
continuation and the deviation is not systematic, we can conclude that the
differences in the normalized DMFT scattering rates at $U=1$ and $U=2$ are
within the error bars.

Figure 2:  DMFT spectral densities for (a) $U=1$ and (b) $U=2$ for different
temperatures

The scattering rates calculated by the BSE without $\mathbb{k}$ are in very
good agreement with the DMFT data for all inverse temperatures except for
$\beta=1.0$. Again, this deviation may well originate from the uncertainties
of the analytic continuation. In any case, the good agreement of the
scattering rates from BSE without $k$ and DMFT along with the $\sim U^{2}$
scaling of the DMFT results, clearly show that even at $U=2t$ we are still in
the perturbative regime. As we show in Appendix C, to second order in $U$ the
scattering rates as calculated in DMFT and BSE without $\mathbb{k}$ are indeed
identical.

Note however that the spectral density333 The DMFT spectral densities were
calculated as $-\mathrm{Im}G_{R}(\omega)/\pi$ with
$G_{R}(\omega)=\int_{-\infty}^{\infty}dx\frac{A_{0}(x)}{\omega-x-\Sigma_{R}(\omega)}$
(19) after analytical continuation of the self-energy. This allows for
resolving features like sharp peaks in the spectral density that would be
smeared out by direct analytic continuation of the local Green’s function in
Matsubara frequencies. in Fig. 2 is already significantly smeared, especially
at the band edges and the Van-Hove singularity, because of the stronger
interaction. This smearing is a direct consequence of the scattering rate in
Fig. 1; and through the DMFT self-consistency it will in turn affect the
scattering rates, but only in higher order in $U$ (when self-consistently
calculating the spectral function as indicated in Appendix C). Possibly this
explains why the BSE without $\mathbb{k}$ in Fig. 1 has a lower scattering
rate at the band edge $\omega=\pm 4$ and a larger one for larger $|\omega|$,
albeit we cannot exclude this to be an artifact of the analytical
continuation.

Both DMFT and BSE without $\mathbb{k}$ show a two-peak structure in the
scattering rates with the peak positions roughly at the band-edges. The width
of these peaks increases with temperature. At the highest temperature
($\beta=0.5$) there is only one peak visible which actually consists of the
two peaks that are strongly overlapping. In Appendix B, we show that the
position of the two peaks can be approximately calculated from the first
moment of the particle- and hole-density. The width and height of the peaks
can be calculated when the zeroth and second moment of the particle-density is
taken into account in addition to the first moment.

After establishing a good agreement between DMFT and BSE without $\mathbf{k}$
at weak coupling, we next turn to the full BSE with momentum conservation. The
thus calculated BSE scattering rates (dots in Fig. 1) deviate from the rates
obtained with the other methods. First of all, as already mentioned, we
highlight that several values, corresponding to different momenta, are present
for each energy. Fig. 1 shows a particularly strong spread at the Fermi level
($\omega=0$). Furthermore, in contrast to BSE without $\mathbb{k}$ and DMFT,
there are no scattering rates outside the non-interacting bandwidth
($|\omega|>4$) any longer, as there is no momentum that has such an energy. In
DMFT due to the aforementioned smearing of the band-edges there are such
states, and in BSE without $\mathbb{k}$ we can at least calculate the
scattering rate a state at such an energy would have.

Another difference is that the BSE scattering rates are generally higher than
DMFT or BSE without $\mathbf{k}$, especially at the band edge
($|\omega|\lesssim 4$ and at higher temperatures also around the Fermi level
($\omega=0$). As DMFT and BSE without $\mathbb{k}$ agree with each other, we
can safely conclude that this difference originate from neglecting the
momentum conservation of the scattering vertex. One can also smoothly
interpolate between the results for the BSE with and without $\mathbf{k}$, by
replacing the momentum conserving $\delta$-function by a Gaussian and
increasing its width (not shown here). The reason for these discrepancies is
that the momentum averaged scattering amplitude does not take into account
that there is e.g. a particularly strong scattering among momenta at the Hove
singularities $(\pm\pi,0)$ and $(0,\pm\pi)$. At low temperatures this
scattering even leads to the formation of a pseudogap [32, 33, 34, 35, 36, 37,
38, 39] which requires a beyond DMFT description [40, 41, 42, 43, 44, 45]. A
precursor thereof is visible here as the strong-momentum dependence of the
scattering rate on the Fermi surface.

##  IV Two-orbital band insulator

Figure 3: Spectral-densities for different temperatures for the case (a)
$U=4$, $\Delta=0$ and (b) $U=2$, $\Delta=2$. For both cases, the effective
band gap is $\Delta_{\textrm{eff.}}\approx 4$.

In this section, we address the case of a band insulator in the weak to
intermediate coupling regime. We consider a two-dimensional Hubbard-type model
with two orbitals ($A$ and $B$) at half-filling, i.e., $n=2$ electrons per
site in the two orbitals. This corresponds to $\mu=0$ for our dispersion
relation below. For simplicity, we assume that electrons may only hop to
neighboring orbitals of the same type and that the hopping amplitude has the
same absolute size but opposite sign for both orbitals ($t_{A}=-1$,
$t_{B}=1=t$). Further, we add a local one-particle energy $\mp(\Delta/2+4t)$
for orbital $A$ and $B$, respectively. This results in a band gap of size
$\Delta$ in the non-interacting DOS, with the top of the valence($A$)-band and
the bottom of the conduction($B$)-band both at the $\Gamma$ point. The
interaction $U$ is local and the same within and between both orbitals such
that the interaction term of the Hubbard model acquires the simple form of Eq.
(1).

We now discuss two different systems, one with $U=4$ and one-particle gap
$\Delta=0$ and one with $U=2$ and $\Delta=2$. Due to the constant Hartree term
in the self-energy, the effective gap in the interacting system is essentially
the same $\Delta_{\textrm{eff.}}\approx U+\Delta=4$ for both setups. This is
because at sufficiently low temperatures, orbital $A$ is almost completely
filled with two electrons per site and orbital B is empty. Hence an electron
in orbital $B$ perceives a Hartree energy $2U$ (interacts with both $A$
electrons); an electron in orbital $A$ instead has a Hartree energy $1U$ (as
it only interacts with the electron of opposite spin in orbital $A$). The
difference enlarges the bandgap to $\Delta_{\textrm{eff.}}=U+\Delta$.

The spectral densities for both cases are displayed in Fig. 3 and follow the
above reasoning. At higher temperatures, we however induce holes in the
valence and electrons in the conduction band. The difference in occupation is
reduced, the bandgap hence smaller. For the highest temperature
($\beta=0.25$), the gap disappears completely for the case $U=4$, $\Delta=0$.
The non-interacting DOS in Fig. 3 is constructed with the above enhanced
effective band gap $\Delta_{\textrm{eff.}}$ instead of $\Delta$.

As this describes the DMFT spectrum at low temperatures reasonably well, we
employ for the BSE the corresponding effective bandstructure

$\displaystyle\epsilon_{A}(\mathbb{k})=$
$\displaystyle-\epsilon(\mathbb{k})-\left(\frac{\Delta_{\textrm{eff.}}}{2}+4t\right),$
(20) $\displaystyle\epsilon_{B}(\mathbb{k})=$
$\displaystyle\epsilon(\mathbb{k})+\left(\frac{\Delta_{\textrm{eff.}}}{2}+4t\right),$
(21)

where $\epsilon(\mathbb{k})$ is defined by Eq. (15). The corresponding DOS of
the non-interacting system corrected by the Hartree shift is used for the BSE
without $\mathbf{k}$ and given by

$\displaystyle A_{0}^{A}(\omega)=$ $\displaystyle
A_{0}\left(\omega+\left(\frac{\Delta_{\textrm{eff.}}}{2}+4t\right)\right),$
(22) $\displaystyle A_{0}^{B}(\omega)=$ $\displaystyle
A_{0}^{A}\left(-\omega\right)$ (23)

with $A_{0}(\omega)$ defined in Eq. (16). Due to particle-hole symmetry and
the simple form of the interaction, the BSE calculation can be simplified as
outlined in Appendix D.

Figure 4: Scattering rates normalized by the squared interaction for an
electron in the upper band of a two-orbital band insulator as calculated with
DMFT, BSE with and without $\mathbf{k}$. Two different sets of parameters are
used: $U=4$, $\Delta=0$ and $U=2$, $\Delta=2$. The gray, vertical lines
indicate the band edges of the non-interacting system.

Fig. 4 shows the scattering rate of the two-band system. The BSE with momentum
conservation shows a seemingly parabolic increase starting with a sizable
value at the lower band edge ($\omega=2$). Superimposed on this trend is an
enhanced scattering rate in the middle of the band at $\omega=6$ with a strong
momentum spread of the scattering rate. This is akin to the behavior at the
Fermi level for the weakly correlated one-band Hubbard model in Fig. 1 and can
again be attributed to the van Hove singularity.

Similar as for the one-band case, the scattering rates in BSE without
$\mathbb{k}$ are slightly smaller than in the BSE with $\mathbb{k}$
conservation and already decay toward the upper band edge ($\omega=10$). They
closely resemble the DMFT values for the $U=2$ case; only the peak of the
scatterings is shifted to slightly higher energies than in DMFT. There are
larger differences to the DMFT data at the intermediate coupling $U=4$, which
have systematically higher scattering rates at low energies. This is because
stronger smearing of the spectral density at $U=4$ leads to a smaller
effective gap and some in-gap spectral weight, see Fig. 4. This, in turn,
leads to more thermal excitations and therefore more scatterings. These
effects can be included in BSE without $\mathbb{k}$ if we use the interacting
spectral density $A^{n}(\omega)$ instead of the non-interacting one
$A_{0}^{n}(\omega)$, which leads to a good agreement with the DMFT results,
see Appendix E.

Eye catching is the strong suppression of the scattering rate upon decreasing
temperature. The reason for this is the dramatic reduction of the number of
thermally excited carriers which are needed to act as scattering partners.
Note that with a density-density Coulomb interaction, the electron in the
conduction($B$)-band either needs (i) another $B$-electron to scatter with
[the final state being again two $B$-electrons], or (ii) a hole in the
valence($A$)-band into which an $A$-electron can scatter [final and initial
state being one $A$\- and one $B$-electron]. Both $B$-electron and $A$-hole
scattering partners however require thermally excited carriers that are absent
at low temperatures.

For the Mott insulator discussed in the next section, the scattering rates are
much higher because of impact ionization processes. Here, an electron in the
upper Hubbard band excites an additional electron-hole pair across the gap. In
the band insulator impact ionization corresponds to a process
${c^{{\dagger}}_{iB\sigma}c^{\vphantom{{\dagger}}}_{iA\sigma}c^{{\dagger}}_{iB\bar{\sigma}}c^{\vphantom{{\dagger}}}_{iB\bar{\sigma}}}$
which is not possible in lowest order perturbation theory in the density-
density interaction, nor are Auger processes
$c^{{\dagger}}_{iB\sigma}c^{\vphantom{{\dagger}}}_{iA\sigma}c^{{\dagger}}_{iA\bar{\sigma}}c^{\vphantom{{\dagger}}}_{iA\bar{\sigma}}$.
In the one-band Mott insulator, the two Hubbard bands have the same orbital
index and such processes hence dominate the scattering rate if $\omega$ is
sufficiently large to allow impact ionization [46, 47, 15, 48, 49].

Even if we generalize the Coulomb interaction to the widely employed Kanamori
form [50] with spin-flip and pair-hopping terms, we still need a thermally
excited second electron or hole for scattering. Only, the full Slater
interaction [51, 52] also contains interaction terms that directly mediate
impact ionization. These interaction terms are however small or even vanish,
which is the reason why they are often disregarded in the first place.
Consider e.g. a material with cubic symmetry and the orbitals $A=d_{xy}$ and
$B=d_{xz}$. Then interaction terms such as
$U_{BAAA}c^{{\dagger}}_{iB\sigma}c^{\vphantom{{\dagger}}}_{iA\sigma}c^{{\dagger}}_{iA\bar{\sigma}}c^{\vphantom{{\dagger}}}_{iA\bar{\sigma}}$
vanish because the integral to calculate the matrix element $U_{BAAA}$ is odd
under the transformaton $z\rightarrow-z$; for a furthergoing discussion, see
e.g. [53, 54]. A more viable route to enhance the scattering rate through
impact or Auger processes in a band insulator is if the bands strongly
hybridize so that the conduction and valence bands are admixtures of the $A$
and $B$ orbitals.

It is interesting to note that the scattering rate preserves its two-band like
structure even in the case $U=4$, $\beta=0.25$ when the spectral density does
not show a gap any longer. The reason for this is again that the density-
density interaction does not allow for impact excitation and Auger emission,
which are very gap-size sensitive. Instead the scattering processes induced by
the density-density interaction are agnostic about the gap-size per-se. The
additional $B$ electron still needs another $B$ (or $A$) electron to scatter
with, and two empty final $B$ states (or an empty $A$ and an empty $B$ state).
The scattering process does not need to overcome the size of the gap, in
contrast to impact ionization.

##  V Strong coupling: Mott-insulator

Figure 5: (a) Spectral density as obtained by DMFT and Fermi-Dirac
distribution for the case $U=12$ and $\beta=5$. (b) Scattering rates as
obtained from DMFT, and BSE without $\mathbf{k}$ using either the non-
interacting density of states (BSE without k $A_{0}(\omega)$) or the
interacting DMFT spectral density shown (BSE without k $A(\omega)$).

Finally, we compare the approaches introduced above in the strong coupling
regime of the single-orbital Hubbard model. Since the BSE is a perturbative
treatment in the interaction, this is certainly the most problematic case for
the BSE. For sufficiently large interaction, the non-interacting DOS splits
into two, the upper and lower Hubbard band, see Fig. 5 (top). We have a Mott
insulator, one of the cornerstones of strongly correlated electron systems
[55].

If we use the BSE with the non-interacting DOS, this dramatic reshuffling of
the DOS is not incorporated. The scattering rate is still the very same with a
two peak structure as for weak coupling—just with the prefactor rescaled by
$U^{2}$, see black-dotted line in Fig. 5. This kind of description assumes
that we have a metal with states at low energies. It is not an appropriate
description of a Mott insulator.

This problem can be mitigated if we consider better suited quasiparticles
instead of the non-interacting ones. This is in general not trivial, and not
always can proper quasiparticles with a long life time and weak interaction be
identified. They might not even exist. Taking the electronic DMFT excitations
of the Hubbard bands as our quasiparticles in the BSE without $\mathbb{k}$, we
have to replace the non-interacting DOS $A_{0}(\omega)$ by the interacting
spectral density $A(\omega)$ of Fig. 5 (top) in Eq. (14). Even if we have no
well defined quasiparticles such a quantum Boltzmann description is possible
[15] if we have a separation of time scales, and the average-time
(distribution function) dynamics is slower than the relative-time dynamics. As
was shown in [15] the thus modified BSE without $\mathbf{k}$ provides a good
description of the DMFT impact ionization processes and redistribution of
spectral weight in non-equilibrium444The calculation of scattering in this
paper is still possible within equilibrium DMFT theory; whereas the non-
equilibrium processes of [15] required the non-equilibrium DMFT [75, 76].

Here, we instead study in Fig. 5 (bottom, blue line) the one-particle
scattering rates in the BSE without $\mathbf{k}$ and interacting $A(\omega)$:
The Mott insulator is described as two split quasiparticle bands with the gap
$\sim 4$ being much larger than temperature $T=1/5$. Hence, if we add an extra
electron in the upper Hubbard quasiparticle band it has no partners to scatter
in BSE, the scattering rate is zero similar to the suppression of the
scattering rate in the band insulator. However, if the added electron has an
excess energy [$\omega-\omega_{LBE}$ relative to the lower band edge of the
upper Hubbard band $\omega_{LBE}\gtrsim 2$ in Fig. 5] which is larger than the
Mott gap [$\Delta_{\rm Mott}\gtrsim 4$], i.e., $\omega\gtrsim 6$, impact
ionization processes with an electron-hole excitation across the gap become
possible. The phase space of such scattering processes increase quadratically
with $\omega-\omega_{LBE}-\Delta_{\rm Mott}$ for a box shaped DOS. This
explains the BSE without $\mathbf{k}$ scattering rate in Fig. 5, which as
already mentioned well describes impact ionization processes, including the
change of the double occupation and redistribution of spectral weight with
time in non-equilibrium [15].

Let us now turn to the DMFT scattering rate as extracted from the self-energy
and shown in Fig. 5 (bottom, red-dashed line) 555As we do not have a linear
quasiparticle renormalization in the self-energy, we plot
$1/\tau(\omega)=2{\rm Im}\Sigma(\omega)$; $Z=1$ in Eq. (10).. The by far
dominating feature (cut-off by the finite $y$-axis scale) is at $\omega=0$
where $\Sigma(\omega)=(U^{2}/4)\;1/(\omega+i\alpha)$ in the large $U$ limit of
the Mott insulator with a Lorentzian broadening $\alpha\sim\pi T$. This pole
is responsible for the splitting of the DOS into two Mott bands and yields the
$\delta$-like peak in ${\rm Im}\Sigma$ at $\omega=0$. As a matter of course we
cannot expect this feature to be described in the BSE without $\mathbf{k}$. It
is also not necessary as $\omega=0$ is in the middle of the Mott gap where
there are essentially no states—essentially since at low temperature the
aforementioned finite broadening leads to a very small spectral weight. This
filling of the Mott gap with temperature [58] is a feature distinct from a
band insulator. These in-gap states have an extremely short life time.

Figure 6: (a) Spectral density as obtained by DMFT for $\beta=10$ and
different interaction strengths $U=10$, 12, 16. The solid line $A(\omega)$’s
are calculated from the analytically continued $\Sigma(\omega)$; the dashed
lines are directly analytically continued from the Matsubara Green’s function.
(b) DMFT scattering rates for the same parameters as in (a).

Let us now turn to the more relevant DMFT scattering rate within the Hubbard
bands. These are orders of magnitude larger in DMFT than those from the BSE
without $\mathbf{k}$ and with interacting $A(\omega)$. Also their shape is
completely different: There is no suppression at the lower edge of the upper
Hubbard bands which, as argued above, was the case if the scattering is due to
impact ionization requiring a threshold energy; neither are the DMFT
scattering rates flat or follow the shape of the upper Hubbard band. Instead
the scattering rates are strongest around $\omega\sim 4$ close to the lower
band edge, and are dramatically reduced for larger $\omega$. Similar as the
pole at $\omega=0$, the maximum at $\omega\sim 4$ leads to a suppression of
the spectral weight. Fig. 5 (top) where we have calculated $A(\omega)$ from
the analytically continued $\Sigma(\omega)$ even shows a two peak structure in
the upper Hubbard band. Such a two peak structure was previously observed on
the metallic side of the Mott transition, immediately before the quasiparticle
peak vanishes [59, 60, 61]. On the Mott insulating side, Refs. 62 and 63 show
an extra peak or a shoulder feature on the inner side of the Hubbard bands,
similar to our findings. In Fig. 6 we also compare the $A(\omega)$ that is
directly continued from the Green’s function on the imaginary axis, which
shows a shoulder rather than a double peak. While we hence cannot resolve
within the maximum entropy uncertainty, whether we actually have a shoulder or
double peak structure, it is clear that there is a feature in the upper
Hubbard band. Mathematically, this is necessitated by the strong scattering
rate in this region. A simple physical picture or understanding of these side
structures in the Hubbard bands is still missing. Note, that also in strong
coupling perturbation theory to second order such a shoulder and hence
asymmetry of the self-energy within the upper Hubbard band is observed [64],
whereas the Hubbard-III approximation [65] and the Falicov-Kimball model [66,
67] do not show such a shoulder. In agreement, Fig. 6 shows this feature for
different values of $U$. Since the scattering in BSE without $\mathbf{k}$ and
with non-interacting $A_{0}(\omega)$ is merely rescaled by $U^{2}$, it is
clear from Fig. 6 that the agreement of the position of the maximal scattering
rate between BSE and DMFT in Fig. 5 (bottom, black-dotted vs. red-dashed line)
was by chance.

We can conclude that the one-electron scattering rate in a Mott insulator is
very different from an impact ionization picture. It is associated with the
formation ($\omega\sim 0$) of the Hubbard bands and even side structures
therein ($\omega\sim 4$ in Fig. 5). The Hubbard bands are created by the
interaction of the same electrons we also use as a test charge for calculating
the scattering rate. If there is a local extra hole or electron, locally the
Hubbard bands deform. Most noticeable this is in the filling of the Mott gap,
which does not only occur with increasing temperature [58] but also if we
drive the system out of equilibrium [46, 47, 15]. If we have an extra electron
in a disordered spin background of the DMFT Mott insulator, it can hop or
cannot hop to a neighboring site depending on the spin orientation of this
neighbor. This leads to a large scattering rate without changing the number of
double occupations. These processes are included in the DMFT but not in the
BSE, they do not contribute to impact ionization (do not change the number of
double occupations) or major energy redistributions.

##  VI Conclusion

We have studied and compared scattering rates using two widely employed
methods: BSE and DMFT. We have employed these methods out of their comfort
zone, where they cannot be applied with mathematical rigor. For DMFT this is
the dimensionality of the systems studied (2D), which is far away from the
limit of infinite dimensions where DMFT become exact. For the BSE it is the
strong interaction regime of the Mott insulator, where a rate equation with
perturbatively determined scattering rates cannot safely be applied. We have
mitigated the latter in part by using the interacting spectral function
instead of the non-interacting DOS as the quasiparticle states whose
occupation dynamics (here scattering rate) is calculated by BSE.

DMFT somewhat underestimates the scattering rates and by construction cannot
resolve their momentum-, only their energy-dependence. This momentum
dependence is particularly strong in the middle of the band where the Van-Hove
singularity is located. The physical reason behind both discrepancies is that
the phase space for the scattering of a quasiparticle with another
quasiparticle explicitly depends on available unoccupied states linked by
momentum conservation. If we replace the momentum-conserving $\delta$-function
by a Gaussian with increasing width or directly disregard momentum
conservation in the BSE without $\mathbf{k}$, the scattering rates are reduced
and the DMFT results reproduced by BSE without $\mathbf{k}$ for the weakly
correlated metal ($U=1$ or 2).

The biggest challenge for the BSE is the strongly interacting Mott insulating
state. Here the DOS is split into two Hubbard bands which we take as the
starting quasiparticle DOS in the BSE without $\mathbf{k}$. In the BSE, the
scattering rate is due to impact ionization. These processes are well
described and in good agreement with DMFT [15]. However, in DMFT additional
scattering processes which can be associated with the formation of the Hubbard
bands and shoulders therein dominate. The same specimen of electrons that
through their interaction form the Hubbard bands are also added as a charge
probe, locally disturbing the spectrum. These huge DMFT scattering rates are
beyond a BSE description with a static DOS.

Scattering in an interacting band insulator bears no similarity at all with
that in the Mott insulator. It is strongly suppressed at low temperatures
since scattering is only possible if there are thermal excitations across the
gap. BSE without $\mathbf{k}$ and DMFT agree, while the BSE with momentum
conservation has, similar as for the weakly correlated metal, somewhat larger
scattering rates. The difference to the Mott insulator does not only lie in
the huge scattering associated with the Hubbard bands, but also in the absence
of impact ionization which dominates the scattering in BSE for a Mott
insulator. Impact ionization and Auger processes are only possible in a band
insulator through higher order in $U$ processes, through quite small Coulomb
matrix elements beyond the Kanamori interaction, or a sizable hybridization
between valence and conduction band. This strongly suggests that Mott
insulators are better suited than band insulators for increasing the
efficiency of solar cells through impact ionization [68, 69, 46, 47, 15, 48,
49].

###### Acknowledgements.

We thank M. Eckstein and P. Werner for discussions, and acknowledged financial
support from the Austrian Science Fund (FWF) through the Doctoral School W1243
Solids4Fun (Building Solids for Function; MW) and project P30997 (M.W., J.K.,
K.H.), and from Nanyang Technological University, NAP-SUG (M.B.). Calculations
have been done in part on the Vienna Scientific Cluster (VSC).

##  Appendix A Scattering rate from the retarded Green’s function

###  A.1 Derivation of the formula

If we linearize the self-energy around the (real part of the) pole
$\tilde{\epsilon}(\mathbf{k})=Z[{\epsilon}(\mathbf{k})+\mathrm{Re}\Sigma_{R}(0)-\mu]$
we get Eq. (7) whose Fourier transformation is

$G(\mathbf{k},t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega\frac{Ze^{-i\omega
t}}{\omega-\tilde{\epsilon}(\mathbf{k})-Z\mathrm{Im}\Sigma_{R}(\omega)}.$ (24)

Note that a linearization of the self-energy around $\omega=0$ might not be
justified any longer if the pole is at large frequencies and that there may be
more than one pole at a given $\mathbf{k}$. For example for the Mott insulator
we have two poles for each $\mathbf{k}$. However this merely means that we
have $\mathrm{Re}\Sigma_{R}(\tilde{\epsilon})$ instead of
$\mathrm{Re}\Sigma_{R}(0)$ and that we have a sum of poles in Eq. (24) instead
of a single one, respectively. With these modifications, we can use the same
procedure as discussed in the following for a single pole.

The integral (24) can be solved by closing the contour on the lower complex
half-plane of frequencies (since $t\\!>\\!0$ the integrand is exponentially
suppressed here). Then, Eq. (24) can be computed by the residue theorem. The
pole is at $[{\rm
Re}\omega_{p},{\rm\imaginary}\omega_{p}]=[\tilde{\epsilon}(\mathbf{k}),Z{\rm\imaginary}\Sigma(\omega_{p}\approx\tilde{\epsilon}(\mathbf{k}))]$.
Here, in principle ${\rm\imaginary}\omega_{p}$ would have to be obtained self-
consistently, but if ${\rm\imaginary}\Sigma$ is small we can (approximately)
calculate it using only the real part the pole. The residue theorem then
yields for the integral in Eq. (24)

$2\pi
i\lim_{\omega\rightarrow{\omega}_{p}}(\omega-{\omega}_{p})\frac{Ze^{-i\omega
t}}{\omega-\tilde{\epsilon}(\mathbf{k})-Z{\rm
Im}\Sigma_{R}(\tilde{\epsilon})}\propto
e^{-it\tilde{\epsilon}}e^{tZ\mathrm{Im}\Sigma_{R}(\tilde{\epsilon})};$ (25)

or for the probability to find a particle that is added at time $0$ to the
quasiparticle state $\mathbf{k}$ still in this state at a later time $t$:

$|G_{R}(\mathbf{k},t)|^{2}\propto
e^{2Z\mathrm{Im}\Sigma_{R}(\tilde{\epsilon}(\mathbf{k}))t},$ (26)

which yields the (inverse) life time Eq. (8).

###  A.2 Analytic example

Figure 7:  Model self-energy of Eq. (27) for $A=0.1$.

It is instructive to consider an example, where everything can be computed
exactly, such that we can test the above approximations. We consider a self-
energy of the form

$\Sigma(\omega)=A\Big{(}\frac{1}{\omega-E+i\alpha}+\frac{1}{\omega+E+i\alpha}\Big{)}.$
(27)

For parameters $A=0.1$, $E=4$, $\alpha=1.5$ it is very similar in size and
shape to our results for the single-orbital weak coupling results for
$\beta\approx 2$, as shown in Fig. 7.

If we insert this self-energy together with $\mu=0$ into the Green’s function
Eq. (6), the locations of the poles are determined by a cubic equation in
$\omega$. It is possible to solve this equation analytically for arbitrary
parameters. Notably, the pole locations will depend on $\epsilon(\mathbf{k})$.
The locations of the poles in dependence of $\epsilon$ are shown in Fig. 8,
and for a larger value of $A=0.8$ in Fig. 9. Pole 2 is the one, we usually
associate with the quasiparticle.

Figure 8: Pole locations with model self-energy. Clearly the imaginary part of
the location of the pole with the smallest imaginary part (pole 2) agrees well
with the imaginary part of the self-energy
${\rm\imaginary}\Sigma(\omega=\tilde{\epsilon})$. The imaginary parts of the
other two poles are much larger. $Z\approx 0.99$ for this case. Figure 9:
Same as Fig. 8 but with $A$ increased by a factor of 8. Also here the
imaginary part of the location of the pole with the smallest imaginary part is
similar to the imaginary part of the self-energy, but not in quantitative
agreement. At the band edge, a second pole is of similar size and will thus
have considerable influence on the scattering rate. $Z\approx 0.94$ for this
case.

Now it is possible to compute the time-dependent Green’s function exactly by
evaluating the Fourier integral in Eq. (24). The results for a few different
values of ${\epsilon}$ are shown as solid lines in Fig. 10. Given the exact
$G(t)$ as a reference, we show the contribution of the residue of the pole
(pole 2) that is closest to the real axis, Eq. (25) as dashed lines. Finally
we also show the exponential decay where the scattering rate was approximated
by the imaginary part of the self-energy. Clearly, for small values of
$\epsilon$ this is an excellent approximation. Closer to the band edge it does
not match so well any more, especially in the case of the large self-energy
(large $A$; right panel). This mismatch may already be anticipated when
looking at Fig. 9.

Figure 10: Absolute square of the time-dependent Green’s function for the
model self-energy with $A=0.1$ (left) and $A=0.8$ (right). Solid lines are
time-dependent Green’s functions that are computed by Eq. (24). Dots show the
contribution of the residue at the pole with smallest
$|\mathrm{Im}\hat{\omega}|$. Crosses show the approximation given by Eq. (26).

##  Appendix B Convolution Method

For the case of a single band at half-filling, we here reformulate the
expression for the scattering rate in BSE without $\mathbb{k}$, Eq. (14), to
gain some further analytical insight. To this end, we define the particle
density as $n_{p}(\omega)\equiv f_{\textrm{FD}}(\omega)A_{0}(\omega)$ and the
hole-density as $n_{h}(\omega)\equiv(1-f_{\textrm{FD}}(\omega))A_{0}(\omega)$.
With these definitions and the scattering amplitude for the one-band system
introduced in Section III we may rewrite Eq. (14) as

$\begin{split}&\frac{1}{\tau_{n}(\epsilon_{0})}=2\pi
U^{2}\int\mathrm{d}\epsilon_{1}\mathrm{d}\epsilon_{2}\mathrm{d}\epsilon_{3}\Big{[}\delta(\epsilon_{0}+\epsilon_{1}-\epsilon_{2}-\epsilon_{3})\\\
&\quad\times\Big{(}n_{h}(\epsilon_{1})n_{p}(\epsilon_{2})n_{p}(\epsilon_{3})+n_{p}(\epsilon_{1})n_{h}(\epsilon_{2})n_{h}(\epsilon_{3})\Big{)}\Big{]}.\\\
\end{split}\\!$ (28)

For a system with particle-hole symmetry it holds that
$n_{p}(\omega)=n_{h}(-\omega)$. Using this property and the definition of the
convolution
$(a*b)(\omega)\equiv\int\mathrm{d}\tilde{\omega}\leavevmode\nobreak\
a(\omega-\tilde{\omega})b(\tilde{\omega})$, Eq. (28) can be further reduced to

$\frac{1}{\tau_{n}(\omega)}=2\pi U^{2}\big{[}g(\omega)+g(-\omega)\big{]},$
(29)

with

$g(\omega)\equiv(n_{p}*n_{p}*n_{p})(\omega)\textrm{ .}$ (30)

The above equation states that the scattering rate consists of the sum of the
particle density convoluted with itself twice, and its mirrored version.

According to the central limit theorem [70], a function with compact support
becomes a Gaussian function in the limit when it is convoluted an infinite
times with itself. If the particle-density is smooth, the result after two
convolutions with itself is already very similar to a Gaussian, see Fig. 11
(a).

This allows us to further reduce complexity and increase understanding: A
general Gaussian function, i.e.

$f_{\textrm{gauss}}(\omega)\equiv\frac{\alpha}{\sigma\sqrt{2\pi}}\times\exp[-\frac{{(\omega-\omega_{0})}^{2}}{2\sigma^{2}}]$
(31)

is completely defined by three parameters: its integral value $\alpha$, its
variance $\sigma^{2}$ and its zero-point $\omega_{0}$. For a given Gaussian
these three parameters can be calculated from its zeroth-, first- and second-
moment,

$\displaystyle\alpha$ $\displaystyle=F_{0}[f_{\textrm{gauss}}],$ (32)
$\displaystyle\omega_{0}$
$\displaystyle=\frac{F_{1}[f_{\textrm{gauss}}]}{F_{0}[f_{\textrm{gauss}}]},$
(33) $\displaystyle\sigma^{2}$
$\displaystyle={\frac{F_{2}[f_{\textrm{gauss}}]}{F_{0}[f_{\textrm{gauss}}]}-{\left(\frac{F_{1}[f_{\textrm{gauss}}]}{F_{0}[f_{\textrm{gauss}}]}\right)}^{2}},$
(34)

where $F_{n}[f]$ is the $n$-th moment of the function $f(\omega)$, i.e.,

$F_{n}[f]\equiv\int\mathrm{d}\omega f(\omega){\omega}^{n}\textrm{ .}$ (35)

Figure 11: (a) Particle-density $n_{p}$ and the particle-density convoluted
one- and two-times with itself. (b) Scattering rate in BSE without
$\mathbf{k}$ calculated with the convolution method compared to the
approximated scattering rate that consists of two Gaussian functions.

We can now approximate the function $g(\omega)$ with a Gaussian function by
calculating the parameters $\alpha$, $\omega_{0}$ and $\sigma$ from its
moments using Eqs. (32)-(34) with $g$ instead of $f_{\textrm{gauss}}$. The
function $g(\omega)$ is calculated from a convolution of the particle-density,
hence, the moments of $g(\omega)$ can be directly inferred from the moments of
the particle-density $n_{p}(\omega)$,

$\displaystyle F_{0}[g]=$ $\displaystyle{\left(F_{0}[n_{p}]\right)}^{3},$ (36)
$\displaystyle F_{1}[g]=$ $\displaystyle
3F_{1}[n_{p}]{\left(F_{0}[n_{p}]\right)}^{2},$ (37) $\displaystyle F_{2}[g]=$
$\displaystyle
3\left[F_{2}[n_{p}]{\left(F_{0}[n_{p}]\right)}^{2}+2{\left(F_{1}[n_{p}]\right)}^{2}F_{0}[n_{p}]\right].$
(38)

With the Eqs. (32)-(34) and the Eqs. (36)-(38) the parameters for the Gaussian
function can eventually be calculated from the moments of particle-density as

$\displaystyle\alpha$ $\displaystyle={\left(F_{0}[n_{p}]\right)}^{3},$ (39)
$\displaystyle\omega_{0}$ $\displaystyle=3\frac{F_{1}[n_{p}]}{F_{0}[n_{p}]},$
(40) $\displaystyle\sigma^{2}$
$\displaystyle=3\left[{\frac{F_{2}[n_{p}]}{F_{0}[n_{p}]}-{\left(\frac{F_{1}[n_{p}]}{F_{0}[n_{p}]}\right)}^{2}}\right],$
(41)

and the scattering rate may be approximated as

$\left(\frac{1}{\tau(\omega)}\right)_{\textrm{gauss}}\equiv 2\pi
U^{2}\left[f_{\textrm{gauss}}(\omega)+f_{\textrm{gauss}}(-\omega)\right].$
(42)

Fig. 11 (b) shows that this approximate Gaussian agrees with the exact BSE
without $\mathbf{k}$ scattering rate to very good approximation. This explains
the origin of the maximal scattering rate and why this maximum lies within the
bandwidth of the DOS not at its edge as in the BSE with $\mathbf{k}$.

##  Appendix C Connection to 2${}^{\text{nd}}$ order perturbation theory

The scattering rate of Eq. (28) is actually equivalent to second order
iterated perturbation theory (IPT) [7, 71, 72]. In IPT, the DMFT self-energy
is calculated in second order in $U$ from the impurity Green’s function $\cal
G$. Directly on the real frequency axis and in terms of the impurity spectral
function ${\cal A}(\omega)=-1/\pi\;{\rm Im}{\cal G}(\omega)$ the IPT self-
energy reads (see e.g. Eq. (22) in [73] where $\epsilon_{1}$ and
$\epsilon_{2}$ are exchanged)

$\displaystyle\Sigma_{R}(\omega)$ $\displaystyle=\lim_{\alpha\rightarrow
0}U^{2}\int\mathrm{d}\epsilon_{1}\mathrm{d}\epsilon_{2}\mathrm{d}\epsilon_{3}{\cal
A}(\epsilon_{1}){\cal A}(\epsilon_{2}){\cal A}(\epsilon_{3})$
$\displaystyle\\!\\!\\!\times\frac{f_{\rm FD}(\\!-\\!\epsilon_{1}\\!)f_{\rm
FD}(\epsilon_{2}\\!)f_{\rm FD}(\epsilon_{3}\\!)\\!+\\!f_{\rm
FD}(\epsilon_{1}\\!)f_{\rm\rm FD}(\\!-\\!\epsilon_{2}\\!)f_{\rm
FD}(\\!-\\!\epsilon_{3}\\!)}{\omega+i\alpha+\epsilon_{1}-\epsilon_{2}-\epsilon_{3}}.$

With $f_{\rm FD}(-\epsilon)=1-f_{\rm FD}(\epsilon)$ and
$\lim_{\alpha\rightarrow 0}{\rm
Im}1/(\omega+\epsilon_{1}-\epsilon_{2}-\epsilon_{3})=-\pi\delta(\omega+\epsilon_{1}-\epsilon_{2}-\epsilon_{3})$
this yields Eq. (28) or Eq. (14) if we replace ${\cal A}(\epsilon)$ by
${A}_{0}(\epsilon)$ which is possible to lowest order in $U$ or the first
iteration of the IPT. Through the DMFT self-consistency condition [74, 9]
${\cal G}(\omega)^{-1}=G(\omega)^{-1}+\Sigma(\omega)$, ${\cal A}(\epsilon)$ is
updated in subsequent iterations.

##  Appendix D Simplifications for the two-orbital case due to particle-hole
symmetry

In this Section, we discuss some simplifications that are possible due to
particle-hole symmetry and a density-density interaction $U$ which is the same
for all orbitals. Indeed, the scattering rate can be actually calculated from
a single band in BSE. The reason for this is as follows: For the BSE with
momentum conservation the scattering rate in the upper band reads

$\begin{split}&\frac{1}{\tau_{B}(\mathbb{k}_{0})}=6\pi
U^{2}\frac{1}{{V_{BZ}}^{2}}\int\mathrm{d}^{2}k_{1}\mathrm{d}^{2}k_{2}\mathrm{d}^{2}k_{3}\Big{[}\\\
&\times\delta(\epsilon_{B}(\mathbb{k}_{0})+\epsilon_{B}(\mathbb{k}_{1})-\epsilon_{B}(\mathbb{k}_{2})-\epsilon_{B}(\mathbb{k}_{3}))\\\
&\times\sum_{\mathbb{G}}\delta(\mathbb{k}_{0}+\mathbb{k}_{1}-\mathbb{k}_{2}-\mathbb{k}_{3}+\mathbb{G})\\\
&\times\Big{(}(1-f_{\textrm{FD}}(\epsilon_{B}(\mathbb{k}_{1})))f_{\textrm{FD}}(\epsilon_{B}(\mathbb{k}_{2}))f_{\textrm{FD}}(\epsilon_{B}(\mathbb{k}_{3}))\\\
&+f_{\textrm{FD}}(\epsilon_{B}(\mathbb{k}_{1}))(1-f_{\textrm{FD}}(\epsilon_{B}(\mathbb{k}_{2})))(1-f_{\textrm{FD}}(\epsilon_{B}(\mathbb{k}_{3})))\Big{)}\Big{]},\end{split}$
(44)

and for the BSE without $\mathbb{k}$ case it reads

$\begin{split}&\frac{1}{\tau_{B}(\epsilon_{0})}=6\pi
U^{2}\int\mathrm{d}\epsilon_{1}\mathrm{d}\epsilon_{2}\mathrm{d}\epsilon_{3}\Big{[}\\\
&\quad\times\delta(\epsilon_{0}+\epsilon_{1}-\epsilon_{2}-\epsilon_{3})A_{0}^{B}(\epsilon_{1})A_{0}^{B}(\epsilon_{2})A_{0}^{B}(\epsilon_{3})\\\
&\quad\quad\quad\times\Big{(}(1-f_{\textrm{FD}}(\epsilon_{1}))f_{\textrm{FD}}(\epsilon_{2})f_{\textrm{FD}}(\epsilon_{3})\\\
&\quad\quad\quad+f_{\textrm{FD}}(\epsilon_{1})(1-f_{\textrm{FD}}(\epsilon_{2}))(1-f_{\textrm{FD}}(\epsilon_{3}))\Big{)}\Big{]}.\end{split}$
(45)

Due to particle-hole symmetry it further holds that
$1/\tau^{A}(-\omega)=1/\tau^{B}(\omega)\equiv 1/\tau(\omega)$. The
multiplicative factor of 3 compared to the one-band case (Section III) in the
scattering amplitude reflects the different scattering processes an electron
in the upper band may perform: an electron with a certain spin $\sigma$ in
band $B$ may scatter with an an electron $B\bar{\sigma}$, $A\sigma$ and
$A\bar{\sigma}$. Since the density-density interaction does not allow for
spin-flips and pair-hopping nor impact excitation which would require an
interaction of the form
$c^{\dagger}_{iB\bar{\sigma}}c^{\phantom{dagger}}_{iA\bar{\sigma}}c^{\dagger}_{iB\sigma}c^{\phantom{dagger}}_{iB\sigma}$
nor Auger excitations, there are no further allowed processes to be taken into
account. Since the interaction between the bands is the same as within the
bands ($U=V$), all scattering processes have the same scattering amplitude
$\propto 2\pi U^{2}$ ($\propto 2\pi\frac{1}{{V_{BZ}}^{2}}U^{2}$), eventually
leading to $\propto 3\times 2\pi U^{2}=6\pi U^{2}$ ($\propto
6\pi\frac{1}{{V_{BZ}}^{2}}U^{2}$).

##  Appendix E Band insulator with renormalized bands

Figure 12: Scattering rate normalized by $U^{2}$ for the two-band case with
$U=4$. Same as Fig. 4 but using the interacting DMFT spectral density
$A(\omega)$ instead of $A_{0}(\omega)$ in Eqs. (22), (23) for the BSE without
$\mathbf{k}$. For the BSE with momentum conservation the dispersion and gap
has been adapted to the DMFT as well. Using the interacting spectral function
leads to a better agreement with DMFT.

As already mentioned in Section IV the deviation of BSE without $\mathbb{k}$
and DMFT can be reduced by using the interacting spectral density instead of
the non-interacting DOS [$A_{0}(\omega)\to A(\omega)$ in Eq. (45)]. For the
theoretical justification and background of this procedure, see Ref. [15].

For the band insulator, the main difference between $A_{0}(\omega)$ and
$A(\omega)$ is that thermal excitation across the gap reduce the difference in
Hartree energy and hence the band gap. Moreover, with more thermal excitations
there are more particles an electron or hole can scatter with. Hence the
scattering rate is enhanced, which in turn leads to broadening effects in
$A(\omega)$, visible in Fig. 3 at the band edges and the van Hove singularity.

In order to make all methods comparable, we are also using a renormalized band
for the BSE with momentum conservation. For that purpose we calculate the
momentum-dependent spectral density $A_{\mathbb{k}}(\omega)$ and then extract
for each given momentum the corresponding energy where the spectral density
has its maximum. In this way one can obtain a renormalized dispersion relation
which we then use for the calculation of the scattering rate.

Fig. 12 shows the thus obtained BSE results at $U=4$ with and without momentum
conservation; the DMFT result is the same as in Fig. 4. Using the interacting
spectral function and correspondingly renormalized bandstructure, reduces the
scattering rate in the middle of the upper band at $\omega\sim 6$. In contrast
the scattering rate at small $\omega\sim 0$ is enhanced at high temperatures.
Here, the interacting spectrum has states in the (pseudo)gap. Overall, using
the interacting DOS as a starting point the BSE can explain most changes of
the $U=4$ DMFT scattering rate compared to $U=2$.

## References

  * Grioni [2001] M. Grioni, J. Electron Spectrosc. Relat. Phenom.. 117-118 (2001), special issue on correlated electron systems.
  * Damascelli  _et al._ [2003] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. 75, 473 (2003).
  * Drude [1900a] P. Drude, Annalen der Physik 308, 369 (1900a).
  * Drude [1900b] P. Drude, Annalen der Physik 306, 566 (1900b).
  * Note [1] Both in dynamical mean-field theory and Boltzmann there are no vertex corrections to the optical conductivity.
  * Metzner and Vollhardt [1989] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989).
  * Georges and Kotliar [1992] A. Georges and G. Kotliar, Phys. Rev. B 45, 6479 (1992).
  * Jarrell [1992] M. Jarrell, Phys. Rev. Lett. 69, 168 (1992).
  * Georges  _et al._ [1996] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996).
  * Boltzmann [1872] L. Boltzmann, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften. 66, 275 (1872).
  * Snoke [2007] D. W. Snoke,  _Solid State Physics Essential Concepts_ , 1st ed. (Pearson Education, 2007).
  * Ziman [1960] J. Ziman,  _Electrons and phonons: the theory of transport phenomena in solids_ (Oxford university press, 1960).
  * Chambers [1990] R. G. Chambers,  _Electrons in Metals and Semiconductors_ (Chapman and Hall, 1990).
  * Note [2] For an overview of the Mott-Hubbard transition and the physics of the Mott insulator, see [55].
  * Wais  _et al._ [2018] M. Wais, M. Eckstein, R. Fischer, P. Werner, M. Battiato, and K. Held, Phys. Rev. B 98, 134312 (2018).
  * Abrikosov  _et al._ [1975] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski,  _Methods of Quantum Field Theory in Statistical Physics_ (Dover, New York, 1975).
  * Held [2007] K. Held, Advances in Physics 56, 829 (2007).
  * Nolting [2015] W. Nolting,  _Grundkurs Theoretische Physik 7: Viel-Teilchen-Theorie_ (Springer Spektrum, 2015).
  * Matsubara [1955] T. Matsubara, Prog. Theor. Phys. 14, 351 (1955).
  * Gull  _et al._ [2011] E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Rev. Mod. Phys. 83, 349 (2011).
  * Kaufmann  _et al._ [2019] J. Kaufmann, P. Gunacker, A. Kowalski, G. Sangiovanni, and K. Held, Phys. Rev. B 100, 075119 (2019).
  * Parragh  _et al._ [2012a] N. Parragh, A. Toschi, K. Held, and G. Sangiovanni, Phys. Rev. B 86, 155158 (2012a).
  * Parragh  _et al._ [2012b] N. Parragh, A. Toschi, K. Held, and G. Sangiovanni, in [22], p. 155158.
  * Jarrell and Gubernatis [1996] M. Jarrell and J. E. Gubernatis, Physics Reports 269, 133 (1996).
  * Geffroy  _et al._ [2019] D. Geffroy, J. Kaufmann, A. Hariki, P. Gunacker, A. Hausoel, and J. Kuneš, Phys. Rev. Lett. 122, 127601 (2019).
  * Kaufmann [2020] J. Kaufmann, ana_cont: Package for analytic continuation of many-body green’s functions, https://github.com/josefkaufmann/ana_cont (2020).
  * Wais  _et al._ [2020] M. Wais, K. Held, and M. Battiato, arXiv:2004.02683 (2020).
  * Ono [2018] S. Ono, Phys. Rev. B 97, 054310 (2018).
  * Ono and Suemoto [2020] S. Ono and T. Suemoto, Phys. Rev. B 102, 024308 (2020).
  * Katanin  _et al._ [2012] A. A. Katanin, V. Y. Irkhin, and P. A. Igoshev,  _Model’nye podhody k magnetizmu dvumernyh zonnyh sistem_ (Fizmatlit, Moscow, 2012).
  * Note [3] The DMFT spectral densities were calculated as $-\mathrm{Im}G_{R}(\omega)/\pi$ with  $G_{R}(\omega)=\intop\ilimits@_{-\infty}^{\infty}dx\frac{A_{0}(x)}{\omega-x-\Sigma_{R}(\omega)}$ (46) after analytical continuation of the self-energy. This allows for resolving features like sharp peaks in the spectral density that would be smeared out by direct analytic continuation of the local Green’s function in Matsubara frequencies.
  * Vilk and Tremblay [1997] Y. M. Vilk and A.-M. S. Tremblay, J. Phys. I France 7, 1309 (1997).
  * Norman  _et al._ [1998] M. R. Norman, H. Ding, M. Randeria, J. C. Campuzano, T. Yokoya, T. Takeuchi, T. Takahashi, T. Mochiku, K. Kadowaki, P. Guptasarma, and D. G. Hinks, Nature 392, 157 (1998).
  * Timusk and Statt [1999] T. Timusk and B. Statt, Reports on Progress in Physics 62, 61 (1999).
  * Keimer  _et al._ [2015] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, Nature 518, 179 (2015).
  * Lee  _et al._ [2006] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006).
  * Sordi  _et al._ [2012] G. Sordi, P. Sémon, K. Haule, and A.-M. S. Tremblay, Phys. Rev. Lett. 108, 216401 (2012).
  * Gunnarsson  _et al._ [2015] O. Gunnarsson, T. Schäfer, J. P. F. LeBlanc, E. Gull, J. Merino, G. Sangiovanni, G. Rohringer, and A. Toschi, Phys. Rev. Lett. 114, 236402 (2015).
  * Wu  _et al._ [2018] W. Wu, M. S. Scheurer, S. Chatterjee, S. Sachdev, A. Georges, and M. Ferrero, Phys. Rev. X 8, 021048 (2018).
  * Sadovskii  _et al._ [2005] M. V. Sadovskii, I. A. Nekrasov, E. Z. Kuchinskii, T. Pruschke, and V. I. Anisimov, Phys. Rev. B 72, 155105 (2005).
  * Zhang and Imada [2007] Y. Z. Zhang and M. Imada, Phys. Rev. B 76, 045108 (2007).
  * Katanin  _et al._ [2009] A. A. Katanin, A. Toschi, and K. Held, Phys. Rev. B 80, 075104 (2009).
  * Gull  _et al._ [2013] E. Gull, O. Parcollet, and A. J. Millis, Phys. Rev. Lett. 110, 216405 (2013).
  * Schäfer  _et al._ [2015] T. Schäfer, F. Geles, D. Rost, G. Rohringer, E. Arrigoni, K. Held, N. Blümer, M. Aichhorn, and A. Toschi, Phys. Rev. B 91, 125109 (2015).
  * Rohringer  _et al._ [2018] G. Rohringer, H. Hafermann, A. Toschi, A. A. Katanin, A. E. Antipov, M. I. Katsnelson, A. I. Lichtenstein, A. N. Rubtsov, and K. Held, Rev. Mod. Phys. 90, 025003 (2018).
  * Werner  _et al._ [2014] P. Werner, K. Held, and M. Eckstein, Phys. Rev. B 90, 235102 (2014).
  * Sorantin  _et al._ [2018] M. E. Sorantin, A. Dorda, K. Held, and E. Arrigoni, Phys. Rev. B 97, 115113 (2018).
  * Maislinger and Evertz [2020] F. Maislinger and H. G. Evertz, preprint , arXiv:2007.16201 (2020).
  * Kauch  _et al._ [2020] A. Kauch, P. Worm, P. Prauhart, M. Innerberger, C. Watzenböck, and K. Held, preprint , arXiv:2007.16035 (2020).
  * Kanamori [1963] J. Kanamori, Progress of Theoretical Physics 30, 275 (1963).
  * Slater [1960] J. C. Slater,  _Quantum Theory of Atomic Structure_ (McGraw-Hill, New York, 1960).
  * Griffith [1971] J. S. Griffith,  _The Theory of Transition-MetalIons_ (Cambridge University Press, Cambridge, 1971).
  * Ribic  _et al._ [2014] T. Ribic, E. Assmann, A. Tóth, and K. Held, Phys. Rev. B 90, 165105 (2014).
  * Bünemann and Gebhard [2017] J. Bünemann and F. Gebhard, Journal of Physics: Condensed Matter 29, 165601 (2017).
  * Gebhard [1997] F. Gebhard,  _The Mott Metal-insulator transition_ (Springer-Verlag (Berlin), 1997).
  * Note [4] The calculation of scattering in this paper is still possible within equilibrium DMFT theory; whereas the non-equilibrium processes of [15] required the non-equilibrium DMFT [75, 76].
  * Note [5] As we do not have a linear quasiparticle renormalization in the self-energy, we plot $1/\tau(\omega)=2{\rm Im}\Sigma(\omega)$; $Z=1$ in Eq. (10).
  * Mo  _et al._ [2004] S.-K. Mo, H.-D. Kim, J. W. Allen, G.-H. Gweon, J. D. Denlinger, J.-H. Park, A. Sekiyama, A. Yamasaki, S. Suga, P. Metcalf, and K. Held, Phys. Rev. Lett. 93, 076404 (2004).
  * Karski  _et al._ [2008] M. Karski, C. Raas, and G. S. Uhrig, Phys. Rev. B 77, 075116 (2008).
  * Ganahl  _et al._ [2015] M. Ganahl, M. Aichhorn, H. G. Evertz, P. Thunström, K. Held, and F. Verstraete, Phys. Rev. B 92, 155132 (2015).
  * Lee  _et al._ [2017] S.-S. B. Lee, J. von Delft, and A. Weichselbaum, Phys. Rev. Lett. 119, 236402 (2017).
  * Granath and Schött [2014] M. Granath and J. Schött, Phys. Rev. B 90, 235129 (2014).
  * Nishimoto  _et al._ [2004] S. Nishimoto, F. Gebhard, and E. Jeckelmann, Journal of Physics: Condensed Matter 16, 7063 (2004).
  * Kalinowski and Gebhard [2002] E. Kalinowski and F. Gebhard, J. Low Temperature Phys. 126, 979 (2002).
  * Hubbard [1964] J. Hubbard, Proc R. Soc. London 281, 401 (1964).
  * van Dongen and Leinung [1997] P. G. J. van Dongen and C. Leinung, Annalen der Physik 509, 45 (1997).
  * Freericks and Zlatić [2003] J. K. Freericks and V. Zlatić, Rev. Mod. Phys. 75, 1333 (2003).
  * Manousakis [2010] E. Manousakis, Phys. Rev. B 82, 125109 (2010).
  * Assmann  _et al._ [2013] E. Assmann, P. Blaha, R. Laskowski, K. Held, S. Okamoto, and G. Sangiovanni, Phys. Rev. Lett. 110, 078701 (2013).
  * Montgomery and Runger [2018] D. C. Montgomery and G. C. Runger,  _Applied Statistics and Probability for Engineers_ , 7th ed. (Wiley, 2018).
  * Kajueter [1996] H. Kajueter, Ph.D. thesis, Rutgers University (1996).
  * Kajueter and Kotliar [1996] H. Kajueter and G. Kotliar, Phys. Rev. Lett. 77, 131 (1996).
  * Potthoff  _et al._ [1997] M. Potthoff, T. Wegner, and W. Nolting, Phys. Rev. B 55, 16132 (1997).
  * Georges and Krauth [1992] A. Georges and W. Krauth, Phys. Rev. Lett. 69, 1240 (1992).
  * Freericks  _et al._ [2006] J. K. Freericks, V. M. Turkowski, and V. Zlatić, Phys. Rev. Lett. 97, 266408 (2006).
  * Aoki  _et al._ [2014] H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, and P. Werner, Rev. Mod. Phys. 86, 779 (2014).