# Polaron spectra and edge singularities for correlated flat bands

Dimitri Pimenov<EMAIL_ADDRESS>Department of Physics, Cornell University,
Ithaca, New York 14853, USA

###### Abstract

Single- and two-particle spectra of a single immobile impurity immersed in a
fermionic bath can be computed exactly and are characterized by divergent
power laws (edge singularities). Here, we present the leading lattice
correction to this canonical problem, by embedding both impurity and bath
fermions in bands with non-vanishing Bloch band geometry, with the impurity
band being flat. By analyzing generic Feynman diagrams, we pinpoint how the
band geometry reduces the effective interaction which enters the power laws;
we find that for weak lattice effects or small Fermi momenta, the leading
correction is proportional to the Fermi energy times the sum of the quantum
metrics of the bands. When only the bath fermion geometry is important, the
results can be extended to large Fermi momenta and strong lattice effects. We
numerically illustrate our results on the Lieb lattice and draw connections to
ultracold gas experiments.

##  I Introduction

A rare example of an analytically solvable quantum problem is the description
of a single immobile impurity embedded in a Fermi sea [1, 2, 3, 4]. Despite
many-body appearances, this problem can be formulated in single-particle
language [5, 6]: if the impurity is structureless, it acts as a time-dependent
scattering potential for the Fermi sea electrons, inducing phase shifts of the
single-particle orbitals. These lead to a vanishing overlap of the Fermi sea
ground state with and without impurity – a phenomenon known as Anderson
Orthogonality catastrophe [7]. The resulting impurity spectra feature
divergent power laws (edge singularities), whose exponents can be expressed
via the phase shifts at the Fermi level.

Due to its exact solvability, the edge singularity setup can be a starting
point to explore related problems which are undeniably many-body. One possible
direction is to consider heavy but mobile impurities. The finite mass adds
recoil, cutting off the singularities in the problem [8, 9, 10]. For equal
masses, one arrives at the so-called “Fermi-polaron” problem which has gained
enormous traction in the context of ultracold gases and cavity semiconductor
experiments in the last years [11, 12, 13].

Another interesting route is the modification of edge singularities due to
lattice effects. In previous studies of edge singularities for lattice models,
only the Fermi sea fermions were subject to specific lattice or trap
enviroment [14, 15, 16]. Here, we propose a new variant of the problem: both
bath fermions and impurity are placed in bands with non-trivial band geometry.
Such a situation can for instance arise if the impurity is a single heavy hole
created by photoexcitation out of a flat band. We aim to determine the
universal leading modification of the edge singularities due to (weak) lattice
effects, independent of the concrete lattice of choice.

Non-trivial flat band systems form an ideal breeding ground for strong-
correlation physics, since interactions dominate over kinetic effects. Typical
examples are given by Landau Levels and modern Moiré materials like TBG [17,
18]. Another venue for generation of flat bands are optical lattices hosting
ultracold atoms, where for instance the Lieb lattice has been realized [19,
20, 21, 22]. While all these systems are most interesting at generic filling,
they are typically inaccessible by controlled theory. The setup we are
considering here is simpler: kinetic effects are still quenched, but the
single-hole limit provides a way to get the interactions under control.

Our theoretical starting point is a multiband model with two active bands, a
flat one (termed $f$ band), which is initially filled, and a dispersive one
(termed $c$ band) filled up to the chemical potential $\mu$. We assume a
short-ranged interaction between $f$ and $c$ particles, which contains
overlaps of Bloch functions (form factors) as a result of the lattice
structure. Our goal is to compute single-hole correlation functions
$A(\nu)\sim\text{Im}\braket{f^{\dagger}f}\\!(\nu)$ and
$\chi(\nu)\sim\text{Im}\braket{cf^{\dagger}fc^{\dagger}}\\!(\nu)$, which
describe photoemission spectra (RF spectra in ultracold atom experiments) or
interband absorption spectra, respectively.

In the standard edge singularity scenario, the spectra scale as
$A(\nu)\sim\nu^{2\alpha^{2}-1},\chi(\nu)\sim\nu^{-2\alpha}$, where $\alpha$ is
the dimensionless $c$-$f$ interaction. On the lattice, the form factors come
into play. To find their leading effect, we need to evaluate the Bloch
overlaps for generic diagrams. This can be achieved exactly if only the
$c$-band geometry is of importance, for general band fillings. If the $f$-band
geometry is important as well, the single-particle character of the problem is
lost in general: For instance, a finite effective mass is generated for the
$f$-hole. On the other hand, the problem can be controlled if the Fermi
momentum $k_{F}$ is small, equivalent to a weak lattice effect: as we show by
diagrammatic analysis, in this case the dominant effect is to reduce the
interaction $\alpha$, while other effects, such as $f$-band mass generation,
are subleading. The interaction correction to $\alpha$ scales as
$k_{F}^{2}\text{tr}(g^{f}+g^{c})$, where $g^{c},g^{f}$ are the quantum metrics
of the respective bands at the $c$-band minimum. If the photocreated hole has
a momentum $|\mathbf{Q}|\gg k_{F}$, the $f$-band metric must be evaluated at
this momentum. The reduced interaction can be interpreted in terms of the
minimal real-space spread of the wavefunctions, which cannot be localized
completely due to the Bloch band geometry: therefore, the effective scattering
potential seen by the $c$ fermions acquires a finite range, reducing the phase
shift at the Fermi level. We check these results numerically for fermions
moving in a weakly doped Lieb lattice, which contains a flat and dispersive
band with a non-trivial metrics.

While Moiré materials can in principle host the lattice edge singularity
physics, experimental observation might be challenging due to the insufficient
energy resolution of spectral measurements to date. On the other hand,
ultracold gas systems provide both a platform to realize topological flat
bands and also come with a well-developed experimental toolbox to resolve
polaronic spectra. In particular, we expect the discussed geometrical effects
to be observable in RF spectroscopy [23, 24].

The remainder of this article is structured as follows: in Sec. II.1, we
introduce the general Hamiltonian, and the Lieb lattice in Sec. II.2. In Sec.
III, we recapitulate the standard continuum edge singularities. In Sec. IV, we
introduce a band geometry for the $c$-band, study its impact on photoemission
and absorption-type spectra, and introduce a perturbative formulation of
lattice effects in terms of the quantum metric. In Sec. V, we apply this
perturbative treatment to the case where both $c$ and $f$ bands have a non-
vanishing band geometry. In Sec. VI we discuss the experimental relevance and
limitations of our results, and close in Sec. VII by providing a summary and
an outlook. Technical details are relegated to Appendices.

##  II Setup

###  II.1 General Hamiltonian

Consider a generic $d$-dimensional multi-band model at $T=0$ with two active
bands: a flat valence band ($f$-band) and dispersive conduction band
($c$-band). In the band basis, the kinetic Hamiltonian reads:

$\displaystyle
H_{\text{kin}}=\sum_{{\mathbf{k}}}-E_{0}f^{\dagger}_{\mathbf{k}}f_{\mathbf{k}}+\sum_{\mathbf{k}}\epsilon_{\mathbf{k}}c^{\dagger}_{\mathbf{k}}c_{\mathbf{k}}\
.$ (1)

We use units such that $\hbar,e=1$, and suppress spins – they only lead to
factors of two which we will reinstall where needed. Energies are measured
from the bottom of the conduction band, and we assume that the band gap
$E_{0}$ is the largest scale in the problem, $E_{0}\rightarrow\infty$. As
sketched in Fig. 1, the dispersive band is occupied up to the chemical
potential $\mu$ 111For results on Orthogonality Catastrophe for band
insulators, see Ref. [63]). The flat band is filled as well, but these filled
$f$ states are inert. We will study processes where a single hole is injected
in the $f$ band. This can e.g. be achieved via photoexcitation in one of two
ways: either, a high-energy light pulse is applied which ejects an
$f$-particle from the system [photoemission, Fig. 1(a)] or the $f$-particle is
lifted into the $c$-band [interband absorption, Fig. 1(b)] by irridating light
with an energy $\gtrsim(E_{0}+\mu)$.

Figure 1: Hole creation by photoexcitation. Filled (empty) circles indicate
holes (electrons), the light pulse is indicated by a wiggly orange line. (a)
Photoemission, where the electron is ejected from the system (shown in gray)
(b) Interband absorption, which creates a hole with momentum $\geq k_{F}$.

Now we include interactions. A general interaction term involving $c$,$f$
fermions can be written down as

$\displaystyle H_{\text{int}}$ $\displaystyle=\frac{1}{2}\int
d^{2}rd^{2}r^{\prime}\Psi^{\dagger}({\mathbf{r}})\Psi^{\dagger}({\mathbf{r}}^{\prime})V({\mathbf{r}}-{\mathbf{r}}^{\prime})\Psi({\mathbf{r}}^{\prime})\Psi({\mathbf{r}})\
,$ (2)

with field operators

$\displaystyle\Psi({\mathbf{r}})=\frac{1}{\sqrt{\Omega}}\sum_{{\mathbf{k}},n\in\\{{f,c}\\}}u_{n,{\mathbf{k}}}({\mathbf{r}})e^{i{\mathbf{k}}\cdot{\mathbf{r}}}a_{n,{\mathbf{k}}}\
.$ (3)

Here, $a_{f,{\mathbf{k}}}=f_{\mathbf{k}},\ a_{c,{\mathbf{k}}}=c_{\mathbf{k}}$,
$u_{n,{\mathbf{k}}}({\mathbf{r}})$ is the normalized cell-periodic Bloch
function, and $\Omega$ is the system volume. In momentum space, the
interaction becomes

$\displaystyle H_{\text{int}}=$
$\displaystyle\frac{1}{2\Omega}\sum_{\begin{subarray}{c}n_{1},n_{2},n_{3},n_{4}\\\
{\mathbf{k}},{\mathbf{k}}^{\prime},{\mathbf{q}},{\mathbf{K}}\end{subarray}}V_{{\mathbf{q}}+{\mathbf{K}}}a^{\dagger}_{n_{1},{\mathbf{k}}+{\mathbf{q}}}a^{\dagger}_{n_{2},{\mathbf{k}}^{\prime}-{\mathbf{q}}}a_{n_{3},{\mathbf{k}}^{\prime}}a_{n_{4},{\mathbf{k}}}$
$\displaystyle\times$
$\displaystyle\braket{n_{1},{\mathbf{k}}+{\mathbf{q}}}{n_{4},{\mathbf{k}}}_{\mathbf{K}}\braket{n_{2},{\mathbf{k}}^{\prime}-{\mathbf{q}}}{n_{3},{\mathbf{k}}^{\prime}}_{-{\mathbf{K}}},$
(4)

where

$\displaystyle\braket{n_{i},{\mathbf{k}}}{n_{j},{\mathbf{p}}}_{\mathbf{K}}=$
(5) $\displaystyle\frac{1}{\text{unit cell vol.\ }}\int_{\text{u.c.}}d^{d}r\
\overline{{u_{n_{i},{\mathbf{k}}}({\mathbf{r}})}}u_{n_{j},{\mathbf{p}}}({\mathbf{r}})\exp(-i{\mathbf{K}}\cdot{\mathbf{r}})\
.$

$V_{{\mathbf{q}}+{\mathbf{K}}}$ is the interaction matrix element in momentum
space. In Eq. (4), ${\mathbf{k}},{\mathbf{k}}^{\prime}$ and ${\mathbf{q}}$ are
restricted to the first Brillouin zone, while ${\mathbf{K}}$ is a reciprocal
lattice vector.

The band geometry is encoded in the Bloch factor overlaps in Eq. (4). To
isolate their effects on the edge singularities, we will simplify the
interaction as

$\displaystyle V_{{\mathbf{q}}+{\mathbf{K}}}\simeq
V_{0}\times\delta_{{\mathbf{K}},0}\ ,$ (6)

dropping the summation over ${\mathbf{K}}$. That is, we assume that the
interaction in reciprocal space is essentially constant for the momentum
transfers of interest (of order $k_{F}$), but decays quickly enough for
momentum transfers on the order of a reciprocal lattice vector, which allows
to neglect Umklapp processes. We assume that there is a clear separation
between these scales, which applies in a weak doping limit. In real space,
this implies that the interaction is constant on the scale of a unit cell, but
decays strongly on the much larger scale $1/k_{F}$. Note that this excludes
interactions with an explicit sublattice structure. For a screened Coulomb-
like attraction between the $f$-hole and the $c$-electrons, $V_{0}>0$ 222 A
discussion of edge singularities in the context of long-ranged Coulomb
interactions can be found in Ref. [64] . To summarize, the most general
interaction studied in this work reads

$\displaystyle H_{\text{int}}=$
$\displaystyle\frac{1}{2\Omega}\sum_{\begin{subarray}{c}n_{1},n_{2},n_{3},n_{4}\\\
{\mathbf{k}},{\mathbf{k}}^{\prime},{\mathbf{q}}\end{subarray}}V_{0}a^{\dagger}_{n_{1},{\mathbf{k}}}a^{\dagger}_{n_{2},{\mathbf{k}}^{\prime}}a_{n_{3},{\mathbf{k}}^{\prime}-{\mathbf{q}}}a_{n_{4},{\mathbf{k}}+{\mathbf{q}}}$
$\displaystyle\times$
$\displaystyle\braket{n_{1},{\mathbf{k}}}{n_{4},{\mathbf{k}}+{\mathbf{q}}}\braket{n_{2},{\mathbf{k}}^{\prime}}{n_{3},{\mathbf{k}}^{\prime}-{\mathbf{q}}}\
.$ (7)

When a hole is created in a photoexcitation process [Fig. 1(a)], the measured
spectra are determined by the correlation functions involving $c,f$ operators.
For the photoemission spectrum $A_{\mathbf{Q}}(\omega)$ (which can essentially
be measured via RF spectroscopy in ultracold gas experiments [23, 24]), the
relevant correlation function is the propagator:

$\displaystyle F_{\mathbf{Q}}(t)$
$\displaystyle=-i\braket{0}{\hat{T}\left\\{f_{\mathbf{Q}}(t)f^{\dagger}_{\mathbf{Q}}(0)\right\\}}{0}$
(8) $\displaystyle A_{\mathbf{Q}}(\omega)$
$\displaystyle=\text{Im}\left[F_{\mathbf{Q}}(-\omega)\right]\ ,$

where $\hat{T}$ is a time-ordering operator. In general, $\ket{0}$ is the
interacting ground state; however, in the limit $E_{0}\rightarrow\infty$ we
are interested in, it is equivalent to the non-interacting ground state with a
filled $f$-band. This implies that $F_{\mathbf{Q}}(t)\sim\theta(-t)$ is purely
advanced when $E_{0}\rightarrow\infty$.

For the interband absorption spectrum $\chi(\omega)$ [Fig. 1(b)], we need the
interband current-current correlation function

$\displaystyle\Pi(t)=(-i)\times$
$\displaystyle\sum_{{\mathbf{k}}_{1},{\mathbf{k}}_{2},\eta}\\!J_{cf}^{\eta}({\mathbf{k}}_{2})J_{fc}^{\eta}({\mathbf{k}}_{1})\braket{0}{\hat{T}\left\\{f_{{\mathbf{k}}_{2}}^{\dagger}(t)c_{{\mathbf{k}}_{2}}(t)c_{{\mathbf{k}}_{1}}^{\dagger}(0)f_{{\mathbf{k}}_{1}}(0)\right\\}}{0}$
$\displaystyle\chi(\omega)=-\text{Im}\left[\Pi(\omega)\right],$ (9)

where the matrix elements of the interband operator can be expressed as [27]

$\displaystyle J_{cf}^{\eta}({\mathbf{k}})$
$\displaystyle=(E_{0}+\epsilon_{\mathbf{k}})\braket{c,{\mathbf{k}}}{\partial_{\eta}}{f,{\mathbf{k}}},\quad\partial_{\eta}\equiv\partial_{{\mathbf{k}}_{\eta}}$
(10) $\displaystyle\simeq
E_{0}\braket{c,{\mathbf{k}}}{\partial_{\eta}}{f,{\mathbf{k}}},$

and the last approximation holds for large $E_{0}$.

As pointed out in Ref. [27], care needs to be taken when evaluating optical
response for a set of active bands, since interband current matrix elements
scale with the band gap, see Eq. (10), and higher “passive” bands may
therefore contribute as well. In our case, processes involving active bands
lead to logarithmic singularities when the external frequency is close to a
specific threshold energy. For off-resonant higher-band contributions, such
singularities should not appear, and we therefore neglect them in the
following.

###  II.2 Exemplary tight binding model: Lieb Lattice

To illustrate our results on a concrete tight binding model, we will consider
the two-dimensional Lieb lattice (see, e.g., Ref. [28]), which has already
been realized with optical lattices [19, 20, 21, 22]. The kinetic part of the
Hamiltonian is given by

$\displaystyle
H_{\text{Lieb}}=-t\sum_{\mathbf{R},s=\pm}a^{\dagger}_{2,\mathbf{R}+s\hat{\mathbf{x}}}a_{1,\mathbf{R}}+a^{\dagger}_{3,\mathbf{R}+s\hat{\mathbf{y}}}a_{1,\mathbf{R}}$
(11)
$\displaystyle+it^{\prime}\sum_{\mathbf{R},s=\pm}a^{\dagger}_{3,\mathbf{R}+s\hat{\mathbf{y}}}a_{2,\mathbf{R}+s\hat{\mathbf{x}}}+a^{\dagger}_{2,\mathbf{R}-s\hat{\mathbf{x}}}a_{3,\mathbf{R}+s\hat{\mathbf{y}}}+\text{h.c.},$

where $a_{1},a_{2},a_{3}$ denote operators on sublattices $1,2,3$ as indicated
in Fig. 2(a). This lattice is characterized by an exactly flat central band,
($f$ band), and two dispersive bands, with the bandgap equal to $t^{\prime}$
[Fig. 2(b)]. We consider a doped upper band, calling it the $c$-band. The
minimum of the $c$-band is located at the $M$-point, which we will choose as
momentum space origin for convenience. For $t^{\prime}>0$, the Lieb lattice
bands are characterized by Chern numbers $(-1,0,1)$ [28]. While the $f$ band
has a vanishing Chern number, its Bloch functions $u_{f}({\mathbf{k}})$ are
strongly varying.

Figure 2: Lieb lattice. (a) Lattice structure; sublattices are indicated by
coloring. (b) Band structure for $t=1,t^{\prime}=0.6$ (bottom band is not
shown). Momenta are shifted by $(\pi,\pi)$.

##  III Recap: Edge singularities for trivial bands

To set the stage for evaluation of the spectra
$A_{\mathbf{Q}}(\omega),\chi(\omega)$ in the lattice case, we recapitulate the
standard continuum solution of the edge singularity problem [1, 2, 3, 4, 5, 6,
11]. It assumes that the hole is featureless and the momentum-dependence of
hole operators can be erased, $f_{\mathbf{k}}\rightarrow f$. For the
conduction band, the form factors are usually suppressed, and the interaction
$H_{\text{int}}$ is approximated as

$\displaystyle H_{\text{int}}^{(0)}=$
$\displaystyle\frac{1}{\Omega}\sum_{{\mathbf{k}},{\mathbf{q}}}V_{0}c^{\dagger}_{{\mathbf{k}}+{\mathbf{q}}}c_{\mathbf{k}}f^{\dagger}f\
.$ (12)

Only $c$-$f$ interactions are retained in Eq. (12). In particular, interaction
terms involving $f$-fermions only are neglected, assuming that they lead to a
renormalization of the $f$-band that can be absorbed in the bare Hamiltonian.
Furthermore, one typically assumes rotational invariance. The problem is
characterized by an effective dimensionless coupling constant

$\displaystyle\alpha\equiv\rho V_{0}\ ,$ (13)

where $\rho$ is the density of states at the Fermi level. In this idealized
situation, the spectra feature divergent power laws (“edge singularities”).
For $\alpha\ll 1$, they read, to the leading order in $\alpha$:

$\displaystyle A(\omega)\sim\nu^{2\alpha^{2}-1}\theta(\nu),$
$\displaystyle\nu\equiv\omega-E_{0}$ (14)
$\displaystyle\chi(\omega)\sim\nu^{-2\alpha}\theta(\nu),$
$\displaystyle\nu\equiv\omega-E_{0}-\mu\ ,$ (15)

where $\nu$ is the energy measured from the respective threshold, neglecting
perturbative threshold shifts, and non-singular dependencies on $\nu$. The
factor $2$ in Eq. (14) [but not in Eq. (15)] comes from spin. Frequencies are
measured in units of a UV cutoff $\Lambda$ of order $\mu$.

These results can be derived by summing the leading logarithmically divergent
(parquet) diagrams. An essential fact is that the asymptotic behavior as
$\nu\rightarrow 0$ is dominated by scattering of $c$-fermions close to the
Fermi surface. For $A\sim\nu^{\alpha^{2}-1}$, the summation of diagrams is
most easily achieved by applying the “linked cluster” theorem [29], which
states that

$\displaystyle F(t)\sim\exp\left[\sum_{n}C_{n}(t)\right]\ ,$ (16)

$C_{n}(t)$ can be related to time-dependent Feynman diagrams. The leading
behavior, Eq. (14), derives from the the second-order diagram shown in Fig.
3(a), which is evaluated as $C_{2}(t)=-\alpha^{2}\log(\Lambda|t|)$ as
$|t|\rightarrow\infty$ 333 up to factors $\theta(-t)e^{iE_{0}t}$ which we
suppress here for brevity. The evaluation works with logarithmic accuracy,
neglecting $O(1)$ terms when compared to large logarithms. Fourier transform
of $\exp[C_{2}(t)]$ leads to the result in Eq. (14).

Figure 3: (a) Left: Leading contribution to the hole propagator $C_{2}(t)$,
which is exponentiated in the linked cluster approach. Right: Self-energy part
in frequency domain. (b) Leading parquet diagrams that determine $\Pi(\omega)$
up to second order in the interaction.

For the derivation of $\chi\sim\nu^{-2\alpha}$, a short-cut as in Eq. (16) is
not available, and the result must be computed by solving a set of coupled
Bethe-Salpeter equations [3]. The first few relevant diagrams in frequency
space are shown in Fig. 3(b); they reproduce a perturbative expansion [1]:

$\displaystyle\Pi(\nu)$
$\displaystyle=\sum_{\eta}\frac{|J^{\eta}(k_{F})^{2}|\rho}{\alpha}\left(\alpha
L-\alpha^{2}L^{2}+\frac{2}{3}\alpha^{3}L^{3}+\ldots\right)$
$\displaystyle\simeq\sum_{\eta}\frac{|J^{\eta}(k_{F})^{2}|\rho}{\alpha}\left[1-\exp(-2\alpha
L)\right],$ (17)

where $L=\log(|\nu|/\Lambda)$; $\chi(\nu)$ can be derived by restoring the
correct imaginary part of the logarithms via Kramers-Kronig relations. To the
leading order in $\alpha$, self-energy diagrams or vertex corrections do not
contribute to $\chi$.

The diagrammatic derivation produces the results Eq. (14), (15) to the leading
order in $\alpha$. For the completely stuctureless $f$-hole, a non-
perturbative solution of the problem can be found as well: Since
$f^{\dagger}f$ can only take the values $0,1$, the interaction
$H_{\text{int}}^{(0)}$ describes a scattering potential for the $c$-fermions
and the problem is quadratic. The non-perturbative solution is asymptotically
exact as $\nu\rightarrow 0$, and the exponent is determined by the phase
shifts $\delta(\alpha)$ of conduction electrons on the Fermi surface. For the
momentum-independent ($s$-wave) interaction of Eq. (12), one obtains [5, 6]:

$\displaystyle A(\nu)\sim\nu^{2(\delta/\pi)^{2}-1},$
$\displaystyle\chi(\nu)\sim\nu^{-2\delta/\pi+2(\delta/\pi)^{2}}\ ,$ (18)

where $\delta(\alpha)\sim\pi\alpha$ for $\alpha\ll 1$, such that Eq. (18)
agrees with the weak-coupling result in this limit (the exact functional form
of $\delta(\alpha)$ depends on dimensionality and UV regularization).

For a spherically-symmetric system, and an interaction which depends on the
relative angle of scattered $c$-particles on the Fermi surface only,
$V(k_{F}(\hat{e}_{{\mathbf{k}}}-\hat{e}_{{\mathbf{k}}+{\mathbf{q}}}))$, where
$\hat{e}_{\mathbf{k}},\hat{e}_{{\mathbf{k}}+{\mathbf{q}}}$ are unit vectors,
this result can be extended by expansion in an angular momentum basis.
Expressed in terms of phase shifts for partial waves $\delta_{l}$, the result
reads

$\displaystyle A(\nu)\sim\nu^{2\sum_{l}(2l+1)(\delta_{l}/\pi)^{2}-1}$ (19)
$\displaystyle\chi(\nu)\sim\sum_{l}\nu^{-2\delta_{l}/\pi+2\sum_{l^{\prime}}(2l^{\prime}+1)(\delta_{l^{\prime}}/\pi)^{2}}\
,$ (20)

where in the last equation we suppressed the prefactors of the power laws
which correspond to the angular momentum components of the dipole matrix
element.

In the preceding discussion, we have implicitly assumed the absence of bound
states. These can lead to additional low-energy singularities in the spectra,
with associated power laws that have a form similar to Eq. (18) [6, 11]; in
this case, the phase shifts are close to $\pi$. To recover these results in a
diagrammatic calculation, the interaction lines in Fig. 3 need to be replaced
by ladders ($T$-matrices), see, e.g., Ref. [31]. To keep the computations
manageable, we will neglect the effect of bound states, which is justified for
sufficiently small binding energies.

##  IV $c$-band geometry only

In the following, our goal is to relax the assumptions that lead from
$H_{\text{int}}$, Eq. (7) to $H_{\text{int}}^{(0)}$, Eq. (12), step by step.
To begin with, we keep the hole structureless, but allow for a non-trivial
$c$-band geometry, using an interaction of the form

$\displaystyle H_{\text{int}}^{(1)}=$
$\displaystyle\frac{1}{\Omega}\sum_{{\mathbf{k}},{\mathbf{q}}}V_{0}c^{\dagger}_{{\mathbf{k}}+{\mathbf{q}}}c_{\mathbf{k}}f^{\dagger}f\braket{c,{\mathbf{k}}+{\mathbf{q}}}{c,{\mathbf{k}}}\
.$ (21)

Such an interaction can be appropriate if the $c$ and $f$ electrons are in
fact different particle species, as is the case in typical polaron-type
ultracold gas experiments.

The interaction term (21) also describes a scattering potential for the $c$
fermions. Therefore, one may attempt to map Eq. (21) to a momentum-independent
potential by performing a Fermi-surface average of the overlap term, in order
to apply the edge singularity expressions in terms of the $s$-wave phase
shift, Eq. (18). Alternatively, one can try a partial wave decomposition à la
Eq. (20). However, such a procedure is a priori difficult to justify: First,
the overlap term by itself is not gauge invariant, and not spherically
symmetric. Second, it is not obvious how to bound the error made by
independent Fermi-surface averaging of each scattering event.

These obstacles can be circumvented by diagrammatic analysis, which we perform
separately for the single-particle and two-particle correlation functions. To
streamline the notation, we will suppress the label $c$ in the Bloch function
for the remainder of this section.

###  IV.1 Single-hole spectrum $A$

To analyze the single-hole (photoemission) spectrum
$A(\nu)=\text{Im}[F(-\nu)]$, consider the self-energy part of the second order
diagram for the hole propagator $F$ evaluated in the imaginary frequency
domain [Fig. 3(a)]:

$\displaystyle\Sigma(-\omega)=-V_{0}^{2}\int\frac{d\omega_{k}}{2\pi}\frac{d\omega_{p}}{2\pi}\frac{d{\mathbf{k}}}{(2\pi)^{d}}\frac{d{\mathbf{p}}}{(2\pi)^{d}}\mathopen{|}\braket{{\mathbf{k}}}{{\mathbf{p}}}\mathclose{|}^{2}$
$\displaystyle
G_{c}(\omega_{k},{\mathbf{k}})G_{c}(\omega_{p},{\mathbf{p}})F^{(0)}(-\omega+\omega_{k}-\omega_{p})\
.$ (22)

where
$G_{c}(\omega,{\mathbf{k}})=(i\omega-(\epsilon_{k}-\mu))^{-1},F^{(0)}(\omega)=(i\omega+E_{0})^{-1}$.
Evaluating the frequency integrals, one obtains

$\displaystyle\Sigma(-\omega)=$ (23) $\displaystyle
V_{0}^{2}\int_{\begin{subarray}{c}\epsilon_{\mathbf{k}}>\mu\\\
\epsilon_{\mathbf{p}}<\mu\end{subarray}}\frac{d{\mathbf{k}}}{(2\pi)^{d}}\frac{d{\mathbf{p}}}{(2\pi)^{d}}\frac{1}{-i\omega+E_{0}+\epsilon_{\mathbf{k}}-\epsilon_{\mathbf{p}}}\mathopen{|}\braket{{\mathbf{k}}}{{\mathbf{p}}}\mathclose{|}^{2}\
.$

After analytical continuation $i\omega\rightarrow\omega+i0^{+}$, we can e.g.
evaluate the imaginary part as

$\displaystyle\text{Im}\left[\Sigma(-\nu)\right]$
$\displaystyle\simeq\pi\alpha^{2}\int_{0}^{\mu}d\epsilon_{\mathbf{p}}\int_{\mu}^{\Lambda}d\epsilon_{\mathbf{k}}\
\delta(\nu-(\epsilon_{\mathbf{k}}-\epsilon_{\mathbf{p}}))\cdot I_{2}$
$\displaystyle={\pi\alpha^{2}\theta(\nu)\nu}\cdot I_{2}\quad\text{as}\
\nu\rightarrow 0.$ (24)

Here, $I_{2}$ corresponds to the gauge-invariant squared overlap of $c$-Bloch
functions averaged over the Fermi surface:

$\displaystyle
I_{2}\equiv\int_{{\mathbf{k}},{\mathbf{p}}}\mathopen{|}\braket{{\mathbf{k}}}{{\mathbf{p}}}\mathclose{|}^{2},\quad\int_{\mathbf{k}}\equiv\frac{1}{\rho}\int\frac{d{\mathbf{k}}}{(2\pi)^{d}}\delta(\mu-\epsilon_{\mathbf{k}})\
.$ (25)

The restriction to the Fermi surface alone is an approximation for general
$\nu$, but it becomes exact as $\nu\searrow 0$ in Eq. (24). By the Kramers-
Kronig relations,
$\text{Re}\left[\Sigma(-\nu)\right]-\text{Re}[\Sigma(0)]\sim\nu\log(|\nu|)$,
showing the emergence of logarithmic factors. Note that internal momenta that
contribute to $\text{Re}[\Sigma(0)]\sim\alpha^{2}\mu\log(\mu)$ do not have to
be restricted to the proximity of the Fermi surface, as can be seen from
direct computation or Fumi’s theorem [29]; however, $\text{Re}[\Sigma(0)]$ is
only essential for determination of threshold energies, but not for the
detailed form of the spectra.

By introducing $P_{\mathbf{k}}^{c}=\ket{{\mathbf{k}}}\\!\bra{{\mathbf{k}}}$,
the projector on the $c$-band Bloch function with momentum ${\mathbf{k}}$, we
can rewrite

$\displaystyle I_{2}$
$\displaystyle=\int_{{\mathbf{k}},{\mathbf{p}}}\text{Tr}\left\\{P_{{\mathbf{k}}}^{c}P_{{\mathbf{p}}}^{c}\right\\}=\text{Tr}\\{\left(P^{c}\right)^{2}\\}<1$
$\displaystyle P^{c}$
$\displaystyle\equiv\int_{{\mathbf{k}}}P^{c}_{\mathbf{k}}\ ,$ (26)

where the trace acts in band space, and $P^{c}$ is the projector averaged over
the Fermi surface. We therefore see that at the level of the second order
diagram, the only relevant change we need to perform is
$\alpha^{2}\rightarrow\alpha^{2}I_{2}$.

The derivation of the linked cluster theorem for $F(t)$, Eq. (16), relies only
the $f$-hole being structureless, which equally applies to the interaction
(21)[29]. In evaluating the relevant expression $C_{2}(t)$, one then obtains
the same result as in the infinite mass case, but again with the replacement
$\alpha^{2}\rightarrow\alpha^{2}I_{2}$. Therefore, to the leading order in
$\alpha$, but treating the $f$-band geometry exactly, the spectrum becomes:

$\displaystyle
A(\nu)\sim\nu^{2\alpha^{2}\text{Tr}\\{\left(P^{c}\right)^{2}\\}-1}\ .$ (27)

The interaction is suppressed by the Bloch-overlaps which penalize large
momentum transfers. This suppression is enhanced for larger doping
$k_{F}\nearrow$, since the overlap integral
$\text{Tr}\\{\left(P^{c}\right)^{2}\\}$ probes larger momenta. Therefore, the
photoemission spectrum gets more singular. Two representative plots for the
Lieb lattice are shown in Fig. 4(a). For
$\alpha^{2}\text{Tr}\\{\left(P^{c}\right)^{2}\\}\rightarrow 0$, one recovers
the non-interacting form $A(\nu)=\delta(\nu)$ when restoring normalizing
factors.

###  IV.2 Particle-hole spectrum $\chi$

Figure 4: Edge singularities on the Lieb lattice, including $c$-band geometry.
Used parameters: $\alpha=0.3,t^{\prime}=0.6$. (a) Single-hole spectrum
$A(\nu)$, Eq. (27) in arbitrary units; frequencies are measured in units of
the UV cutoff. The inset shows the respective Fermi Level. (b) Absorption
spectrum $\chi(\nu)$, Eq. (36). The plot shows the most singular contribution
with the maximal Eigenvalue $\lambda_{\text{max}}$.

For the particle-hole (intraband absorption) spectrum, we consider the
correlation function

$\displaystyle\Pi(t)$
$\displaystyle=(-i)\sum_{{\mathbf{k}}_{1},{\mathbf{k}}_{2},\eta}\\!E_{0}^{2}\braket{{\mathbf{k}}_{1}}{\hat{O}^{\eta}({\mathbf{k}}_{1},{\mathbf{k}}_{2})}{{\mathbf{k}}_{2}}\times$
$\displaystyle\braket{0}{\hat{T}\left\\{f^{\dagger}(t)c_{{\mathbf{k}}_{2}}(t)c_{{\mathbf{k}}_{1}}^{\dagger}(0)f(0)\right\\}}{0}$
$\displaystyle\hat{O}^{\eta}({\mathbf{k}}_{1},{\mathbf{k}}_{2})$
$\displaystyle=-\partial_{\eta}\ket{f,{\mathbf{k}}_{1}}\\!\left({\partial_{\eta}}\bra{f,{\mathbf{k}}_{2}}\right)\
$ (28)

To maintain the assumption of a structureless hole, we assume that the
momentum-dependence of the operator $\hat{O}^{\eta}$ is negligible. One can
now evaluate the $\Pi(\nu)$ in similar manner as $\Sigma$ described in the
previous section, for instance by computing the leading logarithmic diagrams
of Fig. 3(b). In this computation, an independent momentum variable can be
chosen for each $c$-fermion line. As a result, the low-order diagrams incur
the following additional gauge-invariant factors due to the Bloch overlaps:

$\displaystyle\Pi^{(0)}\\!:$
$\displaystyle\sum_{\eta}\int_{{\mathbf{k}}}\braket{{\mathbf{k}}}{\hat{O}^{\eta}}{{\mathbf{k}}}=\sum_{\eta}\text{Tr}\\{\hat{O}^{\eta}P^{c}\\}$
(29) $\displaystyle\Pi^{(1)}\\!:$
$\displaystyle\sum_{\eta}\int_{{\mathbf{k}},{\mathbf{p}}}\braket{{\mathbf{k}}}{\hat{O}^{\eta}}{{\mathbf{p}}}\braket{{\mathbf{p}}}{{\mathbf{k}}}=\sum_{\eta}\text{Tr}\\{\hat{O}^{\eta}(P^{c})^{2}\\}$
$\displaystyle\Pi^{(2)}_{a,b}\\!:$
$\displaystyle\sum_{\eta}\text{Tr}\\{\hat{O}^{\eta}(P^{c})^{3}\\}\ ,$

with $P^{c}$ the Fermi-surface averaged projector introduced in Eq. (26). This
structure continues for all diagrams without additional fermion loops; all
leading parquet diagrams are of this form [1, 3]. Therefore, we can rephrase
the perturbative expansion for $\Pi(\nu)$, Eq. (17) as

$\displaystyle\Pi(\nu)=$ (30)
$\displaystyle\sum_{\eta}\text{Tr}\left[\frac{\hat{O}^{\eta}\rho}{\alpha}\left(\alpha
LP^{c}-(\alpha L)^{2}(P^{c})^{2}+\frac{2}{3}(\alpha
L)^{3}(P^{c})^{3}\ldots\right)\right].$

Summing this series, we obtain

$\displaystyle\Pi(\nu)=\sum_{\eta}\text{Tr}\left[\frac{\hat{O}^{\eta}\rho}{\alpha}\left(\mathbbm{1}-\exp(-2\alpha
L(\nu)P^{c}\right)\right]\ ,$ (31)

where $\mathbbm{1}$ is the identity in band space and the exponential is a
matrix-exponential. Diagonalizing $P^{c}$, we obtain for
$\chi(\nu)=\text{Im}[\Pi(\nu)]$:

$\displaystyle\chi(\nu)\sim\sum_{i}\nu^{-2\alpha\lambda_{i}},$ (32)

where $\lambda_{i}$ are the Eigenvalues of $P^{c}$, and we suppressed the
prefactors of the the power laws which correspond to the diagonal elements of
$\hat{O}^{\eta}$ rotated into the $P^{c}$ Eigenbasis. Note the analogy between
Eqs. $\eqref{partialpsA},\eqref{partialpschi}$ and Eqs. (27) and (32): $\chi$
is a sum of power laws, while $A$ is a single power law, with the sum in the
exponent. Now increasing $k_{F}$ makes the spectrum less singular because the
maximal Eigenvalue $\lambda_{\text{max}}$ in Eq. (32) is reduced, as we
analyze in detail in the next section. This behavior is expected, since $\chi$
simply becomes a step-function in the limit $\nu\rightarrow 0$ for a constant
density of states. In Fig. 4(b), we show $\chi(\nu)$ for two values of the
chemical potential $\mu$, pinpointing this behavior.

###  IV.3 Perturbative expansion for small band geometry

For a simpler interpretation of the results, which is also generalizable to
the case of non-trivial $f$-bands, it is useful to consider the limit of
“weak” $c$-band geometry: we assume that the momentum-variation of Bloch
functions $\ket{{\mathbf{k}}}$ is small. This can always be justified for weak
doping $\mu$, when $k_{F}$ is much smaller than a reciprocal lattice vector
which sets the typical scale for the variation of $\ket{{\mathbf{k}}}$. One
can think of this limit as the leading lattice correction to the continuum
limit. A related perturbative expansion in the context of excitons can for
instance be found in Ref. [32].

Under this assumption, we can expand $\ket{{\mathbf{k}}}$ around
${\mathbf{k}}=\boldsymbol{0}$, where $\boldsymbol{0}$ denotes the momentum
where the $c$-band has its minimum; however, with the same accuracy, any other
momentum within the Fermi volume can be chosen as point of expansion. Up to
second order in ${\mathbf{k}}$, we have

$\displaystyle\ket{{\mathbf{k}}}\simeq\ket{\boldsymbol{0}}+k_{i}\ket{i}+\frac{1}{2}k_{i}k_{j}\ket{ij},$
(33)

with the notation
$\ket{i}\equiv\partial_{k_{i}}\ket{{\mathbf{k}}}\big{|}_{{\mathbf{k}}=\boldsymbol{0}}$.
Applying this expansion to the squared overlap
$|\\!\braket{{\mathbf{k}}}{{\mathbf{p}}}\\!|^{2}$ which appears in
$I_{2}=\text{Tr}\\{(P^{c})^{2}\\}$, Eq. (26), we obtain

$\displaystyle|\\!\braket{{\mathbf{k}}}{{\mathbf{p}}}\\!|^{2}$
$\displaystyle\simeq
1-g^{c}_{ij}({\mathbf{k}}-{\mathbf{p}})_{i}({\mathbf{k}}-{\mathbf{p}})_{j}$
$\displaystyle\equiv 1-||{\mathbf{k}}-{\mathbf{p}}||^{2}_{c}\ .$ (34)

Here, $g_{ij}$ is the quantum (Fubini-Study) metric of the $c$ band at the
point ${\mathbf{k}}=\boldsymbol{0}$, defined by

$\displaystyle
g_{ij}^{c}=\frac{1}{2}\left(\braket{i}{j}+\braket{j}{i}\right)+\braket{\boldsymbol{0}}{i}\braket{\boldsymbol{0}}{j}.$
(35)

To get from Eq. (34) to (35), we used the identities
$\braket{\boldsymbol{0}}{i}+\braket{i}{\boldsymbol{0}}=0$ and
$\frac{1}{2}\left(\braket{i}{j}+\braket{j}{i}\right)=\text{Re}\braket{i}{j}=-\text{Re}\braket{ij}{\boldsymbol{0}}$.

The quantum metric is the real part of the $c$-$c$ component of the quantum
geometric tensor [33, 34, 35, 36, 37], while the imaginary part corresponds to
the Berry curvature $B$. $g^{c}$ is a gauge-invariant positive-semidefinite
measure of the distance of the Bloch states in Hilbert space. The Brillouin-
zone average of $\text{tr}g^{c}({\mathbf{k}})$ can be related to the minimal
real-space spread $\braket{{\mathbf{r}}^{2}}-\braket{{\mathbf{r}}}^{2}$ of
Wannier functions [33]. While $g^{c}$ can be non-vanishing for a topologically
trivial band, for a band with non-zero Berry curvature $B$ it is bounded from
below: $\text{tr}g^{c}({\mathbf{k}})\geq|B({\mathbf{k}})|$ for any
${\mathbf{k}}$ [36]. For the Lieb lattice, $\text{tr}g^{c}({\mathbf{k}})$ has
a broad maximum at the $M$ point for $t^{\prime}\lesssim 1$ (for
$t^{\prime}\rightarrow 0$, the metric is sharply peaked at the $M$ point), see
Fig. 8(a). Various theoretical proposals [35, 38, 39, 40, 41] and successful
experiments [42, 43] to determine the quantum metric have been put forward.

With Eq. (34) at hand, we can rewrite our result for $A(\nu)$, Eq. (27), as

$\displaystyle
A(\nu)\sim\nu^{2\alpha^{2}(1-\int_{{\mathbf{k}},{\mathbf{p}}}||{\mathbf{k}}-{\mathbf{p}}||_{c}^{2})-1}\
.$ (36)

The edge-singularity is preserved, but the effective interaction is reduced by
the Fermi-surface averaged Hilbert space distance of the scattered
$c$-fermions. The expansion in Eq. (36) is controlled to first order in

$\displaystyle x_{c}\equiv k_{F}^{2}\text{tr}g^{c}\overset{!}{\ll}1\ ,$ (37)

where $k_{F}$ is a Fermi-surface averaged Fermi momentum 444Note that
$|g^{c}_{ij}|<\text{tr}g^{c}$ for a positive semidefinite matrix $g^{c}$..
This parameter quantifies the leading lattice effect. For the Lieb lattice,
the Fermi surface becomes nearly circular for small $k_{F}$, and the
perturbative correction simply reads

$\displaystyle\int_{{\mathbf{k}},{\mathbf{p}}}||{\mathbf{k}}-{\mathbf{p}}||_{c}^{2}\
{\rightarrow}\ x_{c}\quad\text{for}\ k_{F}\rightarrow 0,$ (38)

as verified in Fig. 5(a). This implies that, in principle, measuring the
doping dependence of the edge-singularity exponent gives access to
$\text{tr}g^{c}$ provided that the $k_{F}$-dependence of the coupling constant
$\alpha$ is accounted for.

It is interesting to compare the result in Eq. (36) with the effect of a
finite $f$-mass on $A(\nu)$ [9, 10]. Similar to Eq. (36), the finite hole mass
penalizes large momentum transfers which come at a high kinetic energy cost.
This effect can be captured by evaluating an phase space factor similar to
$I_{2}$. However, in the case of a finite hole mass, this factor vanishes as
$\nu\rightarrow 0$ with a power dependent on dimensionality. This leads to a
more drastic reorganization of the spectrum $A(\nu)$ compared to Eq. (36),
with a partial reemergence of the non-interacting delta function.

Figure 5: (a) Evolution of lattice-corrections to the edge singularity
exponents for the Lieblattice vs. chemical potential $\mu$, for
$t^{\prime}=0.6$. (a) Corrections relevant for $A(\nu)$. (b) Corrections
relevant for $\chi(\nu)$.

Similar to $A(\nu)$, an expansion in terms of the metric can also be applied
to $\chi(\nu)\sim\sum_{i}\nu^{-2\alpha\lambda_{i}}$, Eq. (32). We focus on the
most singular contribution, i.e., the largest Eigenvalue
$\lambda_{\text{max}}$. When $k_{F}\rightarrow 0$, $P^{c}$ becomes a
projector, and its largest Eigenvalue is $1$. For small $k_{F}$, second order
perturbation theory (see App. A) leads to

$\displaystyle\lambda_{\text{max}}\simeq
1-\int_{{\mathbf{k}},{\mathbf{p}}}\frac{||{\mathbf{k}}-{\mathbf{p}}||_{c}^{2}}{2}\
,$ (39)

which we check in Fig. 5(b). Thus

$\displaystyle\chi(\nu)\sim\nu^{-2\alpha\left(1-\int_{{\mathbf{k}},{\mathbf{p}}}\frac{||{\mathbf{k}}-{\mathbf{p}}||_{c}^{2}}{2}\right)}\
.$ (40)

Like in the photoemission case, in the absorption case the effective coupling
constant is reduced. Note that, to the leading order in $x_{c}$, the same
effective coupling

$\displaystyle\alpha_{\text{eff}}=\alpha\left(1-\int_{{\mathbf{k}},{\mathbf{p}}}\frac{||{\mathbf{k}}-{\mathbf{p}}||_{c}^{2}}{2}\right)$
(41)

appears in both Eqs. (36), (40). Interestingly, this result agrees with a
Fermi-surface average of the modulus of the overlap term in
$H_{\text{int}}^{(1)}$ at order $O(x_{c})$, with both incoming and outgoing
momenta on the Fermi surface.

With the accuracy of Eq. (41), we can in fact go beyond the leading order in
$\alpha$ in the determination of the edge singularity exponents for both
$A(\nu),\chi(\nu)$: a generic diagram at $n$-th order in $\alpha$ will incur a
factor

$\displaystyle\alpha^{n}\prod_{l}\text{Tr}\\{(P^{c})^{n_{l}}\\}\ ,\quad
n=\sum_{l}n_{l},\quad n_{l}>1\ ,$ (42)

where $l$ is the number of $c$-fermion loops, and the condition $n_{l}>1$
excludes tadpole-type diagrams which can shift the respective thresholds only.
To the leading order in $x_{c}$, we have

$\displaystyle\text{Tr}\\{(P^{c})^{n_{l}}\\}=\sum_{i}(\lambda^{i})^{n_{l}}\simeq
1-n_{l}\int_{{\mathbf{k}},{\mathbf{p}}}\frac{||{\mathbf{k}}-{\mathbf{p}}||_{c}^{2}}{2},$
(43)

where we used Eq. (39), and the fact that $\lambda_{i}=O(x_{c})$ for
$\lambda_{i}<\lambda_{\text{max}}$ (App. A). As a result,

$\displaystyle\alpha^{n}\prod_{l}\text{Tr}\\{(P^{c})^{n_{l}}\\}\simeq
1-n\int_{{\mathbf{k}},{\mathbf{p}}}\frac{||{\mathbf{k}}-{\mathbf{p}}||_{c}^{2}}{2}\simeq\alpha_{\text{eff}}^{n}+O(x_{c}^{2}).$
(44)

Since $\alpha_{\text{eff}}$ enters every diagram, and not just the
logarithmically dominant parquet diagrams, it will determine the Fermi-level
scattering phase shift of the $c$-fermions. Therefore we can apply the results
expressed in terms of the $s$-wave phase shift, Eq. (18), with
$\delta=\delta(\alpha_{\text{eff}})+O(x_{c}^{2})$.

##  V Full band geometry

So far, we have studied a structureless $f$-hole. To make the connection to
correlated flat band materials, we must lift this restriction, and reintroduce
the full interaction $H_{\text{int}}$ from Eq. (7).

This step is more involved than the introduction of $c$-band geometry alone:
if the $f$-hole has a momentum-dependent Bloch function, the problem looses
its single-particle character. However, as long as the band geometry of both
$c$\- and $f$-bands is weak, large logarithms remain, and we can still gain
insight on spectra by diagrammatic analysis. To keep the problem under
control, we will work to leading order $O(x)\equiv O(x_{c},x_{f})$, where
$x_{f}$ quantifies the band geometry of the $f$-band $x_{f}$ analogously to
$x_{c}$.

In general, the interaction $H_{\text{int}}$ contains terms with up to 4
$f$-operators. In the limit $E_{0}\rightarrow\infty$ only terms which conserve
the number of $f$-electrons survive: Processes violating this condition are
strongly off-shell and suppressed by factors of $1/E_{0}$. This leaves terms
with $4,0$ or $2$ $f$-operators. As before, terms with $0$ $f$-operators can
be absorbed in a renormalized $c$-band, while terms with $4$ $f$-operators are
ineffective for a filled $f$-band. We are left with the terms involving two
$f$\- and two $c$-fermions graphically represented in Fig. 6.

Figure 6: (a) Interaction processes that contribute in the limit
$E_{0}\rightarrow\infty$. (b) Diagrams that involve
$\tilde{H}_{\text{int}}^{(2)}$. For the diagram on the right, the time-domain
structure is indicated. Note that interactions are instantaneous.

In addition to the conventional term $H_{\text{int}}^{(2)}$, processes
$\tilde{H}_{\text{int}}^{(2)}$ where $f$\- and $c$-electrons interconvert are
allowed, see Fig. 6(a). These processes can appear in the diagrams for
$F,\Pi$, as shown in Fig. 6(b): when a conventional diagram contains a
$c$-band hole propagating backwards in time, we can replace it with an
$f$-band hole. On the right hand side of Fig. 6(b), we illustrate the time-
domain structure of the diagram: As required, all $f$-holes propagate
backwards in time. While such processes survive the limit
$E_{0}\rightarrow\infty$, they are small for a weak band geometry: one can
easily show (see Appendix B.1) that for ${\mathbf{k}},{\mathbf{p}}=O(k_{F})$,
the squared overlap element between $c$ and $f$ bands fulfills
$|\\!\braket{c,{\mathbf{k}}}{f,{\mathbf{p}}}\\!|^{2}=O(x)$. Each diagram
involving $\tilde{H}_{\text{int}}^{(2)}$ contains at least four overlap
elements, and is therefore of order $O(x^{2})$, which is to be neglected
within our approximation. Therefore, we only need to keep an interaction term

$\displaystyle H_{\text{int}}\simeq H_{\text{int}}^{(2)}=$ (45)
$\displaystyle\frac{1}{S}\sum_{{\mathbf{k}},{\mathbf{k}}^{\prime},{\mathbf{q}}}V_{0}c^{\dagger}_{{\mathbf{k}}}f^{\dagger}_{{\mathbf{k}}^{\prime}}f_{{\mathbf{k}}^{\prime}-{\mathbf{q}}}c_{{\mathbf{k}}+{\mathbf{q}}}\braket{c,{\mathbf{k}}}{c,{\mathbf{k}}+{\mathbf{q}}}\braket{f,{\mathbf{k}}^{\prime}}{f,{\mathbf{k}}^{\prime}-{\mathbf{q}}}\
.$

###  V.1 Single-hole spectrum $A$

Via photoemission, a single hole with external momentum $\mathbf{Q}$ can be
created. To compute the spectrum $A_{\mathbf{Q}}$ as defined in Eq. (8) to
order $O(x)$, it is instructive to examine the generic third order diagram
shown in Fig. 7. Given the interaction $H_{\text{int}}^{(2)}$, the Bloch-
overlap structure of this diagram can be written as

$\displaystyle\text{BO}(\mathbf{Q},{\mathbf{k}},{\mathbf{p}},{\mathbf{q}})\equiv\text{Tr}\\{P^{c}_{\mathbf{k}}P^{c}_{\mathbf{p}}P^{c}_{\mathbf{q}}\\}\text{Tr}\\{P^{f}_{\mathbf{Q}}P^{f}_{\mathbf{Q}+{\mathbf{k}}-{\mathbf{q}}}P^{f}_{\mathbf{Q}+{\mathbf{p}}-{\mathbf{q}}}\\},$
(46)

where $P^{f}_{\mathbf{Q}}=\ket{f,\mathbf{Q}}\\!\bra{f,\mathbf{Q}}$. We can
distinguish two cases: First, $\mathbf{Q}=O(k_{F})$. In this case, we can
expand the projectors $P^{f}$ in the second factor in small momenta around
$\boldsymbol{0}$ (again, small in the sense that $x_{f}\ll 1$). One can then
easily show (App. B.2) that the $\mathbf{Q}$-dependence drops out: at
$O(x_{f})$, the overlap function depends on the transferred momenta only. This
implies that, at order $O(x_{f})$, no dispersion is generated for the $f$-band
on the relevant recoil momentum scale $k_{F}$. As a result, the large
logarithms in the problem are not cut off by the hole recoil, and the power
laws in the spectrum remain; as in the previous section, what is left to do is
to determine the prefactors of the dominant diagrams by evaluating the
overlaps on the Fermi surface.

Figure 7: Third order diagram for the hole-propagator with momentum labels.

We may also consider the case $|\mathbf{Q}|\gg k_{F}$. In this situation, we
cannot expand $P^{f}_{\mathbf{Q}}$ around $\mathbf{Q}=0$ in general. Instead,
we can expand all $f$-propagators around the momentum $\mathbf{Q}$; the only
difference to the preceding case is that the resulting overlap factors depend
on $g^{f}(\mathbf{Q})$ instead of $g^{f}(\boldsymbol{0})$.

To evaluate the overlap factors involving the $P^{f}$-projectors, for
simplicity we consider an inversion-symmetric Fermi surface (as in the Lieb
lattice case), for which

$\displaystyle g_{ij}^{f}\int_{{\mathbf{k}},{\mathbf{p}}}k_{i}p_{j}=0\ .$ (47)

and therefore

$\displaystyle
g_{ij}^{f}(\mathbf{Q})\int_{{\mathbf{k}},{\mathbf{p}}}(k-p)_{i}(k-p)_{j}$
$\displaystyle=2g^{f}_{ij}(\mathbf{Q})\int_{{\mathbf{k}}}k_{i}k_{j}$
$\displaystyle\equiv 2\int_{k}||{\mathbf{k}}||^{2}_{f,\mathbf{Q}}\ .$ (48)

For the Fermi-surface averaged Bloch overlap of the third-order diagram from
Eq. (46) we then obtain (see App. C)

$\displaystyle\int_{{\mathbf{k}},{\mathbf{p}},{\mathbf{q}}}\text{BO}(\mathbf{Q},{\mathbf{k}},{\mathbf{p}},{\mathbf{q}})=1-3\int_{{\mathbf{k}}}\left(||{\mathbf{k}}||^{2}_{c}+||{\mathbf{k}}||^{2}_{f,\mathbf{Q}}\right)+O(x^{2}).$
(49)

Likewise, at $n$-th order we obtain a factor

$\displaystyle
1-n\int_{{\mathbf{k}}}\left(||{\mathbf{k}}||^{2}_{c}+||{\mathbf{k}}||^{2}_{f,\mathbf{Q}}\right)\
.$ (50)

This shows that, similar to the case of a trivial $f$-band, for a weak band
geometry we can introduce an effective interaction constant

$\displaystyle\alpha_{\text{eff}}(\mathbf{Q})\equiv\alpha\left(1-\int_{{\mathbf{k}}}||{\mathbf{k}}||^{2}_{c}+||{\mathbf{k}}||^{2}_{f,\mathbf{Q}}\right),$
(51)

which again characterizes an effective momentum-independent scattering problem
with phase shift $\delta[\alpha_{\text{eff}}(\mathbf{Q})]$ and and associated
spectrum

$\displaystyle
A(\mathbf{Q},\nu)\sim\nu^{2(\delta[\alpha_{\text{eff}}(\mathbf{Q})]/\pi)^{2}-1}\simeq\nu^{2\alpha_{\text{eff}}(\mathbf{Q})^{2}-1}\quad\text{for}\
\alpha_{\text{eff}}\ll 1.$ (52)

The scattering of $f$-fermions to different momentum states on the Fermi
surface further reduces the effective interaction $\alpha_{\text{eff}}$,
parametrized by the distance of the scattered $f$-states in Hilbert space. The
photoemission power law exponent inherits the $\mathbf{Q}$-dependence of the
local $f$-metric. Results for the Lieb lattice are shown in Fig. 8. Note again
that, for $\alpha_{\text{eff}}(\mathbf{Q})$ to be valid, $x_{c},x_{f}\ll 1$
are required. A closely related requirement is that the metric does not change
too strongly on the scale of $k_{F}$. For the Lieb lattice, this condition
breaks down when $t^{\prime}$ becomes too small and
$\text{tr}g^{f}(\mathbf{Q})$ is strongly peaked at the $M$-point.

Figure 8: (a) $\text{tr}g^{f}(\mathbf{Q})=2\text{tr}g^{c}(\mathbf{Q})$ for the
Lieb lattice at $t^{\prime}=0$. The red circle shows the Fermi surface used
for Fig. (b). Momentum-dependent correction to the effective interaction
$\alpha_{\text{eff}}$, see Eq. (51), when $\mathbf{Q}$ is varied along the
unit cell diagonal $\mathbf{Q}=(\ell,\ell)$, for $\mu=0.015$.

In a typical potential scattering problem, the phase shift is decreasing when
potential range is increased for fixed potential depth. Recalling the
connection of the metric and the minimal spread of the Wannier functions, we
can therefore view Eq. (51) as a result of two effects: first, the effective
$f$ potential seen by the $c$ fermions has a finite range
$\sim\sqrt{\text{tr}g^{f}(\mathbf{Q})}$. Second, any wave packed formed by the
$c$ fermions has a finite width $\sim\sqrt{\text{tr}g^{c}}$, which effectively
adds to the potential range. A cartoon of this effect is shown in Fig. 9.

Note that, strictly speaking, it is the Brillouin-zone average of the metric
and not the metric itself which determines the spread of the Wannier
functions; the simple picture above holds if the momentum scale over which
$\text{tr}g$ changes appreciably is much larger than $\sqrt{\text{tr}g}$
itself, which is realized ideally for a system with a uniform metric, e.g., a
model with wavefunctions as in a Lowest Landau Level [45].

Figure 9: Flat band edge singularity as effective potential scattering
problem: a $c$ wave packet of width $\sqrt{\text{tr}g^{c}(\boldsymbol{0})}$ is
scattered by a potential created by an $f$ electron of with effective range
$\sqrt{\text{tr}g^{f}(\mathbf{Q})}$.

###  V.2 Particle-hole spectrum $\chi$

For the particle-hole spectrum $\chi(\nu)$, at order $O(x)$ we can follow the
same approach as in the previous section and evaluate the Bloch-overlaps for
the logarithmic diagrams of Fig. 3(b), including the interband current matrix
elements, see Eq. (10). It is convenient to introduce an operator
$\hat{J}^{\eta}_{cf}({\mathbf{k}})=P_{\mathbf{k}}^{c}\partial_{\eta}P^{f}_{\mathbf{k}}$.
Then, for instance the “crossed” diagram $\Pi_{b}^{(2)}$ incures a factor

$\displaystyle\int_{{\mathbf{k}},{\mathbf{p}},{\mathbf{q}}}\text{Tr}\\{\hat{J}^{\eta}_{cf}({\mathbf{k}})P_{f}({\mathbf{q}}+{\mathbf{k}}-{\mathbf{p}})\hat{J}^{\eta}_{fc}({\mathbf{q}})P_{c}({\mathbf{q}})\\}\
.$ (53)

A challenge in evaluating Eq. (53) is the momentum-dependence of the current
operators. If we set
$\hat{J}^{\eta}({\mathbf{k}})\simeq\hat{J}^{\eta}(\boldsymbol{0})+\Delta\hat{J}^{\eta}({\mathbf{k}})$,
and neglect the momentum-dependent correction
${\Delta\hat{J}^{\eta}({\mathbf{k}})}$, the $O(x)$ evaluation of (53) and
subsequent diagrams, including multiloop processes, proceeds as for the
single-hole spectrum, and we find

$\displaystyle\chi(\nu)\simeq\nu^{-2\delta[\alpha_{\text{eff}}(\boldsymbol{0})]/\pi+2(\delta[\alpha_{\text{eff}}(\boldsymbol{0})]/\pi)^{2}}\simeq\nu^{-2\alpha_{\text{eff}}(\boldsymbol{0})},$
(54)

with $\alpha_{\text{eff}}(\boldsymbol{0})$ as in Eq. (51). A priori, the
missed correction
$\sim\int_{\mathbf{k}}\braket{\boldsymbol{0}}{\Delta\hat{J}^{\eta}({\mathbf{k}})}{\boldsymbol{0}}$
is not smaller than $O(x)$. However, this correction only appears twice in
each diagram, independent of the order in the interaction, while the
corrections derived from the Bloch overlaps for the momenta pile up as in Eq.
(50). As a result, the exponent of the edge singularity in Eq. (54) is not
impacted by the momentum-dependence of $\hat{J}^{\eta}$.

##  VI Discussion

While the absorption $\chi\sim\nu^{-2\alpha_{\text{eff}}(\boldsymbol{0})}$ in
principle is of direct relevance for correlated flat band materials (e.g.
Moiré systems) with a filled flat and doped conduction band, experimental
observation might be challenging due to the insufficient energy resolution of
spectral measurements to date. Furthermore, in such systems processes where
$c$ electrons relax into the $f$ band must be considered, which lead to an IR
tail in the $f$ spectra. Due to energy conservation, these processes require
emission of an excited particle, for instance a photon, phonon or additional
$c$ electron (Auger process); they are often suppressed by small matrix
elements if the band gap between the $f$ and $c$ band is sufficiently large.

The case for the observation of the single-hole edge singularity $A(\nu)$ can
be made more easily, in particular when $c$-particles and $f$-hole correspond
to different particle species and thus cannot interconvert, which prohibits
the relaxation process described above. This is possible if the $f$-hole is an
“impurity” coupled to a $c$-fermion bath, a scenario that can e.g. realized in
experiments involving quantum dots, where edge singularities have been
observed for trivial bands [46]. Likewise, this situation applies to typical
“polaronic” spectral measurements in ultracold gases, where two different
particle species are studied (see e.g. [23, 24, 11] and Refs. therein). In
this case, a mature experimental technique for measuring single-particle
spectra is inverse (injection) RF spectroscopy 555standard RF spectroscopy,
which is akin to ARPES, measures the occupied part of the spectral function,
which complicates the detection of impurity spectra: here, the $f$-particles
have to be initially prepared in a state where the interaction with the $c$
particles can be neglected; by applying a weak RF pulse of frequency $\omega$
($\nu$ plus threshold energy), particles are excited into an interacting
$f$-band. In linear response, the depletion current from the non-interacting
state is proportional to $A(\nu)$, and the spectral resolution is inversely
proportional to the pulse duration. A possible observable is the dependence of
the spectrum on the Fermi surface volume, either via Eq. (27) for general
Fermi momentum, or Eq. (36) for small one (separating out the distinct
$k_{F}$-dependence of $\alpha$). For the ultracold gases, a crucial
experimental challenge is the required small temperature $T\ll\mu$; finite
temperatures will broaden the edge singularities in a well-understood manner
[48].

If the $f$-hole is part of a flat band with non-vanishing band geometry, the
momentum dependence of $A_{\mathbf{Q}}$ can in principle be extracted by
combining the RF measurement techniques such as time-of-flight mapping [49] or
Raman spectroscopy [50]. If the power law exponent
$\alpha_{\text{eff}}(\mathbf{Q})$ can be extracted from such a measurement,
the flat band metric $g^{f}(\mathbf{Q})$ can be mapped out in experiment. When
such momentum resolution is not available, the RF spectrum yields the
Brillouin zone average of the $f$-spectrum; if the non-interacting $f$-state
has flat dispersion as well, simply $A_{\text{RF}}(\nu)\sim\int
d\mathbf{Q}A_{\mathbf{Q}}(\nu)$. At $O(x)$, we can equivalently average the
momentum-dependent exponent $\alpha_{\text{eff}}(\mathbf{Q})$. Therefore, at
weak doping the momentum-averaged RF measurement of the power law exponent
gives access to the momentum-averaged metric, a probe of the minimal real-
space spread of the associated Wannier functions.

The universal effective interaction $\alpha_{\text{eff}}(\mathbf{Q})$ was
obtained for weak doping. At strong doping, the universality breaks down, and
the results will strongly depend on lattice details. In particular, if the
scale on which $\text{tr}g^{f}(\mathbf{Q})$ changes strongly becomes
$O(k_{F})$, effective mass generation becomes important, cutting off
logarithmic singularities. In passing, we note that effective two-body masses
in the flat band can be related to the integral over the quantum metric (which
can be non-zero even for a momentum-independent metric) if a sublattice-
sensitive contact interaction is used [51, 52, 53]. For a finite $f$-band
mass, the $f$-particle becomes a mobile “Fermi polaron”, which can for
instance be described variational methods [54] and has been explored in the
literature in various lattice contexts [55, 56, 57, 58, 59, 60].

##  VII Summary and Outlook

In this work, we derived a universal lattice generalization of edge
singularities: we considered processes where a single degree of freedom (hole)
in a $f$-band interacts with fermions in a dispersive $c$-band, allowing for a
non-trivial Bloch geometry for both bands, and evaluated corresponding single-
hole $\sim\braket{f^{\dagger}f}$ and particle-hole
$\sim\braket{cf^{\dagger}fc^{\dagger}}$ spectra. We found that the leading
effect of the band geometry is to reduce the effective coupling that enters
the edge singularity exponents, which we derived by evaluating Bloch-function
overlaps appearing in the respective singular Feynman diagrams. For $k_{F}$
much smaller than a reciprocal lattice vector, corrections to the exponents
are proportional to the quantum metrics times the Fermi energy, which can be
traced back to the finite range of the effective scattering potential created
by the $f$-hole. Our results for the exponents are summarized in Tables 1, 2.

In our derivation of the spectra, diagrammatic perturbation theory and a
coordinate representation of the metric was employed. It would be interesting
to reformulate the problem purely as a scattering problem on Riemannian
manifold which reflects the non-trival Bloch band geometry [61].

Lastly, we note that the orthogonality catastrophe underlying the edge
singularities can occur in bosonic systems as well [62], which is of
particular relevance for the ultra-cold gas setups. To enrich this bosonic
orthogonality with effects of lattice geometry and topology is a worthwhile
goal for future study.

Acknowledgement. I thank Moshe Goldstein, Dan Mao and Erich Mueller for useful
discussions, and am especially grateful to Debanjan Chowdhury for insightful
suggestions and support during the preparation of this manuscript. I
acknowledge funding by the German Research Foundation (DFG) under Project-ID
442134789.

$\boldsymbol{A(\nu)}$ | Exact in interaction |  $O(\alpha^{2})$
---|---|---
No band geometry | $2[\delta(\alpha)/\pi]^{2}-1$ | $2\alpha^{2}-1$
$c$-band geometry | $2[\delta(\alpha_{\text{eff}})/\pi]^{2}-1$ | $2\alpha^{2}\text{Tr}\\{(P_{c})^{2}\\}-1$
| $\alpha_{\text{eff}}\simeq\alpha\left(1-\text{tr}g^{c}(\boldsymbol{0})\frac{k_{F}^{2}}{d}\right)$ | $P_{c}=\int_{\mathbf{k}}\ket{c,{\mathbf{k}}}\\!\bra{c,{\mathbf{k}}}$
$c$\- and $f$-band geometry | $2[\delta(\alpha_{\text{eff}}(\mathbf{Q}))/\pi]^{2}-1$ | $2\alpha^{2}_{\text{eff}}(\mathbf{Q})-1$
| $\alpha_{\text{eff}}(\mathbf{Q})\simeq\alpha\left(1-\text{tr}g^{c}(\boldsymbol{0})\frac{k_{F}^{2}}{d}-\text{tr}g^{f}(\mathbf{Q})\frac{k_{F}^{2}}{d}\right)$ |
Table 1: Exponents $\gamma$ for the single-hole spectrum $A(\nu)\sim\text{Im}\left[\braket{f^{\dagger}f}(\nu)\right]\sim\nu^{\gamma}$ in $d$ dimensions, relevant for photoemission. Results involving the metric are for a spherical Fermi surface.  $\boldsymbol{\chi(\nu)}$ | Exact in interaction |  $O(\alpha)$
---|---|---
No band geometry | $-2\delta(\alpha)/\pi+2[\delta(\alpha)/\pi]^{2}$ | $-2\alpha$
$c$-band geometry | $-2\delta(\alpha_{\text{eff}})/\pi+2[\delta(\alpha_{\text{eff}})/\pi]^{2}$ | $-2\alpha\lambda_{\text{max}}$
|  | $\lambda_{\text{max}}=\text{max}[\text{EV}(P_{c})]$
$c$\- and $f$-band geometry | $-2\delta(\alpha_{\text{eff}}(\mathbf{Q}))/\pi+2[\delta(\alpha_{\text{eff}}(\mathbf{Q}))/\pi]^{2}$ | $-2\alpha_{\text{eff}}(\mathbf{Q})$
Table 2: Exponents $\gamma$ for the particle-hole spectrum
$\chi(\nu)\sim\text{Im}\left[\braket{cf^{\dagger}fc^{\dagger}}(\nu)\right]\sim\nu^{\gamma}$
in $d$ dimensions, relevant for interband absorption. For definitions of
$\alpha_{\text{eff}},P_{c}$, see Table 1.

##  Appendix A Eigenvalue shift

To determine the maximal Eigenvalue $\lambda_{\text{max}}$ of the Fermi-
surface averaged projector $P^{c}$, we perform an expansion

$\displaystyle P^{c}$ $\displaystyle=\int_{\mathbf{k}}\
\ket{{\mathbf{k}}}\\!\bra{{\mathbf{k}}}\simeq\ket{\boldsymbol{0}}\\!\bra{\boldsymbol{0}}+T_{1}+T_{2}$
(55) $\displaystyle T_{1}$
$\displaystyle=\int_{\mathbf{k}}k_{i}\left(\ket{i}\\!\bra{\boldsymbol{0}}+\ket{\boldsymbol{0}}\\!\bra{i}\right)$
$\displaystyle T_{2}$
$\displaystyle=\int_{{\mathbf{k}}}\frac{k_{i}k_{j}}{2}\left(\ket{ij}\\!\bra{\boldsymbol{0}}+\ket{\boldsymbol{0}}\\!\bra{ij}+\ket{i}\\!\bra{j}+\ket{j}\\!\bra{i}\right)\
,$

where we suppressed the label $c$ in the Bloch functions. Retaining terms up
to second order in the expansion of the Bloch functions, the maximal
eigenvalue reads

$\displaystyle\lambda_{\text{max}}\simeq
1+\braket{\boldsymbol{0}}{T_{1}+T_{2}}{\boldsymbol{0}}+\sum_{\gamma}{|\braket{\gamma}{T_{1}}{\boldsymbol{0}}|^{2}},$
(56)

where $\ket{\gamma}$ are vectors orthogonal to $\ket{\boldsymbol{0}}$. We have
$\braket{\boldsymbol{0}}{T_{1}}{\boldsymbol{0}}=0$ and

$\displaystyle\braket{\boldsymbol{0}}{T_{2}}{\boldsymbol{0}}=-\int_{\mathbf{k}}g_{ij}^{c}k_{i}k_{j}.$
(57)

Furthermore,

$\displaystyle\sum_{\gamma}|\braket{\gamma}{T_{1}}{\boldsymbol{0}}|^{2}=\braket{\boldsymbol{0}}{T_{1}}{\left(\mathbbm{1}-\ket{\boldsymbol{0}}\\!\bra{\boldsymbol{0}}\right)T_{1}|\boldsymbol{0}}=\braket{\boldsymbol{0}}{T_{1}T_{1}}{\boldsymbol{0}}$
$\displaystyle=\int_{{\mathbf{k}},{\mathbf{p}}}k_{i}p_{j}\left(\braket{\boldsymbol{0}}{i}\braket{\boldsymbol{0}}{j}+\braket{\boldsymbol{0}}{i}\braket{j}{\boldsymbol{0}}+\braket{i}{j}+\braket{i}{\boldsymbol{0}}\braket{j}{\boldsymbol{0}}\right)$
$\displaystyle=g_{ij}^{c}\int_{{\mathbf{k}},{\mathbf{p}}}k_{i}p_{j},$ (58)

where in the last step the first two terms cancel and the last two terms can
be symmetrized by relabeling ${\mathbf{k}}\leftrightarrow{\mathbf{p}}$.
Therefore

$\displaystyle\lambda_{\text{max}}$ $\displaystyle\simeq
1-\frac{1}{2}g_{ij}^{c}\int_{{\mathbf{k}},{\mathbf{p}}}(k-p)_{i}(k-p)_{j}$
$\displaystyle\equiv
1-\int_{{\mathbf{k}},{\mathbf{p}}}\frac{||{\mathbf{k}}-{\mathbf{p}}||_{c}^{2}}{2},$
(59)

as in Eq. (39) of the main text.

This result also allows to estimate the remaining Eigenvalues of $P^{c}$ which
appear in the general expression for $\chi(\nu)$, Eq. (32). We note that

$\displaystyle\text{Tr}\\{P^{c}\\}=\int_{\mathbf{k}}\text{Tr}\\{\ket{{\mathbf{k}}}\\!\bra{{\mathbf{k}}}\\}=\int_{\mathbf{k}}=1=\lambda_{\text{max}}+\sum_{\lambda_{i}<\lambda_{\text{max}}}\lambda_{i}\
.$ (60)

Therefore

$\displaystyle\sum_{\lambda_{i}<\lambda_{\text{max}}}\lambda_{i}\simeq\int_{{\mathbf{k}},{\mathbf{p}}}\frac{||{\mathbf{k}}-{\mathbf{p}}||_{c}^{2}}{2}$
(61)

Note also that $P^{c}$ is positive semidefinite, therefore $\lambda_{i}>0\
\forall i$.

##  Appendix B Short Proofs

###  B.1 Upper bound for $|\\!\braket{c,{\mathbf{k}}}{f,{\mathbf{p}}}\\!|^{2}$

We have

$\displaystyle
1=\braket{c,{\mathbf{k}}}{c,{\mathbf{k}}}=\bra{c,{\mathbf{k}}}\left(\sum_{n}\ket{n,{\mathbf{p}}}\\!\bra{n,{\mathbf{p}}}\right)\ket{c,{\mathbf{k}}}\geq$
$\displaystyle|\\!\braket{c,{\mathbf{k}}}{f,{\mathbf{p}}}\\!|^{2}+|\\!\braket{c,{\mathbf{k}}}{c,{\mathbf{p}}}\\!|^{2}=|\\!\braket{c,{\mathbf{k}}}{f,{\mathbf{p}}}\\!|^{2}+1-O(x_{c}),$

therefore

$\displaystyle|\\!\braket{c,{\mathbf{k}}}{f,{\mathbf{p}}}\\!|^{2}=O(x_{c})\ .$
(62)

In the same manner, one can also show that

$\displaystyle|\\!\braket{c,{\mathbf{k}}}{f,{\mathbf{p}}}\\!|^{2}=O(x_{f}),\quad
x^{f}=k_{F}^{2}\text{tr}g^{f}(\boldsymbol{0})\ .$ (63)

###  B.2 $\mathbf{Q}$-independence of overlap factors for
$\mathbf{Q}=O(k_{F})$

Consider the Bloch-overlap factor
$\text{BO}_{f}(\mathbf{Q})\equiv\text{Tr}\\{P^{f}_{\mathbf{Q}}P^{f}_{\mathbf{Q}+{\mathbf{k}}-{\mathbf{q}}}P^{f}_{\mathbf{Q}+{\mathbf{p}}-{\mathbf{q}}}\\}$
from Eq. (46). To shorten notation, we write
$\mathbf{a}={\mathbf{k}}-{\mathbf{q}}$,
$\mathbf{b}={\mathbf{p}}-{\mathbf{q}}$. When $\mathbf{Q}=O(k_{F})$, we can
expand all projectors to second order in momenta with notation similar to Eq.
(55):

$\displaystyle
P^{f}_{\mathbf{Q}}\simeq\ket{f,\boldsymbol{0}}\\!\bra{f,\boldsymbol{0}}+Q_{i}T_{1}^{i}+Q_{i}Q_{j}T_{2}^{ij},$
(64) $\displaystyle
T_{1}^{i}=\left(\ket{f,i}\\!\bra{f,\boldsymbol{0}}+\ket{f,\boldsymbol{0}}\\!\bra{f,i}\right)$
$\displaystyle T_{2}^{ij}=$
$\displaystyle\frac{1}{2}\left(\ket{f,ij}\\!\bra{f,\boldsymbol{0}}+\ket{f,\boldsymbol{0}}\\!\bra{f,ij}+\ket{f,i}\\!\bra{f,j}+\ket{f,j}\\!\bra{f,i}\right)$

Collecting all second-order terms (first order terms vanish), we obtain

$\displaystyle\text{BO}_{f}(\mathbf{Q})\simeq\braket{f,\boldsymbol{0}}{T_{1}^{i}T_{1}^{j}}{f,\boldsymbol{0}}\times$
(65)
$\displaystyle\left(Q_{i}(Q+a)_{j}+(Q+a)_{i}(Q+b)_{j}+(Q+b)_{i}Q_{j}\right)+$
$\displaystyle\braket{f,\boldsymbol{0}}{T_{2}^{ij}}{f,\boldsymbol{0}}\times$
$\displaystyle\left(Q_{i}Q_{j}+(Q+a)_{i}(Q+a)_{j}+(Q+b)_{i}(Q+b)_{j}\right)\
.$

Using that
$\braket{f,\boldsymbol{0}}{\frac{1}{2}\left(T_{1}^{i}T_{1}^{j}+T_{1}^{j}T_{1}^{i}\right)}{f,\boldsymbol{0}}=g_{ij}^{f}$,
see Eq. (58), and
$\braket{f,\boldsymbol{0}}{T_{2}^{ij}}{f,\boldsymbol{0}}=-g_{ij}^{f}$, we
immediately see that all terms involving $Q$ appear in symmetrical
combinations and therefore cancel. From the form of
$\text{BO}_{f}(\mathbf{Q})$ it is clear that this property will also
generalize to higher order.

##  Appendix C Effective interaction for full band geometry

To illustrate the derivation of

$\displaystyle\alpha_{\text{eff}}\equiv\alpha\left(1-\int_{{\mathbf{k}}}||{\mathbf{k}}||^{2}_{c}+||{\mathbf{k}}||^{2}_{f,\mathbf{Q}}\right),$
(66)

we consider the third order Bloch overlap

$\displaystyle\text{BO}(\mathbf{Q})\equiv\int_{{\mathbf{k}},{\mathbf{p}},{\mathbf{q}}}\text{Tr}\\{P^{c}_{\mathbf{k}}P^{c}_{\mathbf{p}}P^{c}_{\mathbf{q}}\\}\times\text{Tr}\\{P^{f}_{\mathbf{Q}}P^{f}_{\mathbf{Q}+{\mathbf{k}}-{\mathbf{q}}}P^{f}_{\mathbf{Q}+{\mathbf{p}}-{\mathbf{q}}}\\}.$
(67)

We evaluate this expression by expanding the projectors up to second order in
momenta as in Eq. (64). Under the assumption of an inversion-symmetric Fermi
surface, where mixed terms in momenta vanish [Eq. (47)], this expansion can be
performed by sequentially keeping one of the momenta
${\mathbf{k}},{\mathbf{p}},{\mathbf{q}}$ non-zero, and setting the remaining
ones to zero; furthermore, we only need to keep the terms of the form
$T_{2}^{ij}$ from Eq. (64). Lastly, at order $O(x)$, the corrections from the
two traces in Eq. (67) add up and can be evaluated independently.

As a result, we obtain

$\displaystyle\int_{{\mathbf{k}},{\mathbf{p}},{\mathbf{q}}}\text{Tr}\\{P_{\mathbf{k}}^{c}P_{\mathbf{p}}^{c}P_{\mathbf{q}}^{c}\\}=1-3\int_{\mathbf{k}}||{\mathbf{k}}||_{c}^{2}+O((x_{c})^{2})$
(68)

For the $f$-projector trace, a possible difference is that the momentum
$-{\mathbf{q}}$ appears in two projectors; however, since these projectors are
adjacent, upon setting ${\mathbf{k}}={\mathbf{p}}=0$ as discussed above, we
have
$P^{f}_{\mathbf{Q}-{\mathbf{q}}}P^{f}_{\mathbf{Q}-{\mathbf{q}}}=P^{f}_{\mathbf{Q}-{\mathbf{q}}}$,
and the ${\mathbf{q}}$-integral therefore gives the same contribution as the
${\mathbf{k}},{\mathbf{p}}$-integrals. Thus

$\displaystyle\text{BO}(\mathbf{Q})=1-3\int_{{\mathbf{k}}}\left(||{\mathbf{k}}||^{2}_{c}+||{\mathbf{k}}||^{2}_{f,\mathbf{Q}}\right)\
.$ (69)

For a generic $n$-th order diagram (excluding tadpole-diagrams), the
evaluation proceeds in the same manner, including diagram which contain
multiple $c$-fermion loops. In particular, one can easily convince oneself
that “non-zero” momenta only appear in adjacent $f$-projectors; see Fig. 10
for an illustration of this fact.

Figure 10: Generic diagram with one $c$-fermion momentum ${\mathbf{q}}$ set to
non-zero only. The momentum ${\mathbf{q}}$ runs in a a loop through the
diagram – the propagators and interaction lines which transport ${\mathbf{q}}$
are marked blue. This ensures that all projectors
$P^{f}_{\mathbf{Q}-{\mathbf{q}}}$ in the trace are adjacent to each other.

Therefore, the corresponding overlap reads,

$\displaystyle\text{BO}_{n}(\mathbf{Q})=1-n\int_{\mathbf{k}}\left(||{\mathbf{k}}||^{2}_{c}+||{\mathbf{k}}||^{2}_{f,\mathbf{Q}}\right)\
.$ (70)

Including the coupling constant $\alpha$, we have, at $n$-th order

$\displaystyle\alpha^{2}\left[1-n\int_{\mathbf{k}}\left(||{\mathbf{k}}||^{2}_{c}+||{\mathbf{k}}||^{2}_{f,\mathbf{Q}}\right)\right]=\alpha_{\text{eff}}^{n}+O(x^{2}),$
(71)

where $\alpha_{\text{eff}}$ is given in Eq. (66).

## References

  * Mahan [1967] G. D. Mahan, Phys. Rev. 163, 612 (1967).
  * Nozières  _et al._ [1969] P. Nozières, J. Gavoret, and B. Roulet, Phys. Rev. 178, 1084 (1969).
  * Roulet  _et al._ [1969] B. Roulet, J. Gavoret, and P. Nozières, Phys. Rev. 178, 1072 (1969).
  * Ohtaka and Tanabe [1990] K. Ohtaka and Y. Tanabe, Rev. Mod. Phys. 62, 929 (1990).
  * Nozières and De Dominicis [1969] P. Nozières and C. T. De Dominicis, Phys. Rev. 178, 1097 (1969).
  * Combescot and Nozieres [1971] M. Combescot and P. Nozieres, Journal de Physique 32, 913 (1971).
  * Anderson [1967] P. W. Anderson, Phys. Rev. Lett. 18, 1049 (1967).
  * Gavoret  _et al._ [1969] J. Gavoret, P. Nozieres, B. Roulet, and M. Combescot, Journal de Physique 30, 987 (1969).
  * Rosch and Kopp [1995] A. Rosch and T. Kopp, Phys. Rev. Lett. 75, 1988 (1995).
  * Pimenov  _et al._ [2017] D. Pimenov, J. von Delft, L. Glazman, and M. Goldstein, Phys. Rev. B 96, 155310 (2017).
  * Schmidt  _et al._ [2018] R. Schmidt, M. Knap, D. A. Ivanov, J.-S. You, M. Cetina, and E. Demler, Reports on Progress in Physics 81, 024401 (2018).
  * Levinsen and Parish [2015] J. Levinsen and M. M. Parish, Strongly interacting two-dimensional fermi gases, in  _Annual Review of Cold Atoms and Molecules_, Vol. Volume 3 (World Scientific, 2015) pp. 1–75.
  * Massignan  _et al._ [2014] P. Massignan, M. Zaccanti, and G. M. Bruun, Reports on Progress in Physics 77, 034401 (2014).
  * Hentschel and Guinea [2007] M. Hentschel and F. Guinea, Phys. Rev. B 76, 115407 (2007).
  * Sindona  _et al._ [2013] A. Sindona, J. Goold, N. Lo Gullo, S. Lorenzo, and F. Plastina, Phys. Rev. Lett. 111, 165303 (2013).
  * Lee [2010] H. C. Lee, Phys. Rev. B 82, 153410 (2010).
  * Cao  _et al._ [2018] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Nature 556, 80 (2018).
  * Andrei  _et al._ [2021] E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani, and A. F. Young, Nature Reviews Materials 6, 201 (2021).
  * Leykam  _et al._ [2018] D. Leykam, A. Andreanov, and S. Flach, Advances in Physics: X 3, 1473052 (2018).
  * Taie  _et al._ [2015] S. Taie, H. Ozawa, T. Ichinose, T. Nishio, S. Nakajima, and Y. Takahashi, Science Advances 1, e1500854 (2015).
  * Ozawa  _et al._ [2017] H. Ozawa, S. Taie, T. Ichinose, and Y. Takahashi, Phys. Rev. Lett. 118, 175301 (2017).
  * Taie  _et al._ [2020] S. Taie, T. Ichinose, H. Ozawa, and Y. Takahashi, Nature Communications 11, 257 (2020).
  * Punk and Zwerger [2007] M. Punk and W. Zwerger, Phys. Rev. Lett. 99, 170404 (2007).
  * Liu  _et al._ [2020a] W. E. Liu, Z.-Y. Shi, M. M. Parish, and J. Levinsen, Phys. Rev. A 102, 023304 (2020a).
  * Note [1] For results on Orthogonality Catastrophe for band insulators, see Ref. [63]).
  * Note [2] A discussion of edge singularities in the context of long-ranged Coulomb interactions can be found in Ref. [64].
  * Tai and Claassen [2023] W. T. Tai and M. Claassen, arXiv preprint arXiv:2303.01597 (2023).
  * Beugeling  _et al._ [2012] W. Beugeling, J. C. Everts, and C. Morais Smith, Phys. Rev. B 86, 195129 (2012).
  * Mahan [2000] G. D. Mahan,  _Many-particle physics_ (Springer Science & Business Media, 2000).
  * Note [3] Up to factors $\theta(-t)e^{iE_{0}t}$ which we suppress here for brevity.
  * Pimenov and Goldstein [2018] D. Pimenov and M. Goldstein, Phys. Rev. B 98, 220302 (2018).
  * Srivastava and Imamoğlu [2015] A. Srivastava and A. m. c. Imamoğlu, Phys. Rev. Lett. 115, 166802 (2015).
  * Marzari and Vanderbilt [1997] N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997).
  * Cheng [2010] R. Cheng, arXiv preprint arXiv:1012.1337 (2010).
  * Neupert  _et al._ [2013] T. Neupert, C. Chamon, and C. Mudry, Phys. Rev. B 87, 245103 (2013).
  * Roy [2014] R. Roy, Phys. Rev. B 90, 165139 (2014).
  * Ozawa and Mera [2021] T. Ozawa and B. Mera, Phys. Rev. B 104, 045103 (2021).
  * Kolodrubetz  _et al._ [2013] M. Kolodrubetz, V. Gritsev, and A. Polkovnikov, Phys. Rev. B 88, 064304 (2013).
  * Ozawa and Goldman [2018] T. Ozawa and N. Goldman, Phys. Rev. B 97, 201117 (2018).
  * Klees  _et al._ [2020] R. L. Klees, G. Rastelli, J. C. Cuevas, and W. Belzig, Phys. Rev. Lett. 124, 197002 (2020).
  * von Gersdorff and Chen [2021] G. von Gersdorff and W. Chen, Phys. Rev. B 104, 195133 (2021).
  * Tan  _et al._ [2019] X. Tan, D.-W. Zhang, Z. Yang, J. Chu, Y.-Q. Zhu, D. Li, X. Yang, S. Song, Z. Han, Z. Li, Y. Dong, H.-F. Yu, H. Yan, S.-L. Zhu, and Y. Yu, Phys. Rev. Lett. 122, 210401 (2019).
  * Yu  _et al._ [2019] M. Yu, P. Yang, M. Gong, Q. Cao, Q. Lu, H. Liu, S. Zhang, M. B. Plenio, F. Jelezko, T. Ozawa, N. Goldman, and J. Cai, National Science Review 7, 254 (2019).
  * Note [4] Note that $|g^{c}_{ij}|<\text{tr}g^{c}$ for a positive semidefinite matrix $g^{c}$.
  * Wang  _et al._ [2021] J. Wang, J. Cano, A. J. Millis, Z. Liu, and B. Yang, Phys. Rev. Lett. 127, 246403 (2021).
  * Latta  _et al._ [2011] C. Latta, F. Haupt, M. Hanl, A. Weichselbaum, M. Claassen, W. Wuester, P. Fallahi, S. Faelt, L. Glazman, J. von Delft, H. E. Türeci, and A. Imamoglu, Nature 474, 627 (2011).
  * Note [5] Standard RF spectroscopy, which is akin to ARPES, measures the occupied part of the spectral function, which complicates the detection of impurity spectra.
  * Knap  _et al._ [2012] M. Knap, A. Shashi, Y. Nishida, A. Imambekov, D. A. Abanin, and E. Demler, Phys. Rev. X 2, 041020 (2012).
  * Cheuk  _et al._ [2012] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012).
  * Ness  _et al._ [2020] G. Ness, C. Shkedrov, Y. Florshaim, O. K. Diessel, J. von Milczewski, R. Schmidt, and Y. Sagi, Phys. Rev. X 10, 041019 (2020).
  * Törmä  _et al._ [2018] P. Törmä, L. Liang, and S. Peotta, Phys. Rev. B 98, 220511 (2018).
  * Peotta and Törmä [2015] S. Peotta and P. Törmä, Nature Communications 6, 8944 (2015).
  * Iskin [2021] M. Iskin, Phys. Rev. A 103, 053311 (2021).
  * Chevy [2006] F. Chevy, Phys. Rev. A 74, 063628 (2006).
  * Halász and Chalker [2016] G. B. Halász and J. T. Chalker, Phys. Rev. B 94, 235105 (2016).
  * Grusdt  _et al._ [2016] F. Grusdt, N. Y. Yao, D. Abanin, M. Fleischhauer, and E. Demler, Nature Communications 7, 11994 (2016).
  * Camacho-Guardian  _et al._ [2019] A. Camacho-Guardian, N. Goldman, P. Massignan, and G. M. Bruun, Phys. Rev. B 99, 081105 (2019).
  * Grusdt  _et al._ [2019] F. Grusdt, N. Y. Yao, and E. A. Demler, Phys. Rev. B 100, 075126 (2019).
  * Liu  _et al._ [2020b] R. Liu, Y.-R. Shi, and W. Zhang, Phys. Rev. A 102, 033305 (2020b).
  * Pimenov  _et al._ [2021] D. Pimenov, A. Camacho-Guardian, N. Goldman, P. Massignan, G. M. Bruun, and M. Goldstein, Phys. Rev. B 103, 245106 (2021).
  * Ahn  _et al._ [2022] J. Ahn, G.-Y. Guo, N. Nagaosa, and A. Vishwanath, Nature Physics 18, 290 (2022).
  * Guenther  _et al._ [2021] N.-E. Guenther, R. Schmidt, G. M. Bruun, V. Gurarie, and P. Massignan, Phys. Rev. A 103, 013317 (2021).
  * Gu and Sun [2016] J. Gu and K. Sun, Phys. Rev. B 94, 125111 (2016).
  * Tupitsyn and Prokof’ev [2019] I. S. Tupitsyn and N. V. Prokof’ev, Phys. Rev. B 99, 245122 (2019).