Kondo breakdown in multi-orbital Anderson lattices induced by destructive
hybridization interference

Fabian Eickhoff1$\star$ and Frithjof B. Anders2$\dagger$

1 Institute For Software Technology, German Aerospace Center, 51147 Cologne,
Germany

2 Department of Physik, condensed matter theory, TU Dortmund University, 44221
Dortmund, Germany

$\star$<EMAIL_ADDRESS>, $\dagger$<EMAIL_ADDRESS>
## Abstract

In this paper we consider a multi band extension to the periodic Anderson
model. We use a single site DMFT(NRG) in order to study the impact of the
conduction band mediated effective hopping of the correlated electrons between
the correlated orbitals onto the heavy Fermi liquid formation. Whereas the
hybridization of a single impurity model with two distinct conduction bands
always adds up constructively, $T_{K}\propto\exp(-\alpha
U/(\Gamma_{1}+\Gamma_{2}))$, we show that this does not have to be the case in
lattice models, where, in remarkable contrast, also an emerging low-energy
Fermi liquid scale $T_{0}\propto\exp(-\alpha U/(\Gamma_{1}-\Gamma_{2}))$ is
possible due to quantum interference effects in multi band models. At high
symmetry points, heavy Fermi liquid formation is suppressed which is
associated with a breakdown of the Kondo effect. This results in an
asymptotically scale-invariant (i.e., power-law) spectrum of the correlated
orbitals $\propto|\omega|^{1/3}$ and a small Fermi surface including only the
light quasi-particles. This orbital selective Mott phase demonstrates the
possibility of metallic local criticality within the general framework of
ordinary single site DMFT.

###### Contents

  1. 1 Introduction
  2. 2 Model and Method
    1. 2.1 Model
    2. 2.2 Method
  3. 3 Theoretical motivation
  4. 4 Results
    1. 4.1 Destructive hybridization interference
    2. 4.2 Generality of destructive hybridization interference
    3. 4.3 Spectral functions and Kondo breakdown QCP
      1. 4.3.1 Particle-hole symmetric $f$-orbitals
      2. 4.3.2 Properties of the QCP for PH symmetric f-sites
      3. 4.3.3 PH asymmetric $f$-sites
  5. 5 Summary and discussion
  6. A DMFT integrals

##  1 Introduction

Condensed matter physics has long relied on the Fermi-liquid framework [1] to
understand the low-temperature behavior of metals. However, certain correlated
quantum materials defy this understanding, showcasing non-Fermi liquid
behavior either as stable phases or through zero-temperature quantum phase
transitions, captivating researchers exploring quantum criticality.

Initially research focused on metallic transitions driven by spin-density-
waves (antiferromagnetism or ferromagnetism) within the Landau-Ginzburg-Wilson
(LGW) framework [2, 3], but the spotlight now turns to even more peculiar
transitions where LGW fails, notably observed in heavy Fermion materials [1,
4]. This novel class of phase transitions cannot be understood in terms of
local order parameter fluctuations of a collective magnetic or charge ordered
phase.

Inelastic neutron scattering experiments have revealed valuable insights into
these strange metallic quantum critical points (QCPs) [5, 6, 7]. These
experiments detect unusual features in the frequency and temperature
dependencies of dynamical spin susceptibility, marked by an anomalous power
law exponent and low-frequency $\omega/T$ scaling. Significantly, this
anomalous behavior is consistently observed throughout the Brillouin zone,
indicating that these QCPs are governed by local critical fluctuations rather
then collective excitations in the lattice.

In the context of heavy fermion materials, local criticality is often
discussed as Kondo breakdown QCP (KB-QCP) [8, 9, 10, 11], where localized and
itinerant electrons effectively decouple at low energies leading to transition
of a large to a small Fermi surface. This scenario is connected to orbital-
selective Mott transitions (OSM) [12, 13, 14], where some electrons experience
Mott localization while others remain itinerant. In these cases the emergence
of magnetic order may occur as an instability of the Mott localized phase in
order to remove the entropy of the local moments present in a paramagnetic
Mott insulator, but is not necessarily required for the occurrence of the QCP.

While impurity problems with boundary phase transitions are by now well
understood [15, 16, 17, 18], the origin of local criticality in lattice
problems still lacks a sufficient theoretical description.

Within the generic framework of single-site dynamical mean field theory
(DMFT), it has been demonstrated, for instance, that the OSM phase is in
general unstable against hybridization of the localized states with a Fermi
surface [19]. This raises questions about the existence of the KB-QCP in the
periodic Anderson model (PAM), which is commonly used to describe the
fundamental properties of heavy fermion compounds and where such hybridization
is always present.

In a single-site DMFT, the KB-QCP can only be stabilized if the hybridization
is assumed to vanish precisely at the Fermi energy [20] or if two-dimensional
magnetic bulk fluctuations are considered to effectively couple to the local
moment degrees of freedom [21, 4], weakening the local nature of the QCP.

Alternatively, two-site cluster extensions of DMFT (CDMFT) have been applied
to the PAM, revealing a KB-QCP stabilized by inter-site correlations [22, 23,
24]. This is a DMFT implementation of the Doniach picture [25] where an
antiferromagnetic ordered phase mediated by RKKY interaction at small
hybridization competes with heavy Fermi liquid whose low energy scale is
connected to the Kondo temperature. While the QCP is destroyed in the two-
impurity model [26, 27, 28] due to particle-hole symmetry breaking it can be
stabilized in a two-site CDMFT [24] due to the lattice self-consistency
condition. Yet, within this extension, the strong inter-site correlations
driving the phase transition are simultaneously neglected between sites of
different clusters. It remains unclear whether the appearance of the KB-QCP is
an artifact of this two-site cluster approximation or if it persists when the
cluster size is systematically increased. A Kondo breakdown occurring in the
magnetic phase of the half-filled KLM on the honeycomb lattice was reported by
Raczkowski et al. [29]. In this scenario the question arrises how a
realization of Dirac electrons leading to a Kondo breakdown transition in the
impurity limit carries over to the half-filled lattice model.

In this paper, we propose a different mechanism promoting quantum criticality
in heavy fermion materials, which solely relies on destructive quantum
interference effects in a multi-band scenario, realized for example by
depleted versions of the PAM.

While in single or few impurity problems the dynamic screening of the emerging
local moments are governed by the Kondo effect, this does not hold for multi-
impurity models [30, 31, 32] with large number of correlated orbitals. In such
models the real part of the hybridization function becomes more important
since there are not enough Kondo screening channels available. We demonstrate
within a single-site DMFT, that in a multi-orbital extension of the common
PAM, destructive interference in hybridization can lead to an exponentially
reduced low-energy screening scale, the Kondo lattice temperature, without
reducing the local Kondo coupling. This effect is unique to lattice models, as
interference is always constructive in the corresponding single-impurity
Anderson model (SIAM).

In addition to the fact, that such a reduction of the Kondo lattice
temperature makes the system highly susceptible to the occurrence of magnetic
order in the spirit of the traditional Doniach picture [25], which is beyond
our single site DMFT analysis, we establish the existence of another critical
point, which aligns with the concept of local criticality. Approaching these
high symmetry points the Kondo lattice temperature can be continuously turned
to zero, and the local moments gradually decouple from the Fermi surface. A
line of local QCP emerge which are characterized by the degree of particle-
hole asymmetry.

The paper proceeds as follows: Section 2 details the model and methodology. In
Section 3, we provide a theoretical motivation and discuss recent developments
in strongly correlated impurity models. Section 4 unveils our results,
including the impact of destructive hybridization interference and the
emergence of a Kondo breakdown quantum critical point. Finally we summarize
and discuss our results in section 5.

##  2 Model and Method

###  2.1 Model

In this study, we investigate a multi orbital extension of the well-known
Periodic Anderson Model (PAM). The model is defined as follows:

$\displaystyle H=H^{\text{c}}+H^{\text{f}}+H^{\text{hyb}}\,.$ (1)

The individual components of this Hamiltonian describe various aspects of the
system: the non-interacting bands denoted as $H^{\text{c}}$, the interacting
flat band as $H^{\text{f}}$, and the hybridization term as $H^{\text{hyb}}$:

$\displaystyle H^{\text{c}}$
$\displaystyle=\sum_{\mu}^{N_{\nu}}\sum_{\vec{k},\sigma}(\epsilon^{\text{c}}_{\vec{k}\nu}+\epsilon^{\text{c}}_{\nu})n^{\text{c}}_{k\sigma\nu}\,,$
(2) $\displaystyle H^{\text{f}}$
$\displaystyle=\sum_{l}^{N_{f}}\epsilon^{\text{f}}(n^{\text{f}}_{l\uparrow}+n^{\text{f}}_{l\downarrow})+Un^{\text{f}}_{l\uparrow}n^{\text{f}}_{l\downarrow}\,,$
(3) $\displaystyle H^{\text{hyb}}$
$\displaystyle=\sum_{\mu}^{N_{\nu}}\sum_{l,\sigma}V_{\nu}f^{\dagger}_{l\sigma}c_{l\sigma\nu}+\text{h.c.}\,,$
(4)

with
$n^{\text{c}}_{\vec{k}\sigma\nu}=c^{\dagger}_{\vec{k}\sigma\nu}c_{\vec{k}\sigma\nu}$
and $n^{\text{f}}_{l\sigma}=f^{\dagger}_{l\sigma}f_{l\sigma}$. The creation
and annihilation operators $c^{\dagger}_{\vec{k}\sigma\nu}$ and
$c_{\vec{k}\sigma\nu}$ create (annihilate) electrons with spin $\sigma$ and
momentum $\vec{k}$ in the non-interacting band with band index $\nu$ and
dispersion $\epsilon^{\text{c}}_{\vec{k}\nu}+\epsilon^{\text{c}}_{\nu}$, while
$f^{\dagger}_{l\sigma}$ ($f_{l\sigma}$) creates (annihilates) an electron with
spin $\sigma$ in the $f$-orbital at the real space lattice site $l$. $N_{\nu}$
denotes the number of conduction bands and $N_{f}$ the number of correlated
f-sites. The correlation effect of the $f$-orbitals is taken into account by
the local Coulomb interaction of strength $U$. It is important to note that
due to the hybridization $V_{\nu}$, the particle number in each band $\nu$ is
not a conserved quantity, distinguishing it from multi-channel models with
conserved channel symmetry [33, 34, 35].

In most studies of these kind of models only a single dispersive band $\nu$ is
considered, however, as we explain in detail in Sec. 4.2, multiple bands
inherently emerge in depleted versions of the standard PAM for example. If the
lattice site index $l$ in Eqs. (3) and (4) only exhausts a subset of $N_{f}$
different lattice sites, the model is known as multi impurity Anderson model
(MIAM), where the special case $N_{f}=1$ corresponds to the well understood
SIAM. If, on the other hand, $l$ exhausts all lattice sites we restore the
translational invariant PAM [36, 37].

###  2.2 Method

To tackle this complex model in the translational invariant form (PAM), we
employ the DMFT [38, 39], which maps the lattice problem onto an effective
single impurity Anderson model. The central DMFT equation therefore reads:

$\displaystyle
G^{\text{f}}_{\text{lat},\sigma}(z)=G^{\text{f}}_{\text{imp},\sigma}(z)\,.$
(5)

Here, $G^{\text{f}}_{\text{lat},\sigma}(z)$ and
$G^{\text{f}}_{\text{imp},\sigma}(z)$ describe the local single-particle
$f$-Greens function of the lattice and the impurity problem, respectively.
These spectral functions are formally defined as:

$\displaystyle G^{\text{f}}_{\text{lat},\sigma}(z)$
$\displaystyle=\sum_{\vec{k}}\frac{1}{z-\epsilon^{\text{f}}-\sum_{\nu}V_{\nu}^{2}G^{\text{0c}}_{\vec{k}\nu}(z)-\Sigma_{\sigma}(z)}\,,$
(6) $\displaystyle G^{\text{f}}_{\text{imp},\sigma}(z)$
$\displaystyle=\frac{1}{z-\epsilon^{\text{f}}-\Delta_{\sigma}(z)-\Sigma_{\sigma}(z)}\,.$
(7)

In these equations,
$G^{\text{0c}}_{\vec{k}\nu}(z)=(z-\epsilon^{\text{c}}_{\vec{k}\nu}-\epsilon^{\text{c}}_{\nu})^{-1}$
represents the propagator of a particle in the $\nu$-th non-interacting band,
$\Sigma_{\sigma}(z)$ is the correlation-induced part of the self-energy, and
$\Delta_{\sigma}(z)$ characterizes the effective medium in which the single
impurity is embedded. Importantly, $\Sigma_{\sigma}(z)$ itself depends on the
effective medium $\Delta_{\sigma}(z)$, making Eq. (5) a self-consistent
problem. The only approximation made by DMFT is to neglect the momentum
dependence of the self energy,
$\Sigma_{\vec{k}\sigma}(z)\approx\Sigma_{\sigma}(z)$, which gets exact for a
lattice with infinite coordination number, i.e. in infinite dimensions [38,
39].

##  3 Theoretical motivation

In this section, we delve into the theoretical motivation behind our study,
focusing on the interplay between dispersive bands and correlated $f$-orbitals
in lattice models. We begin by considering a specific scenario with a single
dispersive band, $N_{\nu}=1$, described by Eq. (2) and $N_{f}$ correlated
$f$-orbitals coupled to selected lattice sites.

The influence of the band onto the dynamics of these $f$-orbitals is
completely determined by the complex hybridization function, denoted as:

$\displaystyle\Delta_{lm}(z)=V^{2}G^{0\text{c}}_{lm}(z),$ (8)

where $G^{0\text{c}}_{lm}(z)$ describes the propagation of an electron from
site $m$ to site $l$ in the non interacting band.

In a previous study [30], we conducted a thorough analysis of the
hybridization matrix in Eq. (8), unveiling intriguing insights. We observed
that as the number of correlated orbitals, $N_{f}$, increases, the relevance
of the imaginary part diminishes, while the significance of the real part
grows. The complex hybridization matrix $\Delta_{lm}(\omega+i\delta)$ can be
split into a real and imaginary part for $\delta\to 0^{+}$:

$\displaystyle\Delta_{lm}(\omega+i\delta)$ $\displaystyle=$
$\displaystyle\Re\Delta_{lm}(\omega)+i\Delta_{lm}(\omega)$ (9)
$\displaystyle=$ $\displaystyle T_{lm}(\omega)+i\Gamma_{lm}(\omega).$

We have shown that the dynamics of a MIAM with $N_{f}$ sites is determined by
the matrices at $\omega=0$ in the wide band limit. While $T_{lm}(0)$ is
interpreted as an effective hopping between two correlated sites $l$ and $m$,
the eigenvalues of $\Gamma_{lm}$ determine the coupling to the effective
conduction bands. The number of such bands is given by the rank of the matrix
$\Gamma_{lm}$. While the rank is $N_{f}$ for small values of $N_{f}$, we have
proven that for $N_{f}$ exceeding a certain threshold which dependent on model
specifics that rank$(\Gamma_{lm})<N_{f}$. This implies that for large $N_{f}$,
many eigenvalues of the matrix $\Gamma_{lm}$ must be zero, the notion of
individual Kondo screening via an effective conduction band coupling does not
make sense any longer. $T_{lm}(0)$ becomes dominant and neglecting the real
part will even alter the nature of the low-energy fixed point (FP) since it is
responsible for the Fermi liquid formation in a similar way as observed in the
Hubbard model. In fact, for rank$(\Gamma_{lm})=0$, the model becomes a Hubbard
model in the wide band limit with a single particle inter-site hopping
$T_{lm}(0)$, as demonstrated in Sec. II.F.3 of Ref. [30]. For an in-depth
exploration of this analysis, refer to Fig. 1 in Ref. [30] and the
accompanying explanations.

Our understanding of these hybridization function intricacies has enabled the
description of local moment formation in Kondo-Hole systems [31] and the
evolution of the hybridization gap in the spectral function of strongly
correlated impurity clusters [32]. These findings are in notable agreement
with rigorous theoretical statements made by the Lieb-Mattis theorem [40, 41].

In this study, we shift our focus to another aspect, prompted by the
aforementioned crossover behavior, which holds particular relevance for
lattice models with more than one $f$\- and one $c$-orbital per unit cell. In
such scenarios, the hybridization function in Eq. (8) undergoes slight
modifications and can be expressed as:

$\displaystyle\Delta_{lm}(z)=\sum_{\nu}V_{\nu}^{2}G^{0\text{c}}_{\nu,l,m}(z).$
(10)

Of note, the sign of the imaginary part of individual propagators remains
fixed due to causality, ensuring that their contributions consistently sum up.
In the case of the SIAM, this implies that the effective hybridization with
the continuum can only increase, resulting in an enhanced Kondo temperature.

Conversely, the sign of the real part remains undetermined, allowing
contributions from different bands to potentially offset or even cancel each
other out. Consequently, if the screening of the local moment is primarily
dominated by the real part of the propagator, additional bands can have strong
influence and might even suppress the low-energy screening scale.

In the subsequent sections, we employ the combination of DMFT and the
numerical renormalization group (NRG) approach to test this hypothesis within
the framework of the PAM described by Eq. (1), while considering the presence
of two conduction bands.

##  4 Results

To address the properties of the effective SIAM arising from the DMFT self-
consistency equation, Eq. (5), we employ the Numerical Renormalization Group
(NRG) technique, as implemented in the TRIQS open-source package [42]. Details
on how we accurately solved the DMFT integrals can be found in appendix A.

Unless explicitly stated otherwise we use a constant density of state (DOS)
for the two dispersive bands, $N_{\nu}=2$ and
$\rho^{\text{c}}(\omega)=\Theta(|\omega|-D_{\nu})\frac{1}{2D_{\nu}}$, where
$D_{1}=D_{2}=D$ represent the individual band widths, which are chosen to be
identical. This ensures consistency and avoids complications in the
interpretation of the results arising from frequency-dependent DOS.

In addition, the centers of the individual bands, $\epsilon^{\text{c}}_{\nu}$,
are always shifted from the Fermi energy, $|\epsilon^{\text{c}}_{\nu}|>0$, in
order to avoid the Kondo insulator regime when recovering the limit of the
standard PAM with only one conduction band [36].

It is important to emphasize that our results maintain their qualitative
validity regardless of the specific assumptions made, including the choice of
the DOS and the uniform band widths.

In order to analyze the influence of the second band onto the screening of the
local moment at each correlated site $l$ without changing the overall strength
of hybridization, we define a parameter $\alpha$

$\displaystyle V^{2}_{2}=\alpha V^{2}_{1}=\frac{\alpha}{1+\alpha}V^{2},$ (11)

such that $V^{2}_{1}+V^{2}_{2}=V^{2}$ is always fulfilled. Consequently, we
can use $\Gamma$ as our unit of energy, which is defined as:

$\displaystyle\Gamma=\Gamma_{1}+\Gamma_{2}=\frac{\pi V^{2}}{2D}.$ (12)

Notably, by choosing the definition in Eq. (11) we ensure independence of the
leading order of the Kondo temperature of the corresponding SIAM with respect
to $\alpha$ [43]:

$\displaystyle T_{\text{K}}\propto\sqrt{\Gamma U}\exp\left(-\frac{\pi
U}{8\Gamma}\right),$ (13)

for a particle-hole symmetric f-site, $\epsilon^{\text{f}}=-U/2$.

This consistent framework allows for meaningful comparisons and facilitates a
clear understanding of the physical behavior in multi-orbital Anderson
lattices.

###  4.1 Destructive hybridization interference

Within the framework of DMFT, the low energy scale in the lattice problem,
denoted as $T_{0}$, shares a similar analytical form to Eq. (13), as we
effectively map the model onto an effective SIAM. However, due to the self-
consistency procedure, the effective hybridization, represented as
$\tilde{\Gamma}$, can undergo significant renormalization. To gain insight
into this renormalization within the lattice context, we systematically
investigate the influence of a second band.

As evident from the DMFT equation (5), the influence of the bands on the
dynamics of the $f$-orbitals is solely determined by the weighted sum of
individual single-particle Green’s functions of the band electrons,

$\displaystyle\sum_{\nu}V_{\nu}^{2}G^{0\text{c}}_{\vec{k}\nu}(z)=\frac{V_{1}^{2}}{z-(\epsilon^{\text{c}}_{\vec{k}1}+\epsilon^{\text{c}}_{1})}+\frac{V_{2}^{2}}{z-(\epsilon^{\text{c}}_{\vec{k}2}+\epsilon^{\text{c}}_{2})},$
(14)

where its analytic low frequency properties, $\omega/D\ll 1$, significantly
determines the low-temperature physics of the problem.

To meticulously investigate the impact of destructive hybridization
interference, we now make the assumption that the dispersion relations of the
two bands satisfy the following relation:

$\displaystyle\epsilon^{\text{c}}_{\vec{k}1}+\epsilon^{\text{c}}_{1}=-(\epsilon^{\text{c}}_{\vec{k}2}+\epsilon^{\text{c}}_{2}).$
(15)

While the choice of the dispersion for the second band, mirroring the first
one about the Fermi energy, might seem arbitrary at this point, we will
clarify in the next subsection (4.2) that such a behavior can indeed be
achieved through a remarkably straightforward real-space geometry.

Inserting the equations (11) and (15) into Eq. (14), and focusing on the zero
frequency contribution, $\omega=0$, we obtain

$\displaystyle\sum_{i}V_{\nu}^{2}G^{0\text{c}}_{\vec{k}\nu}(0)=\frac{1-\alpha}{1+\alpha}\frac{-V^{2}}{\epsilon^{\text{c}}_{\vec{k}1}+\epsilon^{\text{c}}_{1}}-i\pi
V^{2}\delta(\epsilon^{\text{c}}_{\vec{k}1}+\epsilon^{\text{c}}_{1}),$ (16)

where the contributions are separated into real and imaginary components.

Remarkably, the imaginary part remains independent of $\alpha$, while the real
part features a prefactor of $\frac{1-\alpha}{1+\alpha}$. This observation
makes this model perfectly suited to test our hypothesis that in contrary to a
SIAM where the imaginary part of the hybridization function determines the
influence of the conduction band onto local dynamics, the real part becomes
the dominant property in a lattice system where rank$(\Gamma_{lm})<N_{f}$. In
a multi band setup, the physics of the PAM can change due to destructive
interference at the Fermi energy.

In the following, we focus on the paramagnetic solution and enforce a spin
degenerate self-energy, $\Sigma_{\sigma}(z)=\Sigma(z)$, within the DMFT cycle
by assuming conservation of total spin quantum number in NRG. We adhere to
Wilson’s definition [44], and define the lattice low energy scale $T_{0}$ [45]
as the temperature at which the spin susceptibility of the impurity embedded
in the converged effective medium reaches the value of
$(g\mu_{\text{B}})^{2}\chi_{\text{imp}}(T_{0})=0.07$.

Figure 1:  Low energy scale $T_{0}$ as function of the dimensionless parameter
$\gamma=\frac{1+\alpha}{1-\alpha}$ for different band widths $D$, $U/\Gamma=6$
and $\epsilon^{c}_{1}/D=0.6$. The dashed black line corresponds to the
function $f(\gamma)=\exp(-\frac{\pi U}{8\Gamma}(\gamma-1))$.

Figure 1 displays the low energy scale
$T_{0}(\gamma)/T_{0}(\gamma=1)\sqrt{\gamma}$ vs the dimensionless parameter
$\gamma=\frac{1+\alpha}{1-\alpha}$ for various band widths, $U/\Gamma=6$ and
$|\epsilon^{\text{c}}_{\nu}|/D=0.6$. The plots clearly demonstrate the
exponential dependence of the low energy scale $T_{0}$ on $\gamma$. This
corroborate our conjecture that the screening of the local moment in a
periodic model is mainly driven by the real part of the conduction band
propagator and is not dominated by the imaginary part. This surprising and
somehow counterintuitive result is in contrast of the observation in the SIAM
and the two-impurity Anderson model [46, 47].

By approximating the propagator at zero frequency ($\omega/D\approx 0$) in Eq.
(16), we can determine the renormalization of the hybridization from its real
part: $V^{2}(\gamma)=V^{2}/\gamma$. If we assume $\tilde{\Gamma}\approx\Gamma$
and substitute the renormalized hybridization into Eq. (13) for the single
impurity Kondo temperature, the ratio of the low temperature scales should
follow the relation

$\displaystyle\sqrt{\gamma}\,\frac{T_{0}(\gamma)}{T_{0}(\gamma=1)}=\exp\left(-\frac{\pi
U}{8\Gamma}(\gamma-1)\right).$ (17)

Note that $\gamma=1$ implies $\alpha=0$ which corresponds to the conventional
single band PAM. The factor $\gamma-1=2\alpha/(1-\alpha)$ is a measure how
much the second band contributes to the local dynamics.

The r.h.s of Eq. (17) is added as a dashed black line to Figure 1. For small
values of $\gamma-1$, we observe an excellent agreement between the ratio
determined numerically with the DMFT(NRG) and analytic expression stated in
Eq. (17). As $\gamma$ increases, the zero frequency estimate
$V^{2}(\gamma)=V^{2}/\gamma$ tends to overestimate the renormalization.
However, for larger band widths, the deviation from Eq. (17) becomes smaller.
This behavior can be attributed to the fact that Eq. (14) is strictly valid at
$\omega/D\approx 0$, and the frequency dependence of the propagator becomes
more influential for small band widths $D$. For smaller band width, the finite
frequencies start to contribute stronger which leads to deviations from the
analytic estimate.

By tuning the parameter $\alpha$ from 0 to 1, we continuously increase the
admixture of the second conduction band. $\alpha=0$ corresponds to the
conventional single band PAM away from the Kondo insulator case, and
$\alpha=1$ two a two band model with perfect destructive interference of the
real part of the conduction band propagators. We demonstrated in Fig. 1 that
the low temperature scale of the lattice, $T_{0}$, which has the meaning of an
effective Kondo lattice scale, vanishes for $\alpha\to 1$ ($\gamma\to\infty$).
The model undergoes a quantum phase transition: At the coupling $\alpha=1$,
the Kondo screening breaks down and the effective impurity has a stable local
moment (LM) FP.

In summary, the findings impressively demonstrate the potential for
destructive hybridization interference within the PAM. Note, that in our
DMFT(NRG) calculation, we use the full energy dependency of the medium
$\Delta_{\sigma}(z)$, determined by the Eqs. (5)-(7). We used the fundamental
findings of our MIAM study [30, 31, 32] by focusing on the value of
$\Delta_{\sigma}(z)$ at the Fermi energy, only to construct the model and
predict the outcome of the full DMFT. This demonstrates again the powerful
concept that in such problems with infrared divergence physics the low energy
excitations play the crucial role. Since the parameter regimes are
adiabatically connected deviations from the wide-band limit does not alter the
physics but only lead to some perturbative corrections to the simple
analytical predictions as seen in Fig. 1.

###  4.2 Generality of destructive hybridization interference

In the previous subsection 4.1 we worked out the main result of this paper: If
a flat band of interacting electrons hybridizes with more than one conduction
band, the low temperature screening scale $T_{0}$ (Kondo lattice temperature)
can be exponentially reduced due to destructive interference reducing the
effective f-f hopping. However, to achieve this effect, the dispersions of the
conduction bands need to have a different curvature such that quantum
interference in Eq. (14) is destructive at low frequencies. Therefore, with
Eq. (15), we made the somewhat unconventional assumption that the two bands
are precisely mirrored. In this section we are going to demonstrate that such
destructive interference indeed arises if more general multi orbital versions
of the standard PAM are considered.

Perhaps the simplest way of extending the original PAM, which has only one
conduction band ($N_{\nu}=1$), into a multi-orbital model is by regularly
removing some of the $f$-orbitals such that (a) lattice periodicity is
maintained, (b) the lattice spacing of the $f$-lattice is enlarged compared to
the $c$-lattice and (c) the extended unit cell still contains exactly one
single $f$-orbital. This introduces multiple bands as we need to fold the
original dispersion of the $c$-electrons into the reduced Brillouin zone of
the larger unit cell.

By switching off the coupling $V$ between the $c$\- and $f$-subsystem, we can
use the usual Fourier transformation into momentum space for both subsystems
individually to diagonalize the bilinear part. However, due to the different
lattice spacing of the $c$\- and $f$-lattice, the related reduced Brillouin
zones $\text{Bz}^{\text{c}}$ and $\text{Bz}^{\text{f}}$ differ as well. If we
suppress the spin index $\sigma$ for simplicity, the hybridization part reads:

$\displaystyle H^{\text{hyb}}$
$\displaystyle=V\sum_{l}f^{\dagger}_{l}c^{\phantom{\dagger}}_{l}+\text{h.c.}$
(18)
$\displaystyle=\frac{V}{\sqrt{N_{f}N_{c}}}\sum_{\vec{k}\in\text{Bz}^{\text{c}}}\sum_{\vec{q}\in\text{Bz}^{\text{f}}}\sum_{l}e^{i(\vec{k}-\vec{q})\vec{R}_{l}}f^{\dagger}_{\vec{q}}c^{\phantom{\dagger}}_{\vec{k}}+\text{h.c.}$
(19)
$\displaystyle=\frac{V}{\sqrt{N_{c}^{u}}}\sum_{n=1}^{N_{c}^{u}}\sum_{\vec{q}\in\text{Bz}^{\text{f}}}f^{\dagger}_{\vec{q}}c^{\phantom{\dagger}}_{\vec{q}+\vec{p}_{{}_{n}}}+\text{h.c.}$
(20)

where $N_{c}^{u}=N_{c}/N_{f}$ denotes the number of $c$-orbitals within the
enlarged unit cell. We used the relation

$\displaystyle\sum_{l}e^{i(\vec{k}-\vec{q})\vec{R}_{l}}=N_{f}\sum_{n}\delta_{\vec{k},\vec{q}+\vec{p}_{{}_{n}}},$
(21)

where the set of momentum vectors $\vec{p}_{{}_{n}}$ depends on the basis
vectors $\vec{\delta}_{i}$ of the $f$-lattice,
$\vec{R}_{l}=\sum_{i}\alpha_{il}\vec{\delta}_{i}$, and is determined by the
two conditions:

$\displaystyle\vec{p}_{{}_{n}}\in\text{Bz}^{\text{c}}\quad\land\quad\vec{p}_{{}_{n}}\vec{\delta}_{i}=2\pi
n;\,n\in\mathbb{N}.$ (22)

This relates $\vec{k}\in\text{Bz}^{\text{c}}$ to
$\vec{q}\in\text{Bz}^{\text{f}}\subset\text{Bz}^{\text{c}}$ where
$\vec{p}_{{}_{n}}$ are the momentum vectors for backfolding the c-band band
structure into the Brillouin zone Bz${}^{\text{f}}$.

As a result, the propagator of the $f$-electrons becomes:

$\displaystyle G^{\text{f}}_{\vec{q}\sigma}(z)$
$\displaystyle=\left(z-\epsilon^{\text{f}}-\tilde{G}^{\text{c}}_{\vec{q}}(z)-\Sigma_{\vec{q}\sigma}(z)\right)^{-1}$
(23) $\displaystyle\tilde{G}^{\text{c}}_{\vec{q}}(z)$
$\displaystyle=\frac{1}{N_{c}^{u}}\sum_{n}V^{2}G^{0\text{c}}_{\vec{q}+\vec{p}_{{}_{n}}}(z).$
(24)

$G^{0\text{c}}_{\vec{q}+\vec{p}_{n}}(z)=(z-\epsilon_{\vec{q}+\vec{p}_{{}_{n}}})^{-1}$
describes the propagation of an electron in the respective piece $n$ of the
original conduction band, $\epsilon_{\vec{k}}$ with
$\vec{k}=\vec{q}+\vec{p}_{n}\in\text{Bz}^{\text{c}}$, folded into
$\text{Bz}^{\text{f}}$.

Lets quickly discuss three special cases: (I) the standard PAM, (II) the SIAM
and (III) minimal depletion on a cubic lattice:

  1. I.

In this case the first Brillouin zones of the $f$\- and $c$\- lattice are
identical, $N_{c}^{u}=1$ and $\vec{\delta}_{l}$ describes the basis vectors of
both lattices. The only solutions to Eq. (22) are the reciprocal lattice
vectors $\vec{g}_{l}$, for which
$G^{0\text{c}}_{\vec{q}+\vec{g}_{i}}(z)=G^{0\text{c}}_{\vec{q}}(z)$ holds,
and, consequently, Eq. (24) reduces to the well known form for the PAM
$\tilde{G}^{\text{c}}_{\vec{q}}(z)=V^{2}G^{0\text{c}}_{\vec{q}}(z)$.

  2. II.

By increasing the distance $|\vec{\delta}_{l}|$ between the $f$-orbitals we
can reach the limit of the SIAM for $|\vec{\delta}_{l}|\to\infty$. In this
limiting case the first Brillouin zone of the $f$-orbitals contains only a
single point (which means that we can drop the index $\vec{q}$), while Eq.
(22) is fulfilled by every point $\vec{k}$ within the first Brillouin zone of
the $c$-lattice. Therefore, Eq. (24) becomes
$\tilde{G}^{\text{c}}_{\vec{q}}(z)\to\tilde{G}^{\text{c}}(z)=\sum_{\vec{k}}V^{2}G^{0\text{c}}_{\vec{k}}(z)$,
which indeed is the hybridization function of the SIAM.

  3. III.

Since the cubic lattice is bipartite, the minimal periodic depletion on such a
geometry corresponds to removing the $f$-orbitals of one of the sublattices.
Even if this is only one specific realization of an arbitrary depletion
pattern, this model is commonly referred to as the "depleted PAM" [48, 49,
50]. For simplicity we will focus on the one dimensional case, however, the
result does not change in higher dimensions.

The unit cell of the $f$-lattice is twice as large as the one of the
$c$-lattice: $\delta=2a$, and, consequently, Eq. (22) has two solutions:
$p_{0}=2\pi/a$ and $p_{1}=\pi/a$.

Assuming $\epsilon_{k}=D\cos(ka)$ for the $c$-band we obtain
$\epsilon_{q+p_{0}}=\epsilon_{q}$ and $\epsilon_{q+p_{1}}=-\epsilon_{q}$, such
that Eq. (24) reads:

$\displaystyle\tilde{G}^{\text{c}}_{q}(z)=\frac{V^{2}}{2}\left(\frac{1}{z-\epsilon^{\text{c}}_{q}}+\frac{1}{z+\epsilon^{\text{c}}_{q}}\right).$
(25)

The structure of Eq. (25) is identical to the one in Eq. (14), describing the
hybridization with two dispersive bands. In addition, the sign in front of the
dispersive part of Eq. (24) is different for these bands, matching our
assumption in Eq. (15).

The physics of the depleted PAM [48, 49, 50] at half-filling is well
understood. In accordance with the extended Lieb-Mattis theory [41] the ground
state has a ferromagnetic polarisation of $S_{Z}^{\rm tot}=(N_{f}-1)/2$ since
the Kondo effect only screens a single spin of the lattice. We have shown [30]
that the rank of the hybridization function is rank$(\Gamma_{im})=1$, and,
therefore, only a single Kondo screening channel is available which is
irrelevant in the continuum limit. In the paramagnetic phase, spin ordering is
not considered and free local moments are found within a DMFT approximation
that ignores inter-site two-particle correlations. This physics corresponds to
$\alpha=1$ in the previous section, where we have explicitly shown that the
low-energy scale vanishes in this specific form of the PAM treated with the
DMFT.

In summary, Eq. (23) shows, that a regularly depleted version of the PAM with
only a single conduction band $\epsilon_{\vec{k}}$ is equivalent to a model
described by Eq. (1). The different bands $\epsilon_{\vec{k}\nu}$ correspond
to different pieces of the original conduction band $\epsilon_{\vec{k}}$,
which emerge by folding it into the reduced Brillouin zone of the $f$-lattice.
Of course these pieces will have different curvatures in general, such that
destructive interference within the hybridization can lead to a strongly
reduced Kondo lattice temperature $T_{0}$.

In Sec. 4.1 we used a concerted two band model as a simplified example two
study the effect of destructive interference in a systematic way, by tuning
the parameter $\alpha$. However, depletion of $f$-orbitals in the PAM has a
similar effect, leading to a modified $\alpha$: $0\leq\alpha\leq 1$.
Consequently, destructive hybridization interference is a recurring phenomenon
in multi-orbital extensions of the PAM and will hold a pivotal role in
elucidating the intriguing characteristics of heavy Fermion materials, where
the low energy physics is governed by $T_{0}$.

###  4.3 Spectral functions and Kondo breakdown QCP

Having established the potential for destructive hybridization interference in
the multi-orbital extension of the PAM, we will use our auxiliary two band
model to focus on the special scenario characterized by symmetric couplings,
$\alpha=V_{2}/V_{1}=1$, and $\gamma=\infty$, where Eq. (17) predicts a
complete suppression of the low-energy screening scale. We systematically
explored various parameter sets, encompassing diverse shapes for the
$c$-orbital density of states (DOS), the ratio of $D_{1}/D_{2}$, and particle-
hole asymmetric $f$-orbitals.

In the strongly correlated limit, where local moments develop at intermediate
temperature, we could not identify any combination of these parameters that
would yield a solution with a finite $T_{0}>0$. In stark contrast, the
converged DMFT solution for the SIAM consistently resides in the LM FP. This
confirms our initial conjecture that the screening of the local moments on the
f-sites is connected to the effective inter f-site hopping mediated by the
effective hopping matrix elements $T_{lm}(0)$ in a PAM and not by initial
local hybridization $\Gamma_{ll}(0)$ which is only relevant for MIAMs with the
rank$(\Gamma_{lm})=N_{f}$ which excludes the PAM. We would like to emphasize
that this effect is included in the DMFT self-consistency equations, and our
analysis only reveals its underlying mechanism.

We split this subsection into two parts. In the first part, we focus on the
special case of locally particle-hole (PH) symmetric $f$-orbitals, whereas we
allow particle hole asymmetry in the second part.

####  4.3.1 Particle-hole symmetric $f$-orbitals

Here we concentrate on the special case of PH symmetric $f$-orbitals by
setting $\epsilon^{\text{f}}=-U/2$. Note however, that due to
$|\epsilon^{\text{c}}_{\nu}|\not=0$, the Hamiltonian remains PH asymmetric as
long as $\alpha\not=1$. PH-symmetry is achieved for $\alpha=1$.

A word is in order to point out the differences to the single band PH
symmetric model, $\alpha=0,|\epsilon^{\text{c}}_{\nu}|=0$ which is a Kondo
insulator. In the latter case, $T_{lm}(0)\not=0$ and the effective $f$-site
does not only couple to the conduction band media $\tilde{\Gamma}(\omega)$ but
also to an effective free orbital that has $f$-orbital character as pointed
out by Pruschke et al. [36]. Within the DMFT, this coupling shows up as a
$\delta$-peak at $\omega=0$ in $\tilde{\Gamma}(\omega)$ and is essentially
generated by the finite $T_{lm}(0)$ to the neighboring $f$-sites.
Consequently, a local RKKY type singlet emerges despite of the gapped
$\tilde{\Gamma}(\omega)$, which would normally lead to a LM FP. Essentially,
Pruschke et al. [36] divided $\tilde{\Gamma}(\omega)$ into two parts: one
stemming for the conduction electrons which shows a pseudo-gap and, therefore,
does not provide a screening channel, and a $\delta$-peak stemming from the
neighboring f-site which is being responsible for the singlet formation. This
is in perfect alignment with the MIAM theory [30].

We present the spectral functions of the $f$-orbitals in Fig. 2(a), the
$c_{1}$-orbitals in Fig. 2(b), and the $c_{2}$-orbitals in Fig. 2(c) for three
different values of $\alpha$, $\alpha=0.0$ (black line), $\alpha=0.5$ (orange
line), $\alpha=1.0$ (red line), and the parameters: $D/\Gamma=5$,
$U/\Gamma=10$, and $|\epsilon^{\text{c}}_{\nu}|/D=0.4$.

Figure 2:  Single-particle spectral function of (a) the $f$-orbitals,
$\rho^{f}(\omega)$ (b) the $c_{1}$-orbitals, $\rho^{c}_{1}(\omega)$ and (c)
the $c_{2}$-orbitals, $\rho^{c}_{2}(\omega)$, of the paramagnetic DMFT
solution. The orange line represents the case where only the $c_{1}$ orbitals
hybridize with the $f$-orbitals ($\alpha=0.0$), while the black line depicts
results for identical hybridization with both bands ($\alpha=1.0$). The inset
displays the $f$-spectra for small frequencies. The model parameters used are:
$D/\Gamma=5$, $U/\Gamma=10$, and $|\epsilon^{\text{c}}_{\nu}|/D=0.4$.

We begin with the results for $\alpha=0.0$ (black lines), where only the
$c_{1}$-orbitals hybridize with the correlated $f$-orbitals ($V_{2}=0$), and
we recover the standard PAM. Since the $c_{2}$-orbitals are entirely
decoupled, their spectral function is identical to the initially assumed flat
DOS with a width of $D$:
$\rho^{\text{c}}_{2}(\omega)=\Theta(|\omega|-D)\frac{1}{2D}$.

The spectra of the $f$-orbitals exhibit the well-known Hubbard bands around
$\omega/\Gamma\approx\pm U/2$ due to the particle-hole symmetric choice of
$\epsilon^{\text{f}}=-U/2$. Additionally, the Kondo resonance with a width of
approximately $T_{0}$ around the Fermi energy ($\omega=0$) is clearly visible.
The dip near the Fermi energy, present in both the $f$ and $c_{1}$-orbital
spectra, resembles the nature of hybridized bands. This dip evolves into a
complete gap at $\omega=0$ in the limit of the Kondo insulator regime for
$\epsilon^{\text{c}}_{1}\to 0$ [36].

Having reviewed the spectral functions in the context of the standard PAM with
a single conduction band, we now shift our focus to $\alpha=1$ (red lines).

While the Hubbard bands in the $f$-spectra remain largely unchanged, the Kondo
resonance has collapsed. This suppression of the Kondo resonance aligns with
the prediction from Eq. (17), which anticipates the vanishing of $T_{0}$ when
$\alpha=1$, effectively rendering $\gamma=\infty$. Indeed, as $\alpha$
increases from 0, the Kondo resonance rapidly (exponentially) narrows,
exemplarily shown for $\alpha=0.5$ (orange lines).

In stark contrast, the $c$-orbitals manifest a divergence right at the Fermi
energy, indicative of an overall metallic solution such that the Kondo
breakdown falls within the class of orbital selective Mott transitions (OSM)
[12, 13, 14]: The half-filled f-bands become an Mott insulator which is
characterized by one local moment pre correlated site in the paramagnetic
phase. Note that due to the RG flow, these moments are typically extended and
not localized at the f-sites [31, 51]. In essence the f-electrons are excluded
from the Fermi-surface. Including the remaining two-particle interactions
between the f-spins which is beyond the single-particle DMFT, we expect a
magnetic ordering of the $f$-momenta while the metallic properties are coming
from the light quasiparticles.

####  4.3.2 Properties of the QCP for PH symmetric f-sites

After establishing that the critical coupling $\alpha=1$ belongs to a quantum
critical point which separates two strong coupling Fermi liquid FPs we
investigate the power law in the spectral functions right at the KB-QCP. We
employed three different shapes for the original $c$-orbital DOS: constant,
semicircular (Bethe lattice), and Gaussian (hypercubic lattice). In case of
non constant DOS (semicircular and Gaussian) we replace Eq. (12) by
$\Gamma=\pi V^{2}\rho(\epsilon^{\text{c}})$. Furthermore, we explore various
sets of model parameters encompassing $D_{\nu}/\Gamma\in\\{1,5,10\\}$,
$\epsilon^{\text{c}}/D\in\\{0.2,0.7\\}$, and $U/\Gamma\in\\{4,6,10\\}$.

Figure 3:  Single-particle spectral function of the (a) $c_{1}$-orbitals and
(b) $f$-orbitals on a double logarithmic scale for different sets of
parameters. The spectra are normalized to match at $\omega/\Gamma=10^{-5}$. We
employed three different DOS shapes: constant, semicircular, and Gaussian.
These were combined with varying values of $D/\Gamma\in\\{1,5,10\\}$,
$\epsilon^{\text{c}}/D\in\\{0.2,0.7\\}$, and $U/\Gamma\in\\{4,6,10\\}$. The
dashed black lines represent the power law relationships: (a)
$f(\omega)\propto|\omega|^{-0.33}$, (b) $f(\omega)\propto|\omega|^{0.33}$.

Figure 3 illustrates the positive part of the different spectra on double
logarithmic plots. Figure 3(a) displays the $c_{1}$-orbital spectra and Fig.
3(b) the $f$-orbital spectra. For better comparison, we plotted
$\rho(\omega)/\rho(\omega_{0})$, with $\omega_{0}/\Gamma=10^{-5}$. As expected
for the low-temperature physics of a QCP the spectra at high energies
significantly differ at high energies for the different parameter sets but a
universal low energy regime emerges. This low-frequency behavior adheres to a
power law relationship, denoted as $\propto|\omega|^{r}$. We obtain the
approximate exponents of $r^{\text{f}}\approx 0.33$ for the $f$-spectra and
$r^{\text{c}}\approx-0.33$ for both $c$-spectra. Remarkably, this exponent
remains unaffected by variations in the model parameters, emphasizing the
concept of universality.

It’s important to note that the KB-QCP is independent of the Coulomb
interaction strength, effectively rendering $U>U_{c}=0$. However, if we
introduce a finite bandwidth also for the $f$-orbitals, $U_{c}$ is shifted to
larger values, $U_{c}>0$. Within the OSM phase for $U>U_{c}$ we consistently
observe the same universal behavior, consequently, our primary focus remains
on the zero bandwidth limit.

The effective SIAM at the critical point, $\alpha=1$, corresponds to the well
known and heavily studied pseudo-gap Anderson model [15, 16, 17, 18]. As is
customary for this model in the LM FP, both the effective medium,
$\Im\Delta(z)$, and the imaginary part of the correlation self-energy,
$\Im\Sigma(z)$, exhibit the same power-law exponent as the $f$-orbital
spectra, albeit with opposite signs:
$\Im\Delta(z)\propto|\omega|^{r^{\text{f}}}$,
$\Im\Sigma(z)\propto|\omega|^{-r^{\text{f}}}$.

The phase space of the SIAM in the presence of a pseudo-gap DOS, characterized
by an exponent $r<0.5$, is well-documented for hosting an interacting non-
Fermi-liquid FP with significant local moment fluctuations [15, 16, 17, 18].
This non-Fermi-liquid FP exhibits a remarkable $\omega/T$ scaling behavior
[16], consistent with experimental observations in various heavy fermion
compounds at criticality [5, 6, 7].

In a PH pseudo-gap quantum impurity problem with a power-law hybridization
function $\propto|\omega|^{r}$, a critical Kondo coupling strength $J_{c}(r)$
was identified for $r<1/2$ [52] which governs the transition from a stable LM
FP to a the stable SC FP above $J_{c}(r)$. Even though the pseudo-gap of the
effective medium is described by an exponent $r^{\text{f}}=0.33<1/2$, we were
not able to find any stable FP other than the LM FP. This peculiar result can
be attributed to the nature of the pseudo-gap arising as a consequence of
self-consistency rather than being an inherent feature [53]. While studies of
the pseudo-gap SIAM generally assume the power-law dependency to occur over
the whole bandwidth of the conduction band DOS, we only obtain such behavior
at low frequencies $\omega<\tilde{D}$ with the low-energy effective band width
$\tilde{D}<D$. While the exponent $r^{\text{f}}=0.33$ is fixed, the effective
strength of hybridization $\tilde{V}^{2}$ in the small interval
$I=[-\tilde{D}:-\tilde{D}]$,

$\tilde{V}^{2}=-\frac{1}{\pi}\int_{-\tilde{D}}^{\tilde{D}}\Im\Delta(z)d\omega$
(26)

depends on the DMFT self-constancy condition. While in a conventional pseudo-
gap SIAM, the decrease of $U/\Gamma$ increases the Kondo coupling strength $J$
obtained by a Schrieffer-Wolff transformation [54], we observe a decrease in
$\tilde{V}^{2}$ due to the DMFT self-consistency condition effectively
stabilizing the LM FP.

However, even if the non-Fermi-liquid FP of the pseudo-gap Anderson model is
absent in the effective site of the DMFT for the lattice model under
investigation, we might be in its close proximity. As a result, at
intermediate temperatures, various properties of the model might still exhibit
hallmarks of the potentially nearby non-Fermi-liquid FP, including the
$\omega/T$ scaling. This is similar to the apparent quantum-critical Mott
transition in organic salts [55], which, at least from the perspective of one-
band DMFT analysis, originates from approximate local quantum criticality
[53].

####  4.3.3 PH asymmetric $f$-sites

Figure 4:  Same as Fig. 2 but for particle-hole asymmetric $f$-orbitals,
$U/\Gamma=10$, $\epsilon^{\text{f}}/\Gamma=-3$.

We extend our analysis to PH $f$-sites, allowing for
$\epsilon^{\text{f}}\neq-U/2$. Figure 4 illustrates the spectral functions for
the same parameters as in Fig. 2, but with $\epsilon^{\text{f}}/\Gamma=-3$.
The qualitative behavior of the spectra, when increasing $\alpha$, remains
unchanged. There is a notable suppression of the Kondo resonance in the
$f$-spectra, accompanied by the development of a peak in both $c$-spectra.

Remarkably, we again find a KB-QCP at $\alpha=1$, supporting our conjecture
that the QCP is driven by the absence of an conduction band induced inter-site
hopping of the f-electrons in combination with an insufficient number of Kondo
screening channels in the lattice. This KB-QCP, however, leads to modified
low-frequency properties of the spectral functions which differ from the PH
symmetric scenario.

Figure 5(a) displays the spectra of the $f$\- and Fig. 5(b) $c_{1}$-orbitals
at the KB-QCP on a low-frequency interval around the Fermi energy for four
different values of $\epsilon^{\text{f}}$ while keeping $U/\Gamma=10$ fixed
(both $c$-spectra are nearly identical on that scale). Importantly, not only
the $f$\- but also the $c$-spectra display a gap around $\omega=0$, signaling
an insulating ground state. The width of the gap is controlled by the degree
of PH asymmetry and increases with the distance from the PH symmetric point.
All spectra remain asymmetric down to zero frequency and display a
discontinuity right at $\omega=0$, formally indicating deviations from a
Fermi-liquid description.

For $\omega<0$, the spectra are governed by an universal power-law dependency
$\propto|\omega|^{r}$ with $r^{\text{f}}=0.5$ and $r^{\text{c}}=-0.5$
independent of the degree of PH asymmetry or other model specifics like the
shape of the conduction band DOS for less than half-filling. In contrast, for
$\omega>0$, the spectra vanish very quickly, and we were not able to extract
the exponent of a possible power-law dependency. For larger than half-filling
these universal exponents are found for $\omega>0$.

Figure 5:  Small frequency regime of the single-particle spectral function of
(a) the $f$-orbitals and (b) the $c_{1}$-orbitals at the KB-QCP, $\alpha=1$,
for locally particle-hole asymmetric $f$-orbitals. The different spectra
correspond to $\epsilon^{\text{f}}/\Gamma=-2$ (black lines),
$\epsilon^{\text{f}}/\Gamma=-3$ (orange lines),
$\epsilon^{\text{f}}/\Gamma=-4$ (blue lines) and
$\epsilon^{\text{f}}/\Gamma=-7$ (orange dashed lines), while $U/\Gamma=10$
remains fixed. Other model parameters used are: $D/\Gamma=5$, and
$|\epsilon^{\text{c}}_{\nu}|/D=0.4$.

It is essential to note that this insulating ground state differs from the
usual Kondo insulator regime that typically occurs in the standard PAM around
the PH symmetric point: (I) While local moments are screened in the Kondo
insulator below a finite low-temperature crossover scale, $T_{0}>0$, here they
remain unscreened down to zero temperature at the KB-QCP. Including non-local
correlations by calculating two particle propagators [38, 56, 57] may lead to
the occurrence of magnetic order or a spin liquid in the presence of
frustration. (II) The width of the gap in the Kondo insulator is proportional
to $T_{0}$ and, consequently, decreases with increasing strength of
correlation, whereas the width of the gap in our case solely depends on the
degree of particle-hole asymmetry. (III) As demonstrated in Ref. [36], in the
Kondo insulator regime the effective site approaches the stable SC FP at low
temperature which exhibits local Fermi liquid properties. In the KB-QCP
insulator, we obtain a discontinuity right at $\omega=0$, violating such a
description. In addition, the fixed point of the effective impurity is a local
moment fixed point.

##  5 Summary and discussion

In this paper we studied a general multi-orbital extension of the widely
recognized PAM within the framework of the DMFT+NRG. Focusing on the effect of
two conduction bands with inverse dispersions to each other, hybridizing with
a flat band of strongly correlated orbitals, we systematically analyzed the
influence of the second band on the low energy Kondo scale $T_{0}$.

When the coupling to one of the bands dominates, we observe the emergence of
well-known characteristics of the standard PAM [36, 37]. This includes the
presence of a Kondo resonance, with a width of $T_{0}$, right at the Fermi
energy, and, depending on the degree of the PH asymmetry, an emerging
depletion of the f-spectrum which develops into a full gap in the Kondo
insulator regime. However, as the coupling to the second band is increased,
when holding the overall hybridization matrix element to the f-orbital
constant, the low-temperature Fermi-liquid scale $T_{0}$ decreases
exponentially fast. It’s important to note that adding arbitrarily shaped
bands with non-conserved flavor to a SIAM always increases the local
hybridization strength and the corresponding Kondo temperature, while this
destructive interference and the associated drop of $T_{0}$ is a unique
manifestation arising only in the lattice.

This exponential reduction of the Kondo lattice temperature $T_{0}$ can be
explained by quantum interference of effective inter-site $f$-hopping elements
[30] originating from the hybridization with different bands. This underlines
the importance of these effective hopping elements over the single-site Kondo
screening mechanism in the PAM [30].

Whereas the auxiliary two band model enables a clean and systematic analysis
of the effect of destructive hybridization interference, we showed that this
will be a quit general feature in depleted versions of the standard PAM,
relevant for a realistic description of heavy fermion materials where the unit
cell typically comprise several atoms. In these cases the $f$-orbitals
hybridize with multiple bands.

At high symmetry points with perfect destructive interference, the lattice low
energy scale $T_{0}$ gets completely suppressed, leading to a Kondo breakdown
critical point. In the DMFT self consistency equations an effective pseudo gap
Anderson impurity model emerges which has a stable LM FP.

In the presence of locally PH symmetric $f$-orbitals, we observe power-law
dependencies in all single-particle spectral functions, described by
$A(\omega)\propto|\omega|^{r}$. Notably, the conduction band DOS remains
metallic, with a diverging low-frequency spectral function characterized by an
exponent of $r^{\text{c}}=-0.33$, while the correlated orbitals exhibit the
development of a pseudo-gap with an exponent of $r^{\text{f}}=0.33$. At this
KB-QCP, the Fermi surface is reduced since it involves the conduction
electrons only. The exponents do not depend on the the non-interacting
metallic DOS of the conduction electrons or the strength of Coulomb
interaction. Therefore, the critical interaction strength is zero, $U_{c}=0$,
however, a finite bandwidth for the correlated $f$-orbitals increases $U_{c}$
such that $U>U_{c}$ is required in order to reach this OSM phase.

Remarkably we find that the occurrence of the KB-QCP is neither associated
with the PH symmetry or does it depend on the magnitude of the imaginary part
of hybridization function as used in the traditional Doniach picture [25].
However, reducing the electron filling away from half filling, the KB-QCP is
modified:

(I) A gap right at the Fermi energy is present in all spectral functions,
demonstrating insulating nature of the solution. (II) For $\omega<0$, we
observe power law dependency in all spectra with modified universal exponents
$r^{\text{f}}=0.5$ and $r^{\text{c}}=-0.5$. (III) The size of the gap depends
on the distance to the PH symmetric point, at which the gap turns into a true
pseudo gap. For increasing the electron filling above half filling, the
universal exponents are observed for $\omega>0$ as can be understood by a PH
transformation.

We believe that our results are highly relevant for a broad range of strongly
correlated materials. Versions of the PAM customized to specific heavy fermion
materials will generally fall into the class of multi-orbital generalizations
of the standard PAM. In such cases destructive hybridization interference can
play a crucial role in understanding the low temperature behavior like the
low-energy screening scale, determining the onset of universality.

Moreover, the notion of destructive hybridization interference provides a new
avenue to explore Kondo breakdown scenarios within a rather general framework,
without the need for intricate material-specific details. This avenue has the
potential to enrich our existing comprehension of non-Fermi-liquid properties
in strongly correlated materials.

## Acknowledgements

We acknowledge stimulating and fruitful discussions with B. Fauseweh and G.
Camacho.

##### Funding information

This project was made possible by the DLR Quantum Computing Initiative and the
Federal Ministry for Economic Affairs and Climate Action;
qci.dlr.de/projects/ALQU. F..B.A acknowledges funding by Deutsche
Forschungsgemeinschaft through grant Nr. AN-275/10-1.

##  Appendix A DMFT integrals

In order to obtain results with high numerical precision it is very important
to accurately solve the DMFT equation in Eq. (5). While the impurity Greens
function in Eq. (7) are calculated within the NRG, the corresponding lattice
Greens functions need to be calculated by evaluating Eq. (6). In case of the
two band model this equation reads

$\displaystyle G^{\text{f}}_{\text{lat}}(z)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\rho(\epsilon)\;d\epsilon}{z-\epsilon^{\text{f}}-\Sigma(z)-\frac{V_{1}^{2}}{z-(\epsilon+\epsilon^{\text{c}})}-\frac{V_{2}^{2}}{z+(\epsilon+\epsilon^{\text{c}})}}$
(A.1)
$\displaystyle=\int_{-\infty}^{\infty}\frac{\rho(\epsilon)((\epsilon+\epsilon^{\text{c}})^{2}-z^{2})}{F(z,\epsilon)}d\epsilon,$
(A.2)

with

$\displaystyle F(z,\epsilon)=a(z)\epsilon^{2}+b(z)\epsilon+c(z).$ (A.3)

If we make use of the definitions $V^{2}=V_{1}^{2}+V_{2}^{2}$ and $\Delta
V^{2}=V_{1}^{2}-V_{2}^{2}$, the factors $a(z)$, $b(z)$ and $c(z)$ are given by
the following equations:

$\displaystyle a(z)$ $\displaystyle=z-\epsilon^{\text{f}}-\Sigma(z)$ (A.4)
$\displaystyle b(z)$ $\displaystyle=\Delta V^{2}+2\epsilon^{\text{c}}a(z)$
(A.5) $\displaystyle c(z)$ $\displaystyle=zV^{2}+\epsilon^{\text{c}}\Delta
V^{2}+((\epsilon^{\text{c}})^{2}-z^{2})a(z)$ (A.6)

Now we can use partial fraction decomposition in order to rewrite
$F^{-1}(z,\epsilon)$:

$\displaystyle\frac{1}{F(z,\epsilon)}=\frac{A}{\epsilon-\epsilon_{1}}+\frac{B}{\epsilon-\epsilon_{2}}$
(A.7)

where the parameters are given as

$\displaystyle\epsilon_{1}$
$\displaystyle=\frac{\sqrt{b(z)^{2}-4a(z)c(z)}-b(z)}{2a(z)}$ (A.8)
$\displaystyle\epsilon_{2}$
$\displaystyle=\frac{-\sqrt{b(z)^{2}-4a(z)c(z)}-b(z)}{2a(z)}$ (A.9)
$\displaystyle A$ $\displaystyle=\frac{1}{a(z)(\epsilon_{1}-\epsilon_{2})}$
(A.10) $\displaystyle B$ $\displaystyle=-A.$ (A.11)

At this point the original DMFT integral can be written as

$\displaystyle G(z)=\int_{-\infty}^{\infty}d\epsilon$
$\displaystyle\left[A\left(\frac{\rho(\epsilon)\epsilon^{2}}{\epsilon-\epsilon_{1}}-\frac{\rho(\epsilon)\epsilon^{2}}{\epsilon-\epsilon_{2}}\right)\right.+2\epsilon^{\text{c}}A\left(\frac{\rho(\epsilon)\epsilon}{\epsilon-\epsilon_{1}}-\frac{\rho(\epsilon)\epsilon}{\epsilon-\epsilon_{2}}\right)$
$\displaystyle+\left.((\epsilon^{\text{c}})^{2}-z^{2})A\left(\frac{\rho(\epsilon)}{\epsilon-\epsilon_{1}}-\frac{\rho(\epsilon)}{\epsilon-\epsilon_{2}}\right)\right]$
(A.12)

If we now make use of the relations

$\displaystyle\int_{-\infty}^{\infty}d\epsilon\frac{\rho(\epsilon)}{z-\epsilon}$
$\displaystyle=\mathcal{G}(z)$ (A.13)
$\displaystyle\int_{-\infty}^{\infty}d\epsilon\frac{\rho(\epsilon)\epsilon}{z-\epsilon}$
$\displaystyle=z\mathcal{G}(z)-1$ (A.14)
$\displaystyle\int_{-\infty}^{\infty}d\epsilon\frac{\rho(\epsilon)\epsilon^{2}}{z-\epsilon}$
$\displaystyle=z(z\mathcal{G}(z)-1)$ (A.15)

which hold for a particle hole symmetric DOS $\rho(\epsilon)=\rho(-\epsilon)$
we finally obtain

$\displaystyle G(z)=A$
$\displaystyle\left[\left(\epsilon_{1}-\epsilon_{1}^{2}\mathcal{G}(\epsilon_{1})\right)-\left(\epsilon_{2}-\epsilon_{2}^{2}\mathcal{G}(\epsilon_{2})\right)\right]+2\epsilon^{\text{c}}A\left[\left(1-\epsilon_{1}\mathcal{G}(\epsilon_{1})\right)-\left(1-\epsilon_{2}\mathcal{G}(\epsilon_{2})\right)\right]$
$\displaystyle+(z^{2}-(\epsilon^{\text{c}})^{2})A\left[\mathcal{G}(\epsilon_{1})-\mathcal{G}(\epsilon_{2})\right]$
(A.16)

The Hilbert transformation in Eq. (A.13) can be solved with very high
accuracy. For example, in case of a semicircular DOS the analytical solution
is known. The DMFT integrals for the $c$-orbital spectra can be calculated in
a similar way.

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