# Photo-emission signatures of coherence breakdown in Kondo alloys: dynamical
mean-field theory approach

B Poudel1, C Lacroix2, G Zwicknagl3 and S Burdin1 1 Université de Bordeaux,
CNRS, LOMA, UMR 5798, 33400 Talence, France  2 Institut Néel, CNRS and
Université Grenoble-Alpes, Boite Postale 166, 38042 Grenoble Cedex 09, France
3 Technical University Braunschweig, 0531 Braunschweig, Germany

###### Abstract

We study the Kondo alloy model on a square lattice using dynamical mean-field
theory (DMFT) for Kondo substitution and disorder effects, together with
static mean-field approximations. We computed and analyzed photoemission
properties as a function of electronic filling $n_{c}$, Kondo impurity
concentration $x$, and strength of Kondo temperature $T_{K}$. We provide a
complete description of the Angle Resolved Photoemission Spectroscopy (ARPES)
signals expected in the paramagnetic Kondo phases. By analyzing the Fermi
surface, we observe the Lifshitz-like transition predicted previously for
strong $T_{K}$ at $x=n_{c}$ and we discuss the evolution of the dispersion
from the dense coherent to the dilute Kondo regimes. At smaller $T_{K}$, we
find that this transition marking the breakdown of coherence at $x=n_{c}$
becomes a crossover. However, we identify another transition at a smaller
concentration $x^{\star}$ where the effective mass continuously vanishes.
$x^{\star}$ separates the one-branch and the two-branches ARPES dispersions
characterizing respectively dilute and dense Kondo paramagnetic regimes. The
$x-T_{K}$ phase diagrams are also described, suggesting that the transition at
$x^{\star}$ might be experimentally observable since magnetically ordered
phases are stabilized at much lower $T_{K}$. Fermi surface reconstructions in
antiferromagnetic and ferromagnetic phases are also discussed.

Keywords: dynamical mean-field theory, heavy-fermion, disordered Kondo alloys,
coherence breakdown, angle resolved photoemission spectroscopy, Lifshitz
transition, phase transitions

##  1 Introduction

Kondo effect is a fundamental microscopic mechanism which is relevant for
studying strongly correlated electron materials [1, 2]. It is characterized by
the low temperature formation of a highly entangled state between conduction
electrons and quantum magnetic impurities. It was observed in rare-earth based
compounds below the Kondo temperature $T_{K}$, where Kondo impurities are
realized by Kramers doublets of $4f$ or $5f$ orbitals, which may form local
singlets with the spins of conduction electrons. In most cases, the low energy
excitations of Kondo paramagnetic ground states can be described as Fermi
liquids [3].

In concentrated Kondo systems, one remarkable phenomenon emerging from Kondo
effect is the possible realization of a coherent macroscopic Fermi liquid
ground state where localized quantum magnetic impurities contribute to the
formation of non-local, i.e. Bloch waves, fermionic quasiparticle excitations.
This fundamental issue is also very important for microscopic modelling of
rare-earth compounds using renormalized quasi-particle band model [4] where
$f-$electron levels may enter in the description of the Fermi surface.

A standard approach for tuning the physical properties of concentrated Kondo
systems is based on applying mechanical or chemical pressure and invoking
Doniach’s argument [5]. Indeed, magnetic order is stabilized when the strength
of the Kondo interaction, i.e. the corresponding $T_{K}$, is relatively small.
As a consequence, very rich pressure-temperature phase diagrams are obtained
in heavy-fermion Kondo $f-$electron materials, with a large diversity of
unconventional quantum phases and behaviors [6]. These include unconventional
superconductivity [7, 8] as observed, e.g. in CeCu2Si2 [9] or non-Fermi liquid
behavior [10]. In Kondo systems, the concentration of Kondo impurities $x$ and
the electronic filling $n_{c}$ are other relevant parameters besides pressure.

Continuous tuning of $x$ can be realized experimentally in Kondo alloys heavy-
fermions by isostructural substitution of magnetic rare-earth atom with a non-
magnetic one. This induces remarkable changes in macroscopic physical
properties. For example, Ce-La substitution in the series CexLa1-xCu6 [11, 12]
revealed an evolution from coherent dense Kondo lattice regime to dilute Kondo
regime. In the series CexLa1-xNi2Ge2 [13] and CexLa1-xPtIn [14] additional
non-Fermi liquid (nFL) behavior is observed along with dense-dilute transition
whereas in the series CexLa1-xFePO [15] coexistence of spin glass with dilute
Kondo regime is observed. Substitution of Ce (Kondo) by La (non-Kondo) can
also produce antiferromagnetic (AF) to paramagnetic (PM) transitions in
CexLa1-xCu2Ge2 [16] as well as PM to AF phase in CexLa1-xRu2Si2 [17]. Other
types of rare-earth substitutions can also result in rich physics like Fermi-
surface instabilities in unconventional superconductors like CexYb1-xCoIn5
[18] and Nd2-xCexCuO4 [19].

Multiple experimental and theoretical techniques were also used in order to
explore the effect of $n_{c}$ for Kondo lattices (i.e., for $x=1$). Some of
these studies revealed Doniach-like phase diagrams [20, 21, 22, 23]. An
important attention was also focused on the issue of coherence in the
paramagnetic phase of dense Kondo systems. This was first motivated by the
question of ‘exhaustion’ raised by Nozières [24] regarding how a coherent
Fermi liquid Kondo ground state can be formed when the concentration of Kondo
impurities is larger than the concentration of conduction electrons. The
robustness of a coherent Kondo state has then been investigated in the
framework of the Kondo lattice model by several complementary theoretical
approaches [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. Meanwhile,
photoelectron spectroscopy experiments [36, 37, 38] in Cerium and in Ytterbium
heavy-fermion compounds found to be inconsistent with the predictions of the
single impurity Kondo model.

These works motivated further theoretical investigations of the concomitant
roles of $x$ and $n_{c}$ in the issue of coherence in Kondo alloys. A
transition or a crossover between dilute and coherent dense Kondo Fermi liquid
regimes had been suggested by various complementary methods including mean-
field approximation and strong coupling expansions [39, 40, 41, 42, 43, 44,
45], Monte Carlo simulations [46, 47, 48], numerical renormalization group
[49], or local-moment approach [50, 51]. A crucial difference between these
two Kondo Fermi-liquid regimes can be obtained from strong coupling
description: for $x<n_{c}$ each Kondo impurity captures one conduction
electron, forming Kondo singlets, and the Fermi-liquid quasiparticles
correspond to the remaining $n_{c}-x$ electrons. For $x>n_{c}$, all conduction
electrons form Kondo singlets and the Fermi-liquid quasiparticles emerge from
the degrees of freedom of the remaining $x-n_{c}$ unscreened Kondo impurities.
Invoking a particle-hole transformation, doping with Kondo impurities may thus
correspond to particle doping in the dense coherent regime characterized by a
large Fermi-surface enclosing $x+n_{c}$ particles, while it may correspond to
hole doping in the dilute regime, with a reduced Fermi surface enclosing
$n_{c}-x$ particles [42].

While these theoretical works predict a fundamental difference between dense
and dilute Kondo alloys, experimental signatures of such possible transitions
are still not clear or mostly indirect. In this article, we explore one
important property that may distinguish dense from dilute systems: the
contribution (dense coherent Kondo) or the non-contribution (dilute Kondo) of
f-electrons to the formation of the Fermi surface (FS). Here, we investigate
from a theoretical point of view the robustness and breakdown of coherence in
Kondo alloys with a focus on the possible signatures in Photo-emission
spectroscopy (PES) and in Angle Resolved Photo-emission spectroscopy (ARPES).
These experimental technics have already proven to be very useful skills for
investigating the physics of $f-$electron systems, including Kondo alloys [52,
53]. For example, PES experiments revealed limitations of the single impurity
models for describing dense Kondo systems [36, 37, 38]. ARPES experiments
carried on varieties of Kondo lattice systems like CeRu2Si2 [54, 55, 56],
CeRu2Ge2 [57], CeBi [58],CeNiSn [59] and YbRh2Si2 [60, 61, 62] showed large
Fermi surfaces due to the coherent participation of 4$f$ electrons. ARPES
experiments also permitted a direct observation of dispersive Kondo resonance
peaks in CeCoGe1.2Si0.8 [63] and surface and bulk hybridizations in
antiferromagnetic Kondo lattice CeRh2Si2 [64].

In our work, we extend the matrix DMFT/CPA approach developed in [39] to
compute and analyze PES and ARPES signals for Kondo alloys on a square lattice
structure. We consider both paramagnetic Kondo phases and magnetic ordered
ground states. We start this paper by detailing model, methods and
approximations in section 2, results in section 3 followed by conclusion in
section 4.

##  2 Model and method

We consider the Kondo alloy model (KAM) Hamiltonian:

$\displaystyle
H=\sum_{ij\sigma}\left({t}_{ij}-\mu\delta_{ij}\right)c_{i\sigma}^{\dagger}c_{j\sigma}+J_{K}\sum_{i\in{\cal
K}}{\bf S}_{i}\cdot{\bf s}_{i}~{},$ (1)

where $c_{i\sigma}^{(\dagger)}$ denotes creation (annihilation) operator of
electron with spin component $\sigma=\uparrow,\downarrow$ on site $i$ of a
periodic lattice. In this work, we consider a bipartite lattice (square or
Bethe) that we formally split into two sublattices A and B. We consider non-
zero electronic intersite tunneling $t_{ij}\equiv t$ for nearest neighbors $i$
and $j$ only. $\delta_{ij}$ is the Kronecker symbol, and the chemical
potential $\mu$ is determined by fixing the average electronic filling per
site to $n_{c}$. The term on the right hand side represents local Kondo
impurities on a fixed subset ${\cal K}$ of lattice sites, which have been
randomly distributed with a site concentration $x$. The antiferromagnetic
interaction $J_{K}$ couples local Kondo quantum spin $1/2$ operators ${\bf
S}_{i}$ with the local density of spin of conduction electrons ${\bf s}_{i}$.
In the following, ${\cal N}$ (non-Kondo) will refer to the subset of non-Kondo
sites. In this model, each local Kondo spin describes a local 4$f^{1}$
electronic state (Ce based materials) or a 4$f^{13}$ hole state (Yb based
materials) with fixed valence.

###  2.1 DMFT/CPA method

We take into account the randomness of the $\mathcal{K}$ sites distribution in
the KAM using a DMFT/CPA approach [65, 39]. First, the action related with the
KAM Hamiltonian Eq. (1) is expressed as

$S=-\sum_{ij\sigma}\int_{0}^{\beta}d\tau
c_{i\sigma}^{\dagger}(\tau)\left({(\partial_{\tau}-\mu)\delta_{ij}+t_{ij}}\right)c_{j\sigma}(\tau)-\int_{0}^{\beta}d\tau
J_{K}\sum_{i\in\mathcal{K}}\textbf{S}_{i}(\tau)\textbf{s}_{i}(\tau)$ (2)

where $\beta=1/T$ is the inverse temperature, and $\tau$ denotes imaginary
time. Hereafter, we will be following the matrix-DMFT formalism as developed
in [39] for Kondo and non-Kondo paramagnetic solutions and generalized in [45]
to take account the possibility of some magnetically commensurate ordered
phases. Introducing the sub-lattice index $\alpha=$A, B and the site-nature
index $a=\mathcal{K}$ or $\mathcal{N}$, the lattice problem is thus mapped
onto four effective single site problems embedded with dynamical local baths
$\Delta_{a\alpha}^{\sigma}(\tau)=\sum_{ij}t_{\alpha i}t_{j\alpha}\langle
c_{i\sigma}(\tau)c_{j\sigma}^{\dagger}(\tau^{\prime})\rangle^{(\alpha,a)}$
that can be expressed in terms of the cavity ($\alpha,a$) electronic Green’s
functions, i.e., using the KAM action (2) without the contribution of the
local site ($\alpha,a$). Furthermore, we take into acount randomness by
assuming that the local $\mathcal{K}$ or $\mathcal{N}$ configuration of each
site is not correlated with the configuration of neighboring sites. Hence the
dynamical local baths for $\mathcal{K}$ or $\mathcal{N}$ sites are identical,
so
$\Delta_{\mathcal{K}\alpha}^{\sigma}(\tau)=\Delta_{\mathcal{N}\alpha}^{\sigma}(\tau)\equiv\Delta_{\alpha}^{\sigma}(\tau)$.
The local effective actions are expressed as

$\displaystyle S_{\alpha}^{\mathcal{N}}$
$\displaystyle=-\sum_{\sigma}\int_{0}^{\beta}d\tau\int_{0}^{\beta}d\tau^{\prime}c_{\mathcal{N}\alpha\sigma}^{\dagger}(\tau)\mathcal{G}_{\alpha\sigma}^{-1}(\tau-\tau^{\prime})c_{\mathcal{N}\alpha\sigma}(\tau^{\prime})~{},$
(3) $\displaystyle S_{\alpha}^{\mathcal{K}}$
$\displaystyle=-\sum_{\sigma}\int_{0}^{\beta}d\tau\int_{0}^{\beta}d\tau^{\prime}c_{\mathcal{K}\alpha\sigma}^{\dagger}(\tau)\mathcal{G}_{\alpha\sigma}^{-1}(\tau-\tau^{\prime})c_{\mathcal{K}\alpha\sigma}(\tau^{\prime})-J_{K}\int_{0}^{\beta}d\tau\textbf{S}_{\alpha}(\tau)\textbf{s}_{\alpha}(\tau)~{},$
(4)

where the Matsubara frequency $i\omega$ dependencies of the kernel and the
bath are related as $\mathcal{G}_{\alpha\sigma}^{-1}(i\omega)\equiv
i\omega+\mu-\Delta_{\alpha}^{\sigma}(i\omega)$. The local electronic Green’s
functions $G_{a\alpha}^{\sigma}(\tau)\equiv-\langle
c_{a\alpha\sigma}(\tau)c_{a\alpha\sigma}^{\dagger}(0)\rangle$ can be computed
from the dynamical bath. The relation for $\mathcal{N}$-site is explicit since
the corresponding local action (3) is quadratic:

$G_{\mathcal{N}\alpha}^{\sigma}(i\omega)=\frac{1}{i\omega+\mu-\Delta_{\alpha}^{\sigma}(i\omega)}~{}.$
(5)

Computing the local Green function for $\mathcal{K}$-site requires an
appropriate impurity solver for the Kondo interaction term in action (4). The
precise impurity solvers that we employed for the results presented in this
paper are described in A.

Hereafter we present the specific self-consistent relations that we developed
in [45] using the DMFT/CPA approach, allowing to compute the dynamical bath
from the local electronic Green’s functions. We consider two complementary
examples of bipartite lattices: a Bethe lattice, and a square lattice. The
Bethe lattice is very convenient from the computational point of view. Indeed,
considering the limit of a large lattice coordination $Z$ and rescaling the
electronic exchange $t\mapsto t/\sqrt{Z}$, the self-consistency DMFT relation
for a Bethe lattice is explicit:
$\Delta_{\alpha}^{\sigma}(i\omega)=xt^{2}G_{\mathcal{K}\bar{\alpha}}^{\sigma}(i\omega)+(1-x)t^{2}G_{\mathcal{N}\bar{\alpha}}^{\sigma}(i\omega)$,
where $\bar{\alpha}$ denotes the complementary sublattice of $\alpha$.
However, while considering a Bethe lattice structure is very useful for
computing local dynamical correlations, some specificities of the underlying
periodic lattice and its reciprocal lattice are missing, that are crucial for
describing ARPES. In this work we thus considered also the DMFT approach in
its most general framework, having in mind a square lattice as bipartite
structure. We introduce the ($2\times 2$) local Green function matrix:

$\mathbf{G}_{loc,\alpha}^{\sigma}(i\omega)=\begin{pmatrix}xG_{\mathcal{K}\alpha}^{\sigma}(i\omega)&0\\\
0&(1-x)G_{\mathcal{N}\alpha}^{\sigma}(i\omega)\\\ \end{pmatrix}~{},$ (6)

and the disorder-averaged site-dependent Green function matrix which preserves
the sublattice A/B symmetries :

$\mathbf{G}_{ij}^{\sigma}(i\omega)=\begin{pmatrix}G_{ij\sigma}^{\mathcal{K}\mathcal{K}}(i\omega)&G_{ij\sigma}^{\mathcal{K}\mathcal{N}}(i\omega)\\\
~{}~{}&~{}~{}\\\
G_{ij\sigma}^{\mathcal{N}\mathcal{K}}(i\omega)&G_{ij\sigma}^{\mathcal{N}\mathcal{N}}(i\omega)\\\
\end{pmatrix}~{}.$ (7)

The matrix elements are the electronic correlations
$G_{ij\sigma}^{ab}(\tau)\equiv-\langle
c_{i\sigma}(\tau)c_{j\sigma}^{\dagger}(0)\rangle$ that are firstly (i.e.,
before disorder-average) computed formally with the disordered KAM Hamiltonian
(1) assuming the site-natures $a$ and $b$ for $i$ and $j$ respectively. We
also introduce the momentum representation: $\mathbf{G}^{\sigma}_{\bf
kk^{\prime}}(i\omega)\equiv\frac{1}{N}\sum_{ij}e^{i{\bf k}^{\prime}{\bf
R}_{j}-i{\bf k}{\bf R}_{i})}\mathbf{G}_{ij}^{\sigma}(i\omega)$, where $N$ is
the number of lattice sites. In order to take into account the possibility of
ordering that may break lattice symmetry with Néel order, we introduce the
wavevector $\mathbf{Q}=(\pi,\pi)$. We then define the ($4\times 4$) matrix
$\mathbf{\bar{\bar{G}}^{\sigma}_{\mathbf{kk^{\prime}}}}\equiv\begin{pmatrix}\mathbf{G}^{\sigma}_{\bf
kk^{\prime}}&\mathbf{G}^{\sigma}_{\bf kk^{\prime}+Q}\\\
\mathbf{G}^{\sigma}_{\bf k+Qk^{\prime}}&\mathbf{G}^{\sigma}_{\bf
k+Qk^{\prime}+Q}\end{pmatrix}$. Exploiting symmetric properties of the Green
functions for the specific cases of paramagnetic Kondo, ferromagnetic and
staggered antiferromagnetic orders, and invoking the DMFT diagrammatic
expansion, we obtain the following relation:

$\mathbf{\bar{\bar{G}}^{\sigma}_{\mathbf{kk^{\prime}}}}(i\omega)=\delta_{\mathbf{kk^{\prime}}}\left(\begin{pmatrix}\mathbf{\Pi}_{+}^{\sigma}(i\omega)&\mathbf{\Pi}_{-}^{\sigma}(i\omega)\\\
\mathbf{\Pi}_{-}^{\sigma}(i\omega)&\mathbf{\Pi}_{+}^{\sigma}(i\omega)\end{pmatrix}^{-1}-\begin{pmatrix}\mathbf{E_{k}}&0\\\
0&\mathbf{E_{k+Q}}\end{pmatrix}\right)^{-1},$ (8)

where $\mathbf{\Pi}_{\pm}^{\sigma}(i\omega)$ is a local propagator matrix
which emerges in DMFT diagrammatic expansion and

$\mathbf{E_{k}}=\epsilon_{\mathbf{k}}\begin{pmatrix}1&1\\\
1&1\end{pmatrix}~{},$ (9)

with the non-interacting dispersion
$\epsilon_{\mathbf{k}}=2t[\cos{(k_{x})}+\cos{(k_{y})}]$. The DMFT self-
consistency is closed by expressing
$\mathbf{G}_{loc,\alpha}^{\sigma}(i\omega)=\mathbf{G}_{ii}^{\sigma}(i\omega)$
with $i$ on sublattice $\alpha$. When solving this identity in a DMFT self-
consistent scheme, the local Green function is supposed to be known from Eq.
(6) and having solved the local effective action problems given by Eqs. (3)
and (4). In turn, the local effective bath which is involved in these actions
can be obtained with algebraic matrix inversion as follows:

$\mathbf{\Delta}_{\alpha}^{\sigma}(i\omega)\equiv[\mathbf{\Pi}_{\alpha}^{\sigma}(i\omega)]^{-1}-[\mathbf{G}_{loc,\alpha}^{\sigma}(i\omega)]^{-1}~{},$
(10)

where the matrices
$\mathbf{\Pi}_{A}^{\sigma}(i\omega)=\mathbf{\Pi}_{+}^{\sigma}(i\omega)+\mathbf{\Pi}_{-}^{\sigma}(i\omega)$
and
$\mathbf{\Pi}_{B}^{\sigma}(i\omega)=\mathbf{\Pi}_{+}^{\sigma}(i\omega)-\mathbf{\Pi}_{-}^{\sigma}(i\omega)$
can be obtained from the local Green function by invoking the following
identity:

$\mathbf{G}_{loc,\alpha}^{\sigma}(i\omega)=\frac{1}{N_{RBZ}}\sum_{\mathbf{k}\in
RBZ}([\mathbf{\Pi}_{\alpha}^{\sigma}(i\omega)]^{-1}-\mathbf{E_{k}}\mathbf{\Pi}_{\bar{\alpha}}^{\sigma}(i\omega)\mathbf{E_{k}})^{-1}~{}.$
(11)

Here, $N_{RBZ}$ is the volume of the reduced Brillouin zone that corresponds
to one sublattice A/B. Finally, the dynamical local bath
$\Delta_{\alpha}^{\sigma}(i\omega)$ is deduced from the diagonal matrix
elements:

$\Delta_{\alpha}^{\sigma}=\left(\mathbf{\Delta}_{\alpha}^{\sigma}\right)_{\mathcal{K}}=\left(\mathbf{\Delta}_{\alpha}^{\sigma}\right)_{\mathcal{N}}~{}.$
(12)

Once self-consistent solution of the DMFT effective problem is obtained,
physical quantities can be computed from the KAM and provide predictions for
photo-emission experiments in Kondo alloy materials. For example, the average
electronic local density of states (d.o.s.) per spin component can be computed
as:

$\displaystyle\rho(\omega)$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\sum_{\sigma,\alpha}{\cal
I}m[xG_{\mathcal{K}\alpha}^{\sigma}(\omega)+(1-x)G_{\mathcal{N}\alpha}^{\sigma}(\omega)]~{}.$

The d.o.s. can be measured by PES, and it may be related with the ARPES signal
by a sum rule (see section 3.1).

###  2.2 Angle Resolved Photoemission spectroscopy (ARPES) in a paramagnetic
Kondo phase

In this section we present the adaptation of the DMFT/CPA method for modelling
ARPES experimental measurements on Kondo alloy materials. For the sake of
clarity, we assume here that the system is in a paramagnetic phase and we skip
sublattice and spin indices $\alpha$ and $\sigma$, focussing on the
specificities of the random distribution of ${\cal K}$ and ${\cal N}$ sites on
a periodic square lattice. The system is disordered, however the signal which
is measured by ARPES is the spectral weight ${\cal A}({\bf
k},\omega)=-\frac{1}{\pi}{\cal I}m[G_{Alloy}({\bf k},\omega+i0^{+})]$ which is
associated with the intersite one-electron correlation function
$G_{Alloy}({\bf k},i\omega)=\frac{1}{N}\sum_{ij}e^{i{\bf k}({\bf R}_{j}-{\bf
R}_{i})}{G}_{ij}(i\omega)$. Here $\mathbf{k}$ denotes the momentum in the
Brillouin zone of the square lattice. The site-dependent electronic Green
function ${G}_{ij}(i\omega)$ is in principle dependent on the disorder
configuration, i.e., on the specific distribution of ${\cal K}$ and ${\cal N}$
sites. However, the ARPES signal is rather related with $G_{Alloy}({\bf
k},i\omega)$ which involves a sum over all lattice sites $i$ and $j$. We
assume that the disorder is self-averaged when performing this sum. In
principle this is realized if the system is sufficiently big. Therefore, the
ARPES spectral weight of a Kondo alloy system can be obtained from the
disorder-averaged Green-function computed from the KAM using the DMFT/CPA
method:

$G_{Alloy}({\bf k},i\omega)=G_{\bf k}^{\mathcal{K}\mathcal{K}}(i\omega)+G_{\bf
k}^{\mathcal{K}\mathcal{N}}(i\omega)+G_{\bf
k}^{\mathcal{N}\mathcal{K}}(i\omega)+G_{\bf
k}^{\mathcal{N}\mathcal{N}}(i\omega)~{},$ (14)

where $G_{\bf k}^{ab}(i\omega)$ denote the elements of $G_{\bf kk}(i\omega)$
which is the momentum representation of the Green function matrix (7). This
expression reflects the fact that ARPES on a Kondo alloy involves all one-
electron excitations irrespectively from the Kondo or non-Kondo nature of the
sites where a conduction electron is created or annihilated. Invoking the DMFT
self-consistent condition $\sum_{\bf k}G_{\bf
k}^{\mathcal{N}\mathcal{K}}(i\omega)=\sum_{\bf k}G_{\bf
k}^{\mathcal{K}\mathcal{N}}(i\omega)=0$ which reflects the non-ambiguity of
the local nature of a given site, we can check that the sum rule is satisfied
between the ARPES signal and the local density of states:
$\rho(\omega)=\frac{1}{N}\sum_{\bf k}{\cal A}({\bf k},\omega)$.

In a paramagnetic phase, the propagator involved in the DMFT/CPA self-
consistency becomes independent from sublattice and spin indices:
$\mathbf{\Pi}_{\alpha}^{\sigma}(i\omega)\equiv\mathbf{\Pi}(i\omega)$. The
Green function matrix can then be cast into the same form as in ( [39]):

$\displaystyle G_{\bf kk}(i\omega)=\left(\mathbf{\Pi}^{-1}(i\omega)-{\bf
E}_{\bf k}\right)^{-1}~{}.$ (15)

With this effective description, the ARPES spectra are related with the
dispersion relation $det\left(\mathbf{\Pi}^{-1}(\omega)-{\bf E}_{\bf
k}\right)=0$, which may lead to the possible emergence of two branches (in
momentum) since $\mathbf{\Pi}$ and ${\bf E}_{\bf k}$ are $2\times 2$ matrices.
Such a two branches ARPES signature is, for example, obtained in the mean-
field description of a periodic Kondo lattice ($x=1$) in terms of effective
hybridization between conduction electrons and localized electronic
$f-$levels. However the dispersion may also be single branch, like, for
example, in the non-interacting case where $J_{K}=0$ as well as in the dilute
limit $x\to 0$. The DMFT/CPA method applied here for the KAM provides a unique
framework for describing both this single and double branch scenarii.

In order to distinguish the transitions and crossovers between dense and
dilute Kondo alloys, a useful quantity will be the effective mass that could
be related with ARPES signal. First, we introduce the Kondo local self-energy
as:

$\displaystyle\Sigma_{K}(i\omega)\equiv\frac{1}{G_{\mathcal{N}}(i\omega)}-\frac{1}{G_{\mathcal{K}}(i\omega)}~{}.$
(16)

Invoking the Eqs. (10-12) and with some straightforward algebra, the local
propagator matrix involved in the Eq. 15 can be expressed explicitly in terms
of $\Delta(i\omega)$ and $\Sigma_{K}(i\omega)$ as:

$\displaystyle\mathbf{\Pi}^{-1}(i\omega)=\begin{pmatrix}\frac{i\omega+\mu-\Sigma_{K}(i\omega)-(1-x)\Delta(i\omega)}{x}&\Delta(i\omega)\\\
~{}~{}&~{}~{}\\\
\Delta(i\omega)&\frac{i\omega+\mu-x\Delta(i\omega)}{1-x}\end{pmatrix}~{},$
(17)

Using this expression together with Eqs. (14-15), the one-body disorder-
averaged Green-function involved in ARPES measurements can be expressed as:

$\displaystyle G_{Alloy}({\bf
k},i\omega)=\frac{1}{i\omega+\mu-\Sigma_{Alloy}(i\omega)-\epsilon_{\mathbf{k}}}~{},$
(18)

where the effective self-energy is defined as:

$\Sigma_{Alloy}(i\omega)=\frac{x}{\frac{1}{\Sigma_{K}(i\omega)}-\frac{(1-x)}{i\omega+\mu-\Delta(i\omega)}}~{}.$
(19)

We can easily verify that this expression reproduces standard results at
several limits. At extreme dilute limit ($x\to 0$), the effective self-energy
$\Sigma_{Alloy}$ vanishes linearly with $x$ and the non-interacting electronic
Green function is recovered $G_{Alloy}({\bf k},i\omega)\to
1/(i\omega+\mu-\epsilon_{\mathbf{k}})$, with its single branch signature. In
the Kondo lattice limit $x\to 1$, $\Sigma_{Alloy}\to\Sigma_{K}$ and a two-
branches signature can be obtained within a Kondo mean-field approximation
that introduces a pole singularity for $\Sigma_{K}(i\omega)$. We may estimate
from expression (19) that $\Sigma_{Alloy}\approx x\Sigma_{K}$ when $x\approx
1$. This linearity with Kondo spin concentration is approximately valid only
at high and low concentrations. This suggests that the two-branches signatures
of the Kondo lattice coherent state should be preserved upon slight Kondo
dilution. Similarly, the single branch signature should also be maintained in
the very diluted limit. However, this linearity is not valid anymore at
intermediate concentrations and the breakdown of coherence that was predicted
elsewhere around $x=n_{c}$ [42] might lead to ARPES signatures. In order to
address this issue more quantitatively, we introduce the effective mass
$m^{*}$ as:

$\frac{m^{*}}{m_{0}}=1-\left.{\frac{d{\cal
R}e(\Sigma_{Alloy}(\omega))}{d\omega}}\right|_{\omega=0}~{},$ (20)

where $m_{0}$ is the mass of electron without interaction.

###  2.3 Impurity solver used for DMFT/CPA: mean-field approximations

One key step in solving the DMFT/CPA self-consistent equations as described in
section 2.1 is the resolution of the many-body problem defined by the local
effective action on a Kondo site, Eq. (4). Here, we employ mean-field
approximations as local impurity solvers for describing non-Kondo purely
magnetically ordered phases and pure Kondo paramagnetic phases. Indeed, the
aim is to investigate the effect of Kondo alloying with two complementary
perspectives: i) Describing the Fermi-surface folding of antiferromagnetic
Néel ordered phases, ii) Describing the possible connection between the one-
branch versus two-branches ARPES signatures of the Kondo alloys. We thus use
the Weiss mean-field approximation for addressing point i (see A.1), and we
use the Kondo mean-field approximation for addressing point ii (see A.2) since
it is able to describe the one-branch and two-branches ARPES in the dilute
limit ($x=0$) and for the coherent Kondo lattice ($x=1$) respectively. The
competition between these various states as predicted by Doniach [5] and the
resulting ground state phase diagram of the KAM was analyzed in details in
[45] as function of $x$, $n_{c}$, and $J_{K}$. Here, we want to focus onto the
ARPES and PES signatures of these different phases that may emerge from the
KAM [45].
In our mean-field approach, the Kondo crossover at Kondo temperature $T_{K}$
is replaced by a transition where $f$ and $c$-electrons locally hybridize. The
corresponding mean field equation for $T_{K}$ is

$\frac{2}{J_{K}}=\int_{-\infty}^{+\infty}\frac{\rho_{0}(\omega+\mu)}{\omega}\tanh(\frac{\omega}{2T_{K}})d\omega$
(21)

where $\rho_{0}(\omega)$ is non-interacting density of states. It is
worthwhile noting here that $T_{K}$ is independent on $x$, as discussed
in[39]. In the following, the strength of the Kondo interaction will be
adjusted by comparing $T_{K}$ with the non-interacting electronic bandwidth
$W$.

##  3 Results

In this section, we present and discuss our numerical results focusing on
photo-emission signatures obtained for the KAM on a square lattice. In order
to confirm the validity of our results, we also checked if some effects might
be related to the specific square lattice-structure. For that, we also
performed DMFT self-consistency over Bethe lattice (BL). No qualitative
significant differences were found between Bethe and square lattices regarding
local self-energies in the paramagnetic phases.
For both lattices, self-consistent solutions for the dynamical bath from the
DMFT equations (10, 11) and the local mean-field approximations ( 22, 23, 24
or 25, 26, 27) were obtained numerically using standard optimization methods
that we adapted for this specific problem. In this article, we considered pure
paramagnetic Kondo, ferromagnetic (F) and staggered antiferromagnetic (AF)
phases and the results below are presented for 2D square lattice with
electronic bandwidth $W\equiv 8t$.

###  3.1 ARPES and electronic density of states

Figure 1: First Brilloin zone of the square lattice, with indications of the
points $\Gamma=(0,0)$, $X=(0,\pi)$, and $M=(\pi,\pi)$.

In this section, we analyze the ARPES signals obtained through the spectral
function ${\cal A}({\bf k},\omega)$ and the local electronic density of states
$\rho(\omega)=\frac{1}{N}\sum_{\bf k}{\cal A}({\bf k},\omega)$ with a focus on
the specific signatures of the transitions and crossovers in the paramagnetic
Kondo phases. In particular we address the issues of one-branch versus multi-
branches dispersions in ARPES along with gap-less, pseudogapped, or gapped
local electronic density of states.

We solved the self-consistent equations for various electronic fillings and we
obtained similar qualitative results in all cases. Therefore here, we choose
$n_{c}=0.70$ to present an overview of different situations depending on $x$
and $T_{K}$. Overall we identify two different scenarii depending on the
strength of the Kondo interaction: the relatively high $T_{K}$ regime (see
figure 2) and the relatively small $T_{K}$ regime (see figure 3), typically
separated by a threshold corresponding to $T_{K}$ around $W/10$.

Figure 2:  ARPES spectra for $n_{c}$ = 0.70 at relatively strong coupling
$T_{K}/W$ = 0.169 for Kondo impurity concentrations $x=0.01$, $0.30$, $0.70$,
and $1.00$. The wavevector ${\bf k}$ axis corresponds to the high symmetry
lines $\Gamma-X-M-\Gamma$ in the square lattice first Brillouin zone (see
figure 1). The corresponding electronic density of states $\rho(\omega)$ is
plotted on the right side.  Figure 3:  ARPES spectra for $n_{c}$ = 0.70 at
relatively small coupling $T_{K}/W$ = 0.019 for Kondo impurity concentrations
$x=0.01$, $0.08$, $0.50$, and $1.00$ with $x^{\star}=0.08$. The wavevector
${\bf k}$ axis corresponds to the high symmetry lines $\Gamma-X-M-\Gamma$ in
the square lattice first Brillouin zone (see figure 1). The corresponding
electronic density of states $\rho(\omega)$ is plotted on the right side.

In both cases, the dilute Kondo limit ($x\ll 1$) and the dense Kondo limit
($x=1$) generate usual results. The dilute Kondo limit ($x\ll 1$) reproduces
the non-interacting electronic structure, which is characterized here by a
well defined single branch electronic dispersion and a density of states with
a van-Hove singularity usual for the square lattice. The Kondo lattice limit
($x=1$) also presents universal signatures with two branches resulting from
the effective hybridization between conduction electrons and the local levels
associated with Kondo spins. The resulting density of states also presents a
gap. For intermediate concentrations $x$, the situation depends on the
strength of the Kondo interaction.

####  3.1.1 Evidence for a Lifshitz-like transition in the Kondo phase at
large $T_{K}$

The numerical results obtained for various Kondo impurity concentrations at
relatively large Kondo coupling $T_{K}/W=0.169$ are presented in the figure 2.
In the intermediate concentration regime, as long as $x>n_{c}$, the dilution
of Kondo impurities does not close the hybridization gap characterizing the
coherent dense regime. Since we have fixed $n_{c}<1$ in the dense case, the
Fermi level is inside the lower band. When decreasing $x$, a third band starts
to be formed inside the gap and a transition from dense to dilute Kondo
regimes occurs at $x=n_{c}$. This transition is marked by the shift of the
Fermi level from the lower band to the third band. Upon further dilution of
Kondo impurities in the regime $x<n_{c}$, the hybridization gap is filled
giving rise to three branches structure separated by two pseudogaps.

Let us consider at first $x>n_{c}$ regime in order to understand the formation
of the third band and the breakdown of the coherence of the Kondo impurities.
In this dense regime and at strong Kondo coupling, the KAM can be mapped onto
an effective Hubbard model where quasiparticles are the unscreened Kondo
impurities [66, 25, 67, 68]. The corresponding “Coulomb repulsion” is in this
case of the order $T_{K}$ which is very large compared to $W$. Apart from
half-filling which corresponds to a Kondo insulator for $x=1$, the system is a
strongly correlated metal. Considering particle-hole general symmetry, let us
depict a situation where the effective Fermi level is inside the lower band.
The gapped local d.o.s characterizing Kondo lattices at large $T_{K}$ reflects
the two Hubbard bands separated by an energy of the order of $T_{K}$ and the
states in the upper Hubbard band correspond to singlet-triplet excitations.
Dilution of Kondo atoms in the dense regime ($n_{c}<x<1$) changes the number
of carriers and the Fermi level gets closer to the upper edge of the effective
Hubbard band. The transition realized at $x=n_{c}$ for strong Kondo
interaction may thus be analogous to a doping-induced Mott transition [69],
which also presents the formation of a quasiparticule peak inside the Hubbard
gap. However, the effective model is different for $x<n_{c}$: in this dilute
regime and for strong Kondo interaction, quasiparticles emerge from the
supernumerary conduction electrons which do not form Kondo singlets. The third
central band may be associated with the motion of these conduction electrons
on the non-Kondo sites. Further, the states in the fully occupied lower band
represent the electrons forming singlets on Kondo sites whereas the upper
unoccupied band corresponds to excitations of a second electron on a Kondo
site.

####  3.1.2 Evidence for a new critical concentration at low $T_{K}$

We now focus on the ARPES signal obtained for the relatively small Kondo
coupling case. The figure 3 illustrates the results obtained for
$T_{K}/W=0.019$. We observe that the two branches structure characterizing the
dense Kondo state is preserved upon dilution even for $x<n_{c}$. Furthermore,
the effect of disorder-related decoherence is maximum at around $x=n_{c}$
which leads to a broadening of the branches as well as a reduction of the
quasiparticle lifetime. This maximum decoherence also results in a partial
filling of the hybridization gap leaving a pseudogap near the Fermi level.
Upon further dilution of Kondo atoms, the two ARPES branches merge to form a
single branch structure along with the disappearance of the pseudogap. This
occurs at a critical concentration $x^{\star}$ which depends on the strength
of the Kondo interaction. We find that $x^{\star}\ll n_{c}$ at very small
$T_{K}$, and $x^{\star}\to n_{c}$ when $T_{K}$ approaches around $W/10$.

###  3.2 Fermi surfaces

In this section we analyze the experimental signatures in the Fermi surfaces
obtained from the ARPES signal ${\cal A}({\bf k},i\omega=i0^{+})$. Figure 4
depicts the Fermi surface spectra computed for a paramagnetic Kondo ground
state within broad ranges of $x$ and $n_{c}$. For all values considered for
the Kondo interaction and the electronic filling, we find that the Fermi
surface of the Kondo lattice ($x=1$) is large and it includes the
contributions from both the conduction electrons and the Kondo spins. This
universal feature is in good agreement with previous theoretical and
experimental results [52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62]. It can be
well understood in terms of Luttinger theorem, which stipulates that all
fermionic degrees of freedom participate in the formation of the Fermi liquid
ground state.

Figure 4: FS spectra assuming a Kondo paramagnetic ground state for $x=0.01$,
$0.20$, $0.30$, $0.40$, $0.50$, $0.60$, $0.70$, $0.80$, $0.90$, $1.00$ (from
left to right) and $n_{c}=0.20$, $0.30$, $0.40$, $0.50$, $0.60$, $0.70$,
$0.80$, $0.90$, $1.00$ (from bottom to top). Each square corresponds to the
First Brillouin zone of the square lattice (see figure 1). From top to bottom,
$T_{K}/W=0.175$, $0.082$, and $0.0058$: the Lifshitz-like transition around
$x=n_{c}$ is observed for a sufficiently strong Kondo interaction, and it
becomes a gradual crossover for smaller values of the interaction.

####  3.2.1 Fermi surface in the Kondo phase: coherence breakdown

We now focus on the possible breakdown of this coherent Kondo lattice state
when decreasing $x$. For a relatively strong Kondo interaction
($T_{K}/W=0.175$) we observe the Lifshitz-like transition at $x=n_{c}$ that
was predicted in [42]. In this case, the ARPES signal corresponds to
relatively long lifetime quasiparticles with a well defined Fermi surface
(excepted at the transition). The volume of the Fermi-surface shrinks when
increasing $x$ in the dilute regime $x<n_{c}$, and it increases with $x$ in
the dense regime $x>n_{c}$. This feature is consistent with the fact that
Kondo impurities behave as hole-dopant for $x<n_{c}$ and particle-dopant for
$x>n_{c}$. For $T_{K}/W=0.082$, and $0.0058$, the evolution of the Fermi
surface ARPES spectra with concentration $x$ is more gradual, and the
Lifshitz-like transition around $x=n_{c}$ seems to become a crossover at lower
values of $T_{K}$. Furthermore, the broadening of the Fermi surface spectra
around the Fermi wavevectors is maximal around $x=n_{c}$ due to disorder-
related decoherence.

Fermi surface spectra provide apparent signatures of the Kondo lattice
coherence breakdown transition at $x=n_{c}$, especially for systems with
relatively strong Kondo interaction, corresponding to $T_{K}$ typically larger
than $W/10$. This Lifshitz like transition separating dense and dilute Kondo
phases at $x=n_{c}$ becomes a crossover for smaller $T_{K}$. At small $T_{K}$,
we identified another critical concentration from the ARPES signals (see
section 3.1) at $x^{\star}$ characterizing the merging of two ARPES branches.
However, the Fermi surfaces at figure 4 do not present clear signatures of any
characterized feature at or around this critical concentration $x^{\star}$.
This is not surprising since the electronic excitation spectrum is not
accessible from the Fermi surfaces.

Figure 5: Fermi surface spectra for $n_{c}=0.30$ and $x=0.05$, $0.35$, $0.65$
and $0.95$. (a) $T_{K}/W=0.012$ with Kondo ground states: we observe
signatures of a breakdown of coherence associated with a change of topology in
the the Fermi-surface. (b) $T_{K}/W=0.0031$ with ferromagnetic ground states:
we observe Zeeman splitting effect only.  Figure 6: Fermi surface spectra for
$n_{c}=0.90$, and $x=0.05$, $0.35$, $0.65$ and $0.95$. (a) $T_{K}/W=0.0143$
with Kondo ground states: we observe signatures of a breakdown of coherence
associated with a change of topology in the the Fermi-surface. (b)
$T_{K}/W=0.0045$ with antiferromagnetic ground states: we observe only the
folding of the Fermi-surface which results from the staggered Néel ordering.

####  3.2.2 Fermi surface in the magnetically ordered phases

We also studied the Fermi surfaces in the magnetically ordered phases of Kondo
alloys, as depicted in figure 5 (for $n_{c}=0.30$, where the small Kondo
coupling ground state is ferromagnetic) and in figure 6 (for $n_{c}=0.90$,
where the small Kondo coupling ground state is antiferromagnetic). The Zeeman
splitting is obtained and we are also able to reproduce the folding of the
Brillouin zone in the Néel antiferromagnetic state. For both magnetically
ordered states, we find that the evolution of the Fermi surface is either
absent or very smooth and gradual upon varying $x$. This result is in contrast
with the coherence breakdown which is predicted in the Kondo phases. A
possible explanation for this difference may be obtained by considering the
Kondo lattice limit ($x=1$). In this case, Kondo spins contribute to the
formation of a large Fermi surface for the Kondo coherent state, while they do
not contribute for the magnetically ordered states. The breakdown of coherence
which is depicted here in Fermi surfaces spectra for the Kondo phase is thus
related with the contribution of Kondo ions to forming strongly correlated
fermionic quasiparticles. This breakdown of coherence is different from the
breakdown of Kondo effect that distinguishes Kondo phases from pure
magnetically ordered phases. Indeed, Kondo effect is still present in the non-
coherent dilute Kondo phase. In our calculations, we did not consider the
possibility of mixed states where magnetic order might coexists with Kondo
effect. For such states, we expect coherence breakdown signatures that might
be concomitant with Fermi surface reconstructions resulting from magnetic
order.

###  3.3 Phase diagram: emergence of two transitions in the paramagnetic Kondo
phases

The phase diagram of the KAM is depicted in figure 7. It was obtained by
comparing the energies of each phase considered: paramagnetic Kondo,
antiferromagnetic and ferromagnetic phases. From theses phase diagrams, we
observe that in the region with small values of Kondo coupling, the long range
magnetically ordered phases are stabilized. Their competitions and their
stabilities for the KAM, which is consistent with Doniach argument, was
discussed elsewhere [45]. For small to intermediate values of $T_{K}$, a
paramagnetic Kondo phase with three distinct zones are identified: a dilute
Kondo at $x\ll 1$, a dense Kondo at $x\approx 1$ separated by a large zone of
intermediate state $x^{\star}<x<n_{c}$. When increasing the Kondo coupling,
$x^{\star}$ tends towards $n_{c}$ and for strong coupling only dilute and
dense Kondo phases are obtained, separated by a Lifshitz transition.

Figure 7:  Ground state phase diagram of the KAM as functions of $x$ and
$T_{K}/W$ for (a) $n_{c}=0.30$, (b) $n_{c}=0.70$ and (c) $n_{c}=0.90$. In the
Kondo phases the solid line indicates the discontinuity of the self-energy
observed at $x=n_{c}$ for sufficiently strong $T_{K}$. This transition from
the dense coherent Kondo phase becomes a crossover marked by an inflection in
the self energy at smaller $T_{K}$ (doted line), and a significant increase in
the intensity of imaginary part of the self-energy (color or black and white
gradient). A continuous vanishing of the effective mass $m^{\star}$ is
obtained at concentration $x^{\star}$ (dashed line), and we find $m^{\star}<0$
in the intermediate region $x^{\star}<x<n_{c}$. We also solved the DMFT
equations obtained for a Bethe lattice, considering the Kondo paramagnetic
solution only and the same values of model parameters as depicted here. We
were not able to distinguish the figures corresponding to the Bethe lattice
from the ones depicted here for the 2D square lattice. This strong similarity
excludes several interpretations that might invoke specificities of the
lattice structure.

Now, we analyze the different regions of paramagnetic Kondo phases by the
means of effective self-energy $\Sigma_{Alloy}(\omega)$ and its corresponding
effective mass $m^{\star}$ with a focus on ARPES. The transition observed in
the Fermi surfaces (see section 3.2) at $x=n_{c}$ for sufficiently large
values of Kondo coupling is characterized by a discontinuity of
$\Sigma_{Alloy}(\omega=0)$ (solid line in figure 7). Since the real part of
the self-energy is related with a rescaling of the Fermi level, we interpret
this transition as a signature of the Lifshitz transition that was predicted
elsewhere [42] from a strong coupling approach of the Kondo alloy: for
$x>n_{c}$, all magnetic degrees of freedom from conduction electrons are
frozen by Kondo singlets formation. The Fermi liquid quasiparticles in this
coherent dense regime are formed by the remaining degrees of freedom from
unscreened Kondo impurities. In the dilute regime $x<n_{c}$, the microscopic
nature of quasiparticles is different and emerges from unscreened conduction
electrons. Our result shows that this strong coupling picture may be realized
for $T_{K}\gtrsim\frac{W}{10}$. Therefore, in order to observe a signature of
coherence breakdown at $x=n_{c}$ from ARPES experiments in Kondo alloys, one
would need to consider $f-$electron compounds with relatively large Kondo
temperatures. In this case a valence fluctuation or valence transition might
also become relevant as well but this issue is beyond the scope of the present
work.

Hereafter, we analyze different situations with smaller Kondo coupling. The
transition predicted at $x=n_{c}$ separating dense and dilute Kondo regimes
becomes a crossover at smaller coupling and the quasiparticle lifetime is
significantly shorten due to disorder incoherence effects. Indeed, the self-
energy is found to be continuous and characterized by an inflection around
this crossover $x=n_{c}$ (doted line in figure 7, see also figure 8).

Figure 8:  (a) Real part of the self-energy $\Sigma^{\prime}_{Alloy}(0)/W$,
(b) Effective mass $m^{\star}/m_{0}$ as functions of $x$, for $n_{c}=0.70$.
Different Kondo temperatures have been used for the numerics, illustrating the
transition (at strong $T_{K}$) and crossover (at smaller $T_{K}$) obtained
around $x=n_{c}$. The inset in (b) is a focus around the critical
concentration $x^{\star}$ which is characterized by a vanishing of $m^{\star}$
when $T_{K}$ relatively small. At intermediate concentrations
$x^{\star}<x<n_{c}$ we find $m^{\star}<0$.

Photo-emission signatures of coherence breakdown in Kondo alloys: DMFT
approach Whereas, the imaginary part, $\Sigma^{\prime\prime}_{Alloy}(0)$,
which is relatively small at strong coupling, becomes large around this
crossover at smaller coupling (see color/black-white gradient in figure 7).

In agreement with the ARPES spectral analyse of section 3.1 for intermediate
values of Kondo coupling, we found a third intermediate phase, which is
separated from the dilute phase by a transition line in the $x-T_{K}$ phase
diagram: at the critical concentration $x^{\star}$, the effective mass
$m^{\star}$ vanishes continuously. This transition separates the very dilute
regime ($x<x^{\star}$) with $m^{\star}>0$ from an intermediate regime
($x^{\star}<x<n_{c}$) characterized by $m^{\star}<0$ (see inset of figure 8
and dashed line in figure 7). From our numerical datas, we found a powerlaw
relation $T_{K}/W\propto(x^{\star})^{\gamma}$ with an exponent $\gamma<1$ and
the exponent itself depends on the filling $n_{c}$. We found $\gamma=$ 0.71,
0.78 and 0.83 for $n_{c}=$ 0.30, 0.70 and 0.90 respectively. This transition
produces signatures in ARPES signal (see section 3.1) where two branches merge
to one branch at $x^{\star}$ (see figure 3). Meanwhile, Fermi surface
structures (see figure 4) do not show a clear signature of $x^{\star}$.
Keeping in mind that the dispersion relation is
$\omega+\mu-\Sigma_{Alloy}(\omega)=\epsilon_{\bf k}$, the frequency dependence
of the real part
$\omega-\Sigma^{\prime}_{Alloy}(\omega)+\Sigma^{\prime}_{Alloy}(0)$ is
represented in figure 9. We can thus interpret the transition at $x=x^{\star}$
as the gradual formation (or extinction) of extra branches in the one-electron
excitations, as suggested by the low frequency dependence of this quantity,
which changes from locally monotonous (at $x<x^{\star}$) to locally non-
monotonous (at $x>x^{\star}$). The second branch could then be associated with
coherent and dispersive singlet-triplet excitations that may propagate. Such
triplet excitations are gaped out at strong Kondo coupling but they might also
reveal a pseudogap in PES at smaller Kondo coupling as analyzed in the section
3.1. This intermediate state is thus precursor to the coherent state which is
realized around $x\approx 1$. It is very interesting to see that such a pre-
coherent state may start being formed at a relatively small concentration
$x^{\star}$.

Two very important features could make the transition at $x^{\star}$
observable experimentally: first, the imaginary part of
$\Sigma^{\prime\prime}_{Alloy}(0)$ remains relatively small around $x^{\star}$
(see color/black-white gradient in figure 7). We can thus expect that the
excitations are long lifetime quasiparticles which could be revealed by photo-
emission. Secondly, $x^{\star}$ depends on the strength of the Kondo
interaction. We may thus expect that this transition could be tuned by
applying pressure on a compound with fixed concentration $x$. Of course, we
are aware that mechanical pressure is not fully compatible with ARPES
experiments. However, since the underlying phenomenon is a transition in the
one-electron excitation spectrum, we may expect signatures in other sorts of
experiments that could be realized under pressure, e.g., Raman spectroscopy.

Figure 9: Frequency dependence of the real part of the self-energy,
$\omega-\Sigma^{\prime}_{Alloy}(\omega)+\Sigma^{\prime}_{Alloy}(0$) for
$n_{c}=0.70$ and $T_{K}=0.019$. Top: for $x$ in the vicinity of the critical
point $x^{\star}=0.08$ which is characterized by $m^{\star}=0$, we observe the
emmergence of a non-monotonicity at low energy. This leads to the gradual
formation of a multiple-branches dispersion for $x>x^{\star}$. Center: for $x$
in the vicinity of $n_{c}$, the maximum is realized at $\omega<0$ for
$x<n_{c}$ and at $\omega>0$ for $x>n_{c}$ resulting in a second change of sign
of $m^{\star}$. Bottom: for $x$ close to $1$, we observe signatures of the
singularity $\Sigma_{K}(\omega)=\frac{r^{2}}{\omega+\lambda}$ which is
obtained in the mean-field approximation for the Kondo lattice. The non-
monotonicity obtained at lower concentrations is reminiscent of this
singularity and we expect this feature to survive qualitatively beyond the
mean-field approximation.

##  4 Conclusion

In this theoretical paper we have investigated photo-emission spectra of
disordered Kondo alloys. For this purpose, we have adapted the DMFT/CPA
formalism that was developed elsewhere [65, 39] to include the possibility of
various ground states of the KAM: paramagnetic Kondo, antiferromagnetic and
ferromagnetic phases. Using static mean-field theories as local impurity
solvers for DMFT, we investigated a large range of values of electronic
filling $n_{c}$, impurity concentrations $x$, and strength of Kondo coupling
$T_{K}$. We studied both a square lattice and a Bethe lattice structure and we
obtained consistent results when comparison was possible. For the sake of
clarity, the ARPES signals are analyzed only for the square lattice, but we
may expect that they can be generalized to other lattice structures.

Our results confirm the existence of a transition at $x=n_{c}$ between a
coherent dense Kondo regime (for $x>n_{c}$) and a dilute Kondo regime (for
$x<n_{c}$) [42]. This Lifshitz-like transition may be observed in ARPES
experiments through the analyse of Fermi surface for Kondo alloys with a Kondo
temperature higher than about $1/10$ bandwidth. A shrinking of the Fermi-
surface is expected when increasing $x$ in the dilute regime, while the Fermi
surface is enlarge with $x$, in connection with Luttinger theorem, in the
coherent dense Kondo regime. It could be obtained experimentally in the
materials with a relatively large $T_{K}$. We have also shown that this
transition at $x=n_{c}$ becomes a crossover at smaller values of Kondo
interaction.

While Doniach argument is recovered, with magnetically ordered ground states
stabilized at very small Kondo coupling, we identified a broad region of
parameters where an intermediate paramagnetic Kondo state can be stabilized.
This corresponds to values of $T_{K}$ between $W/100$ and $W/10$ with the
regime of concentrations $x^{\star}<x<n_{c}$ which can be realized in a large
variety of heavy-fermion Kondo systems. Unlike the dilute and the dense Kondo
states, this intermediate phase is characterized by a negative effective mass.
Here, negative effective mass does not mean instability, but it corresponds to
the emergence of an extra branch in the electronic dispersion and a formation
of a pseudogap in the local density of states at $x>x^{\star}$.

We also analyze the effects of decoherence resulting from disorder, which may
spoil possibilities of analyzing ARPES signals. We find that these effects are
relatively small at strong Kondo coupling, which may make possible the
observability of the transition at $x=n_{c}$. However this requires compounds
with a relatively large value of $T_{K}$. At smaller values of Kondo
interaction, disorder-induced decoherence may become significant, with a
maximal effect around $x=n_{c}$. This disorder-induced decoherence reduces the
opportunities of observing signatures in Fermi surface at $x=n_{c}$ for small
coupling. We still expect relatively well defined quasiparticles around
$x=x^{\star}$ even through contrary to the transition at $x=n_{c}$, the
transition at $x=x^{\star}$ cannot be observed directly from the Fermi surface
structures. Indeed, the experiments involving low energy excitations (e.g.
ARPES) are more appropriate. This opens rich perspectives for experimental
investigations of the breakdown of coherence in Kondo alloys. Since we predict
that $x^{\star}$ varies with the strength of the Kondo interaction, this
transition might be realized not only by atomic substitution but also by
applying pressure in a Kondo alloy with fixed stoichiometry (i.e. fixed $x$
and $n_{c}$). On a fundamental point of view, this work also suggests the
possible emergence of exceptional points [70, 71] at $x^{\star}$ where two
dispersives branches merge and $m^{\star}$ vanishes. The physical excitations
of the quasiparticles should be investigated at $x=x^{\star}$ theoretically
and experimentally by direct or indirect probes.

We acknowledge useful discussions with Andrés Santander-Syro, Christoph
Geibel, Clemens Laubschat, Cornelius Krellner, Denis Vyalikh, Ilya Sheikin,
Jochen Wosnitza, Jérémy Sourd, Marcelo Rozenberg, Marie-Aude Measson, Sophie
Tencé and Véronique Brouet. This work has been supported by Agence National de
Recherche (ANR) and Deutsche Forschungsgemeinschaft (DFG) Grant Fermi-NESt
(ANR-16-CE92-0018).

## Appendix

##  Appendix A Mean-field approximations

###  A.1 Weiss mean-field approximation for ferromagnetic (F) and
antiferromagnetic (AF) phases

Here we describe the mean-field impurity solver that we used for solving the
local effective action on a Kondo site, which is given by Eq. (4), assuming a
magnetically ordered state. We restrict our study to pure ordered phases, and
we ignore the possibility of mixed phases where Kondo effect and magnetic
order might coexist. Furthermore, we assume that the magnetic order is either
ferromagnetic (F) or staggered antiferromagnetic (AF). Invoking the
$A\leftrightarrow B$ symmetries of these phases, the single site effective
DMFT/CPA action can be solved assuming this site belongs to the sublattice
$A$. Consequently, hereafter, in this section we consider only $\alpha=$A and
we omit the $\alpha$ index.

The Kondo interaction is decoupled using the standard Weiss mean-field
approximation as $\mathbf{S}\cdot\mathbf{s}\approx\langle S^{z}\rangle
s^{z}+S^{z}\langle s^{z}\rangle-\langle S^{z}\rangle\langle s^{z}\rangle$. The
order parameters are the local magnetizations on a Kondo site belonging to the
sublattice $A$: $m_{f}\equiv\langle S^{z}\rangle$, and $m_{c}\equiv\langle
s^{z}\rangle=\frac{1}{2}\sum_{\sigma}\sigma_{z}\langle
c_{\sigma}^{\dagger}c_{\sigma}\rangle$. The local effective action (4) becomes
quadratic and the local Green function on a $\mathcal{K}$-site can be
expressed explicitely in terms of the bath and order parameters as
$G_{\mathcal{K}\sigma}(i\omega)=\frac{1}{i\omega+\mu-\Delta_{\sigma}(i\omega)-\sigma
J_{K}\frac{m_{f}}{2}}$.

Finally, to complete the general DMFT/CPA self-consistent equations described
in section 2.1 which relate the local Green functions and the dynamical bath,
the chemical potential $\mu$ and the order parameters $m_{f}$ and $m_{c}$ are
determined by solving the following self-consistent equations:

$\displaystyle n_{c}$ $\displaystyle=$ $\displaystyle
x\frac{1}{\beta}\sum_{i\omega,\sigma}G_{\mathcal{K}\sigma}(i\omega)+(1-x)\frac{1}{\beta}\sum_{i\omega,\sigma}G_{\mathcal{N}\sigma}(i\omega)~{},$
(22) $\displaystyle m_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{2\beta}\sum_{i\omega,\sigma}\sigma
G_{\mathcal{K}\sigma}(i\omega)~{},$ (23) $\displaystyle m_{f}$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\tanh(\frac{\beta
m_{c}J_{K}}{2})~{}.$ (24)

###  A.2 “Slave boson”, “large-n”, mean-field approximation for the Kondo (K)
phase

Here we describe the mean-field impurity solver that we used for solving the
local effective action on a Kondo site, which is given by Eq. (4), assuming a
paramagnetic Kondo correlated state. We neglect the possibility that magnetic
order might co-exist with Kondo local strong correlations. Consequently,
hereafter, in this section we omit the $\alpha$ sublattice index. We follow
the standard mean-field approximation as introduced by Lacroix and Cyrot in (
[20]), which is analogous to the “large N” expansion or slave boson
approximation developed by Coleman in ( [72]) and Read and Newns in ( [73]).

At first, we represent Kondo spin operator within Abrikosov’s fermionic
representation
$\mathbf{S}^{\sigma\sigma^{\prime}}=f_{\sigma}^{\dagger}f_{\sigma^{\prime}}-\delta_{\sigma\sigma^{\prime}}/2$.
Thereafter the Kondo interaction in Eq. (4) is mapped as ${\bf S}\cdot{\bf
s}\to\frac{1}{2}\sum_{\sigma\sigma^{\prime}}c_{\mathcal{K}\sigma}^{\dagger}c_{\mathcal{K}\sigma^{\prime}}f_{\sigma^{\prime}}^{\dagger}f_{\sigma}$.
Finally, the Kondo interaction is approximated invoking the mean-field
decoupling
$c_{\mathcal{K}\sigma}^{\dagger}c_{\mathcal{K}\sigma^{\prime}}f_{\sigma^{\prime}}^{\dagger}f_{\sigma}\approx\langle
c_{\mathcal{K}\sigma}^{\dagger}f_{\sigma}\rangle
f_{\sigma^{\prime}}^{\dagger}c_{\mathcal{K}\sigma^{\prime}}+\langle
f_{\sigma^{\prime}}^{\dagger}c_{\mathcal{K}\sigma^{\prime}}\rangle
c_{\mathcal{K}\sigma}^{\dagger}f_{\sigma}-\langle
f_{\sigma^{\prime}}^{\dagger}c_{\mathcal{K}\sigma^{\prime}}\rangle\langle
c_{\mathcal{K}\sigma}^{\dagger}f_{\sigma}\rangle$. This mean-field description
results in an emergent effective hybridization between the conduction
electrons and the Abrikosov fermions, $r=\frac{J_{K}}{2}\sum_{\sigma}\langle
c^{\dagger}_{\mathcal{K}\sigma}f_{\sigma}\rangle$, which can be identified as
an order parameter for the Kondo phase. An additional constraint
$\sum_{\sigma}f_{\sigma}^{\dagger}f_{\sigma}=1$ restricts the number of
Abrikosov fermions to one, which are imposed by introducing a Lagrange
parameter $\lambda(\tau)$. With the mean-field approximation, $\lambda$
assumed to be constant and determined self-consistently in order to satisfy
the $f$ occupancy constraint on average.

Since here we are considering paramagnetic Kondo phase, for the sake of
clarity, hereafter we skip the spin index $\sigma$. The local effective action
(4) becomes quadratic and the local Green function on a $\mathcal{K}$-site can
be expressed explicitly in terms of the bath and order parameters as
$G_{\mathcal{K}}(i\omega)=\frac{1}{i\omega+\mu-\Delta_{\sigma}(i\omega)-\frac{r^{2}}{i\omega+\lambda}}$.
The Kondo self-energy involved in Eq. (16) thus has a pole singularity
$\Sigma_{K}(i\omega)=\frac{r^{2}}{i\omega+\lambda}$ which captures several
relevant aspects of Kondo physics, including the two-branches ARPES signature
of the Kondo lattice ($x=1$) coherent state.

Finally, to complete the general DMFT/CPA self-consistent equations described
in section 2.1 which relate the local Green functions and the dynamical bath,
the chemical potential $\mu$, the order parameter $r$, and the Lagrange
multiplier $\lambda$ are determined by solving the following self-consistent
equations:

$\displaystyle n_{c}$ $\displaystyle=$ $\displaystyle
x\frac{2}{\beta}\sum_{i\omega}G_{\mathcal{K}}(i\omega)+(1-x)\frac{2}{\beta}\sum_{i\omega}G_{\mathcal{N}}(i\omega)~{},$
(25) $\displaystyle\frac{r}{J_{K}}$ $\displaystyle=$
$\displaystyle-\frac{r}{\beta}\sum_{i\omega}\frac{G_{\mathcal{K}}(i\omega)}{i\omega+\lambda}~{},$
(26) $\displaystyle 1$ $\displaystyle=$
$\displaystyle\frac{2}{\beta}\sum_{i\omega}\frac{r^{2}G_{\mathcal{K}}(i\omega)}{(i\omega+\lambda)^{2}}~{}.$
(27)

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