(67)

where $T_{0}^{\text{AF}}$ corresponds to $K_{hh}^{\text{ex}}>0$ and

$\displaystyle
S_{\text{imp}}(T_{0}^{\text{FM}})=\frac{1}{2}k_{\text{B}}[\ln(4)+\ln(3)],$
(68)

where $T_{0}^{\text{FM}}$ corresponds to $K_{hh}^{\text{ex}}<0$.

Figure 19: Low energy scales for the models depicted in Fig. 17 but with
$\Delta R_{h}=3a$ (light blue linepoints) and $\Delta R_{h}=4a$ (black
linepoints). In case of $\Delta R_{h}=3a$ the ground state is a singlet such
that the low energy scale is given by $T^{\text{AF}}_{0}$ defined in Eq. (67),
whereas it is given by $T_{0}^{\text{FM}}$ defined in Eq. (68) for a triplet
ground state for $\Delta R_{h}=4a$. The inset shows the data for $\Delta
R_{h}=3a$ on a logarithmic scale for $T_{0}^{\text{AF}}$ together with the fit
function $f(x)=ax^{b}$ for $a=0.0025$ and $b=1.6169$. The parameters of the
model are chosen to be PH symmetric, i.e. $\epsilon^{c}=0$ and
$U=-2\epsilon^{f}_{l}$.

The results for $T_{0}^{\text{AF}}$ and $T_{0}^{\text{FM}}$ vs. $U$,
calculated for $\Delta R_{h}=3a$ and $\Delta R_{h}=4a$ respectively, are
depicted in Fig. 19 (light blue linepoints for $T_{0}^{\text{AF}},\,\Delta
R_{h}=3a$ and black linepoints for $T_{0}^{\text{FM}},\,\Delta R_{h}=4a$). The
overall curves of $T_{0}^{\text{AF}}$ and $T_{0}^{\text{FM}}$ matches those of
$T_{0}$ for the $N_{f}=4$ and $N_{f}=3$ model in Fig. 15(a). For small
$U/\Gamma_{0}$ the low energy scales $T_{0}^{\text{AF}}$ and
$T_{0}^{\text{FM}}$ increase with increasing Coulomb interaction. However, in
the strongly interacting limit, $U/\Gamma_{0}\gg 1$, $T_{0}^{\text{AF}}$
corresponds to the AF RKKY scale $\propto 1/U$ since the model consists of an
even number of $N_{f}=4$ orbitals, whereas $T_{0}^{\text{FM}}$ can be linked
to the exponentially suppressed Kondo temperature due to an odd number of
$N_{f}=5$ orbitals in the model. These different mechanisms are also encoded
in the different NRG FP spectra for the two cases (not depicted here.)

In the weakly interacting limit, $U/\Gamma_{0}\to 0$, the exchange interaction
$K_{hh}^{\text{ex}}$ vanishes as expected from the supercell analysis in Sec.
II.2. The inset shows the $T_{0}^{\text{AF}}$ data on a logarithmic scale and
a corresponding fit to a power law function of the form $f(x)=ax^{b}$, with
$a=0.0025$ and $b=1.6169$. In contrast to the conventional RKKY interaction
which falls of as $1/U$ in the wide band limit, $K_{hh}^{\text{ex}}$ increases
strongly with increasing Coulomb interaction. Note that we can not resolve the
small energy scale for $U/\Gamma_{0}<1.8$ and $\Delta R_{h}>3$. Hence we can
not quantify the dependence of $K_{hh}^{\text{ex}}$ on the distance $\Delta
R_{h}$, however, $|K_{hh}^{\text{ex}}(\Delta
R_{h}=3)|>|K_{hh}^{\text{ex}}(\Delta R_{h}=4)|$ is fulfilled for all $U$. As
the wave function of the magnetic bound states falls of exponentially [22] and
$K_{hh}^{\text{ex}}$ is proportional to the overlap of different magnetic
bound states, we expect $K_{hh}^{\text{ex}}$ to fall of exponentially with
increasing distance $\Delta R_{h}$.

##  V Viewpoint onto the Kondo-hole physics puzzle in
$\text{Ce}_{1-x}\text{La}_{x}\text{Pd}_{3}$

Stable magnetic bound states induced by charge neutral substitution of single
Ce by La are expected to exist in systems whose parent compound shows Kondo
insulating behavior [18]. However, in experiments [13, 11, 12, 15, 16, 17]
with a finite hole concentration $x$ these local bound states haven’t been
observed so far since long range interactions between the bound states, which
are already present at the lowest experimentally accessible $x$ (for example
$x=0.006$ in (Ce1-xLax)3Bi4Pt3 [11]), lead to the formation of a very narrow
Kondo-hole band in the hybridization gap.

The long-range nature of the interaction between the hole induced bound states
in Kondo insulators is in qualitative agreement with our results for Kondo-
holes in finite impurity arrays. As discussed in the context of Fig. 11(b),
the hole induced magnetic moment is increasingly delocalized with increasing
Coulomb interaction $U$ and spreads over several lattice spacings in the
strongly interacting limit $U/\Gamma_{0}\gg 1$.

In contrary, the insertion of Kondo-holes in heavy-fermion materials, such as
Ce1-xLaxCu6 [14] are mainly discussed as examples for a smooth crossover from
a lattice material to single impurity physics [14, 5, 60].

Pristine CePd3 is a heavy fermion metal that is considered to be close to a
Kondo insulator but still maintains Fermi liquid properties at low
temperatures. Therefore, the electronic degrees of freedom relevant for the
low energy physics should violate PH symmetry and the filling of the
conduction bands must deviate from half-filling, however, the model parameters
are at the brink of becoming a Kondo insulator.

Substituting a small amount of nonmagnetic La for the Ce in
$\text{Ce}_{1-x}\text{La}_{x}\text{Pd}_{3}$ leads to a logarithmic increase of
the resistivity [6, 7] with decreasing temperature after the material is
cooled far below the lattice coherence temperature of CePd3 typical for a
system with magnetic impurities in a metal. Early on a negative
magnetoresistance was taken as another indication that the second resistively
increase is connected to secondary Kondo effect [7] but the microscopic origin
of the local moment formation remain a puzzle.

Connecting our investigation of Kondo-hole physics with experiments, the main
message is that Kondo hole insertion generate stable local moments near the PH
symmetric point (which corresponds to Kondo insulators in the PAM) that can be
screened by two mechanisms which require PH asymmetry: (i) hybridizing the
localized single particle state with the itinerant electrons due an asymmetric
conduction band and (ii) indirect coupling via the unfilled La $4f$-orbitals.
We showed in Fig. 13 that taking into account a finite orbital energy
$\epsilon^{f}_{h}<\infty$ of the unoccupied La $4f$-orbitals and a band center
$\epsilon^{c}$ away from a half-filled band yields a Kondo screening of the
Kondo hole induced local moment.

The question of whether exotic physical properties are observable in material
is directed linked to the question of separating the energy or temperature
scales of a two-step screening mechanism: First a formation of a correlated HF
or Kondo insulator phase is required and then, at much lower temperatures, the
screening of the Kondo-hole induced local moments can occur.

In real materials, the PH-symmetry in the conduction bands is broken even for
half-filling. Furthermore, La substitution changes the band structure of the
conduction band slightly as shown by LDA calculations [17] due to the
Lanthanide contraction. Instead of introducing a complicated tight-binding
band structure adequate for only one material, we use the value for the band
center $\epsilon^{c}$ as a control parameter for the degree of PH symmetry
breaking.

Although a finite size MIAM does not include the full lattice physics, we have
demonstrated [32] that major self-screening mechanism of magnetic moment is
correctly captured already in a MIAM of second type. Pruschke at al. [66]
showed already with the DMFT for the PAM that the singlet-formation in the
Kondo-insulators is not due to the Kondo effect since the conduction band as
well as continuum of effective media shows a pseudo-gap DOS but caused by the
effective f-f orbital hybridization that is explicitly included in the matrix
$\Re\bm{\Delta(0)}$ in our mapping.

Figure 20: NRG calculations for the 1d model depicted in Fig. 10 with
$N_{f}=7$ $f$-orbitals. The properties with index ’hole’ were calculated for
an uncorrelated hole-orbital (red in Fig. 10), $U_{h}=0$ and
$\epsilon^{f}_{h}/\Gamma_{0}=25$, in the center of an otherwise strongly
correlated impurity array, $U_{l}=-2\epsilon^{f}_{l}=10\Gamma_{0}$, whereas
all properties indicated by the index ’corr’ correspond to the homogenous
model where all $N_{f}=7$ $f$-orbitals are identical with
$U=-2\epsilon^{f}=10\Gamma_{0}$. (a) impurity induced entropy $S_{\text{imp}}$
as function of the dimensionless temperature $T/\Gamma_{0}$ for the model with
a hole orbital (light blue linepoints) and without hole orbital (black
linepoints) for three different values $\epsilon^{c}/D=0.15$, $0.181$ and
$0.25$, indicated by different types of points. (b) ratio
$T^{\text{hole}}/T^{\text{corr}}$ for the low energy scales $T_{0}$ (light
blue linepoints), defined in Eq. (55), and $\tilde{T}_{1}=\kappa T_{1}$ (black
linepoints), defined in Eq. (69), plotted against the band center
$\epsilon^{c}/D$.

Since we believe that the Kondo-hole physics in the dilute limit is locally
driven and independent of the details of the conduction band other than PH
symmetry breaking, we used a 1d MIAM representation in order to ensure that
the number of screening channels are less than the number of correlated
orbitals in the vicinity of the Kondo hole to mimic the physics of a larger
system. For our NRG calculation we use the MIAM with $N_{f}=7$ $f$-orbitals
shown in in Fig. 10.

We need to define the appropriate low energy scales of the MIAM to make
connection with the experiment. Within a DMFT(NRG) it has been shown that the
lattice coherence temperature is identical to the Kondo temperature of the
effective site up to some universal number depending on the definition [75].
Using that knowledge, we extracted the crossover scale to the low-energy
singlet fixed point in the homogenous model where all $N_{f}=7$ $f$-orbitals
are identical with $U=-2\epsilon^{f}=10\Gamma_{0}$ as measure for the lattice
coherence temperature $T^{\text{corr}}$ in our approach and neglect the DMFT
lattice correction.

To connect the low-temperature scale of the fully correlated array of
$N_{f}=7$ correlated orbitals with those of the $N_{f}=6$ Kondo hole MIAM, we
plotted the temperature dependent impurity entropy $S_{\text{imp}}$ for the
model with hole (light blue linepoints) and without hole (black linepoints)
for three different values $\epsilon^{c}/D=0.15$, $0.181$ and $0.25$ in Fig.
20(a). In the full MIAM the $N_{f}$ spin-$1/2$ local moments are quenched on a
single low-temperature crossover scale $T^{\rm corr}$ as expected from a Kondo
lattice problem. This crossover scale is nearly independent of the band center
$\epsilon^{c}$ for the full MIAM. In the Kondo hole case we, however, observe
three different regimes. For large $\epsilon^{c}$, all $N_{f}-1$ local moments
are screened on the same energy scale $T_{1}$ that qualitatively agrees with
that of the full MIAM. For $\epsilon^{c}\approx 0.18$, only $N_{f}-2$ local
moments are screened at on a similar scale than the full problem, and a
unstable LM $S=1/2$ FP emerges whose local moment started to disappear on a
secondary crossover scale $T_{0}$. This LM FP is stable for
$\epsilon^{c}=0.15$.

To distinguish between the potential two energy scales in the present of of
the Kondo hole, we define an additional low energy scale $T_{1}$,

$\displaystyle
S_{\text{imp}}(T_{1})=1/2k_{\text{B}}\left[\ln(2)+\ln(3)\right],$ (69)

which governs the crossover to the unstable $S=1/2$ LM FP – see also Fig.
20(a). $\ln(3)$ has no physical meaning and could be replaced by any other
reasonable value, since $T_{1}$ only defines a crossover scale. Per
definition, the relation $T_{0}(\epsilon^{c})<T_{1}(\epsilon^{c})$ always
holds. In the large $\epsilon^{c}$ regime, where there is only one crossover
scale $T_{0}(\epsilon^{c})\propto T_{1}(\epsilon^{c})$ holds while for
intermediate $\epsilon^{c}$ the temperatures $T_{0}$ and $T_{1}$ refer to
physically different scales. Therefore we introduce the scaling factor
$\kappa=T_{0}(\epsilon^{c}/D=0.25)/T_{1}(\epsilon^{c}/D=0.25)=0.462$ that
compensates the mismatch between $T_{1}$ and $T_{0}$ at $\epsilon^{c}/D=0.25$
where both refer to the same energy scale, and introduce the rescaled
$\tilde{T}_{1}(\epsilon^{c})=\kappa T_{1}(\epsilon^{c})$ such that
$\tilde{T}_{1}(\epsilon^{c}/D=0.25)=T_{0}(\epsilon^{c}/D=0.25)$. For the
$N_{f}=7$ full MIAM, $\tilde{T}_{1}(\epsilon^{c})=T_{0}(\epsilon^{c})$ always
holds since there is only one crossover scale as illustrated in Fig. 20(a),
and we set $T^{\rm corr}(\epsilon^{c})=T_{0}(\epsilon^{c})$.

In Fig. 20(b) we plotted the ratio $T^{\text{hole}}/T^{\text{corr}}$ for the
low energy scales $T_{0}$ (light blue linepoints) and $\tilde{T}_{1}$ (black
linepoints), plotted as function of the band center $\epsilon^{c}/D$. We used
an $N_{f}=7$ MIAM with a single uncorrelated hole (red in Fig. 10), $U_{h}=0$
and $\epsilon^{f}_{h}/\Gamma_{0}=25$, in the center of an otherwise strongly
correlated impurity array, $U_{l}=-2\epsilon^{f}_{l}=10\Gamma_{0}$. In
contrary to the Kondo hole literature, we included an uncorrelated and
unoccupied $4f$-orbital at an energy of $\epsilon^{f}_{h}\approx 2U$ in the
calculation.

We note that the ratio $\tilde{T}^{\text{hole}}_{1}/T^{\text{corr}}$ remains
almost independent of the band center $\epsilon^{c}$ and the ratio remains of
$O(1)$. The scale $\tilde{T}_{1}$ takes the role of the lattice coherence
scale in the system with and without the Kondo hole. The change is not really
significant and in the experiments on Ce0.97La0.03Pd3 the reduction of
$T^{\text{corr}}$ in the presence of Kondo holes has been interpreted as
signature for the lanthanide contraction [5, 7, 12, 13].

We can identify three different regimes.

(I) For $\epsilon^{c}/D\geq 0.2$ we have $T_{0}\approx\tilde{T}_{1}$ and
$T^{\text{hole}}\approx T^{\text{corr}}$, indicating that there is only one
relevant low energy scale which is nearly identical for the model with and
without a Kondo hole in its center. We believe that this regime is relevant
for Kondo hole substitution in Ce1-xLaxCu6 where we start from a heavy-fermion
compound that is significantly away from half-filling.

(II) $T_{0}^{\text{hole}}/T^{\text{corr}}$ is zero for $\epsilon^{c}/D\leq
0.179$, and a stable LM FP is found as exacted to occur in Kondo insulators in
the dilute limit. Free local moments are present for $T\to 0$ and the finite
entropy is probably removed by magnetic polaron formation [13]. It has been
suggested that disorder and the spatially extended bound states apparently
start to overlap in Kondo insulators already at relative low concentrations
leading to a very narrow band formation in the Kondo insulator gap [19, 20,
21, 16, 13].

(III) In between these two regimes at around $\epsilon^{c}/D\approx
0.179+|\delta|$ and small $\delta$, there is a clear hierarchy,
$0<T_{0}<\tilde{T}_{1}$, indicating that there is an intermediate unstable LM
FP that crosses over to the SC phase on a energy scale clearly below the
coherence temperature $T_{1}$. We believe that this intermediate regime is
relevant for the exotic behavior in doped heavy Fermion materials which are at
the brink of being a Kondo insulator such as CePd3. In this regime the lowest
energy scale, $T^{\text{hole}}_{0}$, corresponds to the onset of magnetic
scattering associated with an increase in the electrical resistivity due to a
Kondo effect of the hole induced magnetic moment.

##  VI Summary and discussion

In this paper we reviewed the effect of Kondo holes in lattice and in impurity
models from two complementary perspectives using (i) a supercell analysis in
the uncorrelated limit and (ii) the Lieb-Mattis theorem in the strongly
interacting regime with well defined local moments.

Without any restrictions concerning the spatial dimension and the geometry of
the underlying lattice, the supercell analysis predicts the existence of
localized decoupled orbitals when Kondo holes are introduced in the PAM or
MIAM, without presupposing PH symmetry, i.e. a half filled conduction band.
Additionally, for the subset of bipartite lattice structures and PH symmetric
models, the Lieb-Mattis theorem makes some rigorous statements about these
Kondo hole Hamiltonians, predicting local cluster magnetic moments which can
be either stable or subject to further screening processes.

In consequence of these generic statements which are in accordance with the
established theory about Kondo holes [18, 19, 20, 21, 22, 23], we expect the
occurrence of hole induced bound states to be a generic feature for a wide
class of HF materials. Indeed, Hamidian et al [8] observed these bound states
inside the hybridization gap in Th-doped URu2Si2 by comparing the differential
conductance far away from a Th atom with that right at a Th atom site. We
demonstrated that these bound states are the consequence of pseudo-gap
formation which can be understood from a local point of view.

Using the NRG in combination with a recently developed wide band approximation
for multi impurity models [32], we solved the MIAM with up to $N_{f}=7$
$f$-orbitals in the strongly interacting regime. Inserting Kondo holes in such
dense impurity arrays our NRG results are in perfect agreement with all exact
statements available, ranging from the non-interacting limit (supercell
analysis) to the strongly interacting regime (Lieb-Mattis theorem). We
identified three different classes at PH symmetry: The effective Kondo
coupling $J^{\text{eff}}_{\text{K}}$ between the hole induced bound state and
the environment can be (i) FM, leading to a stable LM FP, or (ii) AF,
resulting in a low energy scale at which the intermediate LM FP becomes
unstable. Remarkably, this low energy scale is of a counterintuitive Kondo
form due to $J^{\text{eff}}_{\text{K}}\propto U^{n}$, where $n>1$ depends on
the number $N_{f}$ of correlated $f$-orbitals in the model. Alternatively, in
case of several Kondo holes, the hole induced bound states can (iii) interact
directly with each other which may lead to magnetic order in lattice models
such that the ordered magnetic moment per unit cell is significantly reduced
compared to the number of local moments per unit cell.

Going beyond the PH symmetric limit where the Lieb-Mattis theorem is
applicable, by including the nearly unoccupied hole orbital or shifting the
band center of the conduction electrons, we further demonstrated that the LM
FP can be destabilized. In this case the local moment phase is replaced by two
types of singlet phases that are adiabatically connected. At a KT type QCP the
physics is governed by an exponentially suppressed Kondo scale approaching the
strong coupling phase that can be replaced by a singlet formation via
antiferromagnetic RKKY interaction for large deviation from the critical
values.

In heavy Fermions that are close to a Kondo-insulator such as CePd3 we believe
that our analysis can be applied for low concentrations of Kondo-holes.
Breaking particle-hole symmetry in the conduction band as well as including
the unoccupied $4f$-orbitals of La for virtual excitation leads to a coupling
of the Kondo-hole induced local moment such that the localized free spin
$S=1/2$ is screened. This provides the microscopic mechanism for an Kondo
effect as reported in Ce1-xLaxPd3. Deviations of single-ion Kondo behavior is
expected when the Kondo-lattice coherent temperature is not well separated
from the single-impurity Kondo temperature of the residual local moments, as
well as when the concentration is high enough that Kondo-hole induced local
moments start to interact with each other.

## References

  * Haas and Berg [1936] W. D. Haas and G. V. D. Berg, Physica 3, 440 (1936).
  * Kondo [1964] J. Kondo, Progress of Theoretical Physics 32, 37 (1964).
  * Hewson [1993] A. C. Hewson,  _The Kondo Problem to Heavy Fermions_, Cambridge Studies in Magnetism (Cambridge University Press, 1993).
  * Grewe [1984] N. Grewe, Solid State Communications 50, 19 (1984).
  * Grewe and Steglich [1991] N. Grewe and F. Steglich, in  _Handbook on the Physics and Chemistry of Rare Earths_ , Vol. 14, edited by K. A. Gschneidner, Jr. and L. Eyring (North-Holland, Amsterdam, 1991) p. 343.
  * Lawrence  _et al._ [1985] J. M. Lawrence, J. D. Thompson, and Y. Y. Chen, Phys. Rev. Lett. 54, 2537 (1985).
  * Lawrence  _et al._ [1996] J. M. Lawrence, T. Graf, M. F. Hundley, D. Mandrus, J. D. Thompson, A. Lacerda, M. S. Torikachvili, J. L. Sarrao, and Z. Fisk, Phys. Rev. B 53, 12559 (1996).
  * Hamidian  _et al._ [2011] M. H. Hamidian, A. R. Schmidt, I. A. Firmo, M. P. Allan, P. Bradley, J. D. Garrett, T. J. Williams, G. M. Luke, Y. Dubi, A. V. Balatsky, and J. C. Davis, Proceedings of the National Academy of Sciences 108, 18233 (2011).
  * Rosa  _et al._ [2016] P. F. S. Rosa, A. Oostra, J. D. Thompson, P. G. Pagliuso, and Z. Fisk, Phys. Rev. B 94, 045101 (2016).
  * Shimozawa  _et al._ [2012] M. Shimozawa, T. Watashige, S. Yasumoto, Y. Mizukami, M. Nakamura, H. Shishido, S. K. Goh, T. Terashima, T. Shibauchi, and Y. Matsuda, Phys. Rev. B 86, 144526 (2012).
  * Pietrus  _et al._ [2008] T. Pietrus, H. v. Löhneysen, and P. Schlottmann, Phys. Rev. B 77, 115134 (2008).
  * Rotundu  _et al._ [2007] C. R. Rotundu, B. Andraka, and P. Schlottmann, Phys. Rev. B 76, 054416 (2007).
  * Ślebarski  _et al._ [2010] A. Ślebarski, J. Spałek, M. Fijałkowski, J. Goraus, T. Cichorek, and L. Bochenek, Phys. Rev. B 82, 235106 (2010).
  * Onuki and Komatsubara [1987] Y. Onuki and T. Komatsubara, in  _Anomalous Rare Earths and Actinides_, edited by J. Boucherle, J. Flouquet, C. Lacroix, and J. Rossat-Mignod (Elsevier, 1987) pp. 281–288.
  * Malik  _et al._ [1995] S. K. Malik, L. Menon, K. Ghosh, and S. Ramakrishnan, Phys. Rev. B 51, 399 (1995).
  * Adroja  _et al._ [1996] D. Adroja, B. Rainford, A. Neville, P. Mandal, and A. Jansen, Journal of Magnetism and Magnetic Materials 161, 157 (1996).
  * Ślebarski  _et al._ [1998] A. Ślebarski, A. Jezierski, S. Mähl, M. Neumann, and G. Borstel, Phys. Rev. B 58, 4367 (1998).
  * Sollie and Schlottmann [1991a] R. Sollie and P. Schlottmann, Journal of Applied Physics 69, 5478 (1991a).
  * Sollie and Schlottmann [1991b] R. Sollie and P. Schlottmann, Journal of Applied Physics 70, 5803 (1991b).
  * Schlottmann [1992] P. Schlottmann, Phys. Rev. B 46, 998 (1992).
  * Schlottmann [1996] P. Schlottmann, Czech. J. Phys. 46, 1895 (1996).
  * Yu [1996] C. C. Yu, Phys. Rev. B 54, 15917 (1996).
  * Figgins and Morr [2011] J. Figgins and D. K. Morr, Phys. Rev. Lett. 107, 066401 (2011).
  * Baruselli and Vojta [2014] P. P. Baruselli and M. Vojta, Phys. Rev. B 89, 205105 (2014).
  * Xie  _et al._ [2017] N. Xie, D. Hu, and Y.-f. Yang, Scientific Reports 7 (2017).
  * Sen  _et al._ [2015] S. Sen, J. Moreno, M. Jarrell, and N. S. Vidhyadhiraja, Phys. Rev. B 91, 155146 (2015).
  * Kumar and Vidhyadhiraja [2014] P. Kumar and N. S. Vidhyadhiraja, Phys. Rev. B 90, 235133 (2014).
  * Zhu  _et al._ [2012] J.-X. Zhu, J.-P. Julien, Y. Dubi, and A. V. Balatsky, Phys. Rev. Lett. 108, 186401 (2012).
  * Maruyama  _et al._ [2002] I. Maruyama, N. Shibata, and K. Ueda, Phys. Rev. B 65, 174421 (2002).
  * Wermbter  _et al._ [1996] S. Wermbter, K. Sabel, and G. Czycholl, Phys. Rev. B 53, 2528 (1996).
  * Schweitzer and Czycholl [1990] H. Schweitzer and G. Czycholl, Solid State Commun. 74, 735 (1990).
  * Eickhoff and Anders [2020] F. Eickhoff and F. B. Anders, Phys. Rev. B 102, 205132 (2020).
  * Krishna-murthy  _et al._ [1980a] H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, Phys. Rev. B 21, 1003 (1980a).
  * Krishna-murthy  _et al._ [1980b] H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, Phys. Rev. B 21, 1044 (1980b).
  * Andrei  _et al._ [1983] N. Andrei, K. Furuya, and J. H. Lowenstein, Rev. Mod. Phys. 55, 331 (1983).
  * Schlottmann [1989] P. Schlottmann, Physics Rep. 181, 1 (1989).
  * Jones and Varma [1987] B. A. Jones and C. M. Varma, Phys. Rev. Lett. 58, 843 (1987).
  * Eickhoff  _et al._ [2018] F. Eickhoff, B. Lechtenberg, and F. B. Anders, Phys. Rev. B 98, 115103 (2018).
  * Schrieffer and Wolff [1966] J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491 (1966).
  * Schlottmann [1995] P. Schlottmann, Physica B: Condensed Matter 206-207, 816 (1995).
  * Lieb and Mattis [1962] E. Lieb and D. Mattis, Journal of Mathematical Physics 3, 749 (1962).
  * Shen [1996] S.-Q. Shen, Phys. Rev. B 53, 14252 (1996).
  * Titvinidze  _et al._ [2014] I. Titvinidze, A. Schwabe, and M. Potthoff, Phys. Rev. B 90, 045112 (2014).
  * Wilson [1975] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).
  * Bulla  _et al._ [2008] R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. Phys. 80, 395 (2008).
  * Note [1] Note that we cannot make a statement about the nature of the FP: In the TIAM, for instance, two adiabatically connected FP are found, one originating from a RKKY interaction the other driven by the Kondo effect [37, 56].
  * Vojta [2006] M. Vojta, Philosophical Magazine 86, 1807 (2006).
  * Pereira  _et al._ [2006] V. M. Pereira, F. Guinea, J. M. B. L. dos Santos, N. M. R. Peres, and A. H. C. Neto, Phys. Rev. Let. 96, 036801 (2006).
  * Castro Neto  _et al._ [2009] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
  * Nanda  _et al._ [2012] B. R. K. Nanda, M. Sherafati, Z. S. Popović, and S. Satpathy, New Journal of Physics 14, 083004 (2012).
  * May  _et al._ [2018] D. May, P.-W. Lo, K. Deltenre, A. Henke, J. Mao, Y. Jiang, G. Li, E. Y. Andrei, G.-Y. Guo, and F. B. Anders, Phys. Rev. B 97, 155419 (2018).
  * Jiang  _et al._ [2018] Y. Jiang, P.-W. Lo, D. May, G. Li, G.-Y. Guo, F. B. Anders, T. Taniguchi, K. Watanabe, J. Mao, and E. Y. Andrei, Nature Communications 9, 2349 (2018).
  * Fritz and Vojta [2004] L. Fritz and M. Vojta, Phys. Rev. B 70, 214427 (2004).
  * Lechtenberg  _et al._ [2017] B. Lechtenberg, F. Eickhoff, and F. B. Anders, Phys. Rev. B 96, 041109 (2017).
  * Jones  _et al._ [1988] B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988).
  * Affleck  _et al._ [1995] I. Affleck, A. W. W. Ludwig, and B. A. Jones, Phys. Rev. B 52, 9528 (1995).
  * Esat  _et al._ [2016] T. Esat, B. Lechtenberg, T. Deilmann, ChristianWagner, P. Krüger, R. Temirov, M. Rohlfing, F. B. Anders, and F. S. Tautz, Nature Physics 12, 8 (2016).
  * Note [2] Since the trivial irrep of any point group is one dimensional we drop the index $\alpha$ in this case.
  * Mitchell and Bulla [2015] A. K. Mitchell and R. Bulla, Phys. Rev. B 92, 155101 (2015).
  * Grenzebach  _et al._ [2008] C. Grenzebach, F. B. Anders, G. Czycholl, and T. Pruschke, Phys. Rev. B 77, 115125 (2008).
  * Aulbach  _et al._ [2015] M. W. Aulbach, I. Titvinidze, and M. Potthoff, Phys. Rev. B 91, 174420 (2015).
  * Lebanon  _et al._ [2003] E. Lebanon, A. Schiller, and F. B. Anders, Phys. Rev. B 68, 155301 (2003).
  * Nevidomskyy and Coleman [2009] A. H. Nevidomskyy and P. Coleman, Phys. Rev. Lett. 103, 147205 (2009).
  * Schrieffer [1967] J. R. Schrieffer, Journal of Applied Physics 38, 1143 (1967).
  * Doniach [1977] S. Doniach, Physica B 91, 231 (1977).
  * Pruschke  _et al._ [2000] T. Pruschke, R. Bulla, and M. Jarrell, Phys. Rev. B 61, 12799 (2000).
  * Note [3] We have shown in Ref. [32] that the rank$[\bm{\Gamma}]$ is limited by the number of Fermi wave vectors hence by 2 in 1d.
  * Silva  _et al._ [1996] J. B. Silva, W. L. C. Lima, W. C. Oliveira, J. L. N. Mello, L. N. Oliveira, and J. W. Wilkins, Phys. Rev. Lett. 76, 275 (1996).
  * Jones and Varma [1989] B. A. Jones and C. M. Varma, Phys. Rev. B 40, 324 (1989).
  * Cornaglia and Grempel [2005] P. S. Cornaglia and D. R. Grempel, Phys. Rev. B 71, 075305 (2005).
  * Tanaka  _et al._ [2012] Y. Tanaka, N. Kawakami, and A. Oguri, Phys. Rev. B 85, 155314 (2012).
  * Wójcik and Weymann [2015] K. P. Wójcik and I. Weymann, Phys. Rev. B 91, 134422 (2015).
  * Žitko [2010] R. Žitko, Phys. Rev. B 81, 115316 (2010).
  * Lechtenberg and Anders [2018] B. Lechtenberg and F. B. Anders, Phys. Rev. B 98, 035109 (2018).
  * Grenzebach  _et al._ [2006] C. Grenzebach, F. B. Anders, G. Czycholl, and T. Pruschke, Phys. Rev. B 74, 195119 (2006).