Marie - Andrée

Evrard

Kevin

Niklas

Alexandros, Benjamin Lena

Cornelia

De

Antoine Alexandre

Audrey

Florent, Joséphine Camille Brice

Julie Kévin

Mady, Maëlis, Mikael Natacha Laurent

Pauline Nousk

Thomas Je

Claire-Lise Herdrich

Je

Maëlys Etienne Sophie

Benjamin Alexandros

Keywords: corrélés, matériaux quantiques, transition de phases quantiques, fermion lourd, théorie strongly correlated systems, quantum matter, quantum phase transitions, heavy-fermion, dynamical mean-field theory, rotationally invariant slave-bosons, disordered Kondo alloys, coherence breakdown, angle-resolved photoemission spectroscopy, Lifshitz transition, Fermi surface topology, duality, orbital-selective partial localization

Cohérence du réseau et local non-local dualité dans les matériaux à éléctrons f Résumé : Dans cette thèse, nous étudierons les systèmes d'électrons f sous deux aspects différents : d'une part la formation et la brisure de cohérence du réseau, et d'autre part la nature des électrons f qui peuvent être localisé, itinérant ou dual. Dans la première partie, nous aborderons le sujet de la cohérence du réseau dans les systèmes 4f sous l'angle de la substitution atomique des atomes magnétiques par des atomes non-magnétiques. Nous traiterons du désordre généré par la substitution par la théorie de champ moyen dynamique. Nous commençons par généraliser le diagramme de phase de type Doniach avec la substitution en considérant les phases : ferromagnétique, antiferromagnétique et paramagnétique Kondo. Nous étudions également la pertinence de nos diagrammes de phases vis-à-vis des données expérimentales des alliages à la base de cérium. Par la suite, nous nous concentrerons sur la phase paramagnétique Kondo avec un réseau carré afin d'étudier les signatures de la brisure de cohérence du réseau en diluant les impuretés magnétiques. Pour cela, on analyse les signaux de photoémission, les masses effectives, le local potential scattering, et l'ordre de charges. Nous confirmerons les précédentes prédictions de transition type Lifshitz entre les systèmes dilués et denses. De plus, nous détecterons une nouvelle concentration critique pour une instabilité au liquide de Fermi. Ce dernier est caractérisé par une annulation de la masse effective. La deuxième partie de cette thèse traite le caractère dual apparent des électrons 5f dans les fermions lourds à la base d'actinides, où les degrés de liberté 5f itinérants et localisés semblent coexister. Nous utiliserons la méthode des bosons esclaves invariants par rotation pour étudier l'influence de corrélations intra-atomiques, i. e., de type de règle de Hund. Nos résultats confirmerons la conjecture selon laquelle les corrélations intra-atomiques peuvent renforcer les anisotropies dans l'intégrale de saut effectif 5f -5f , et conduisent à une localisation partielle sélectif en orbitale. Enfin, nous analysons les différentes phases partiellement localisées en fonction de la masse de quasi-particules, l'occupation, d'aimantation et la configuration de valence dépendantes en orbitale.

Mots-clés : matériaux fortement

Foreword

The investigation of the electronic properties of a material is one of the key elements to understanding the state of matter. The electronic properties in a solid get defined either by the delocalization of the electrons by forming coherent Bloch states through the overlapping of their wave functions of neighboring sites or by its localization due to strong Coulomb repulsion. A usual way to study them is to use the band theory, which captures the physical properties defined by freely moving electrons in the periodic potential of a lattice. This picture of free electrons is supported by Landau's Fermi liquid theory [START_REF] Landau | The theory of a Fermi liquid[END_REF], which adiabatically connects low-energy excitations to the non-interacting one with one-to-one correspondence. These low-energy excitations are also called quasiparticles, and they are characterized by the renormalized mass with a finite lifetime. Even though this Fermi liquid theory correctly defines the various macroscopic properties like resistivity of varieties of metals, it tends to fail to capture the underlying physics of complex systems with strong electron-electron interactions [START_REF] Stewart | Non-Fermi-liquid behavior in 𝑑-and 𝑓 -electron metals[END_REF]. For these systems, one might do band-structure calculations using theories based on density functional theory (DFT) [START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF][START_REF] Kohn | Self-Consistent Equations Including Exchange and Correlation Effects[END_REF]. Again, DFT fails to capture the low-energy excitations when the electron-electron interactions become strong. This gave rise to alternative methods like dynamical mean-field theory (DMFT) [START_REF] Georges | Dynamical meanfield theory of strongly correlated fermion systems and the limit of infinite dimensions[END_REF][START_REF] Pavarini | From Infinite Dimensions to Real Materials tech[END_REF] to study the strongly correlated systems. The study of these strongly correlated systems is the core subject of this thesis.

Research in strongly correlated electronic systems presents a challenge on both experimental and theoretical sides. Experimental studies of these systems are not straightforward and might require extreme experimental conditions [START_REF] Avella | Strongly correlated systems: experimental techniques[END_REF] such as very low temperature, high pressure, and/or high magnetic field. Therefore, a forward theoretical understanding of the correlation effect is useful. But, the theoretical treatment of the correlations can be tedious and can require particular theoretical [START_REF] Avella | Strongly correlated systems: theoretical methods[END_REF] and numerical [START_REF] Avella | Strongly Correlated Systems[END_REF] techniques. In the simplest scenario, we can define models embedded with the terms defining the interactions between electrons. However, the exact resolution of these models remains a difficult task and is possible only under certain contexts such as low dimensionality. In order to have qualitative results on the thermodynamic limit, we often use methods with some approximations on electronic interactions or on the dimensionality of the system.

In this thesis, we focus on two distinct systems in this thesis: rare-earth-based compounds and uranium-based compounds, where both have strong electron-electron vii correlations with partially filled 𝑓 -shell. Even though our research into these two systems is separate, they provide a complementary view of the physics that may emerge from the nature of 𝑓 -electrons, which can be localized, itinerant, or both (dual) [START_REF] Zwicknagl | The dual nature of 5f electrons and the origin of heavy fermions in U compounds[END_REF].

This thesis contains two independent parts for 4 𝑓 and 5 𝑓 systems. Every part contains an introduction, methods and approximations, results, and conclusions. Below, we present a summary of each chapter:

Chapter 1 is a common opening chapter for the two halves, in which we place our subject in the context of several strongly correlated electron systems. We give an overview of a variety of strongly correlated systems, including their notable features and possible microscopic origins. We also go over various choices of theoretical methods available. Finally, we will discuss the motivations that led to this thesis and our goals.

Part I: Kondo alloys

Chapter 2 is the introductory chapter of the first part of this thesis. In this part, we present an introduction to Kondo physics by tracing its historical developments. The Kondo effect occurs when a localized magnetic impurity immersed in a metal host interacts with the conduction electrons. In lattice systems with many impurities, this effect could lead to a coherent macroscopic manifestation. One of the experimental indicators of this lattice coherence is the enlargement of the Fermi surface vis-à-vis the single impurity system. This lattice coherence and its breakdown as a function of the concentration of magnetic impurities is the core problem of this part. Here, We present experimental evidence and theoretical treatment of lattice coherence and its breakdown on various Kondo systems, and finally, we express our motivation for this study.

Chapter 3 is dedicated to the theoretical aspects of this part related to the treatment of disorder and Kondo interaction decoupling. The dynamical mean-field theory is used extensively in strongly correlated systems, and it becomes exact at the limit of infinite coordination. Here, we start by detailing our model Hamiltonian, and then we introduce matrix dynamical mean-field theory (DMFT) for the paramagnetic phase. Thereafter, we extend former DMFT formalism for the antiferromagnetic phase. The Kondo interaction is decoupled through mean-field approximations. At last, we present the system of self-consistent equations and our method of their numerical resolution.

Chapter 4 discusses our findings from DMFT on the phase diagrams of Kondo alloys. To put our study in context, we present an overview of the pressure and the atomic substitution as tuning factors for Kondo alloys. Then we show our results on a lattice with one magnetic impurity per site, followed by phase diagrams with magnetic impurity dilution. Finally, we explore the relevance of our findings by comparing them to experimental data. viii Chapter 5 is devoted to our findings on lattice coherence upon the magnetic impurity dilution in the paramagnetic Kondo phase. Through photoemission signals, we show and analyze the hallmarks of lattice coherence breakdown in Fermi surfaces, the density of states, and electronic band structure. The influence of disorder on the effective mass and quasiparticle lifetime is also investigated. Finally, we extend the phase diagram obtained in the previous chapter.

Chapter [START_REF] Pavarini | From Infinite Dimensions to Real Materials tech[END_REF] continues the study of the paramagnetic Kondo phase. Here, to characterize further the coherence breakdown in Kondo alloys, we look at local potential scattering and charge order.

Chapter 7 summarize the important findings of our study of Kondo alloys and discusses future directions for both experimental and theoretical research.

Part II: 5f electrons in Uranium alloys

Chapter 8 is the introductory chapter of the second part. The f-electron can acquire both itinerant and localized nature in this chapter. This paradigm was motivated by the direct evidence of this dual nature of 𝑓 -electron was observed in UPd 2 Al 3 through photoemission. However, other direct or indirect experimental evidence pointing towards the dual nature of 𝑓 -electron in uranium-based compounds was also observed. This duality might be driven by the interplay between multiple microscopic origins: Hund's coupling, spin-orbit coupling, and Coulomb interaction. Furthermore, it could potentially result in orbital-selective partial localized phases. To motivate this study, we present a summary of prior experimental findings, microscopic origins, and theoretical investigations.

Chapter 9 covers the most key theoretical aspects of this section. At first, we present our model Hamiltonian, and thereafter, we present the rotationally invariant slave boson (RISB) approach with mean-field approximations. Finally, we will present the system of self-consistent equations that need to be solved numerically.

Chapter 10 presents the numerical aspects of our study. Solving RISB equations is time-consuming and prone to numerical errors. Thus, they require special attention. During our study, we have employed multiple numerical schemes to obtain the correct physical ground-state solutions. Here, we describe our local and global reduction schemes in detail.

Chapter 11 presents the results on UPt 3 upon varying the orbital-dependent electronic bandwidth. We will first show and discuss our results using isotropic bandwidth, followed by an analysis of the effect of anisotropies on orbital-dependent electronic bandwidth. Finally, we present our phase diagram with orbital-selective partially localized phases, and we characterize them.

Chapter 12 summarizes the key aspects of this part and discusses the future prospects.

ix Chapter 1

General introduction

Electrons in a material define the physical properties like magnetic, optical, transport, and electronic, embedded in them. Moreover, these electrons can further interact with each other to give rise to some extraordinary features in so-called strongly correlated material systems (SCES). This strong electron-electron interaction may lead them to have either an itinerant character contributing to the chemical bonding, a localized character with no contribution to chemical bonding or even both, making them dual. Throughout this thesis, this localized versus the itinerant character of electrons will be our central theme.

Within band theory [START_REF] Ashcroft | Solid state physics[END_REF], a material is a conductor when it has partially filled energy bands. This condition is fulfilled when it has an odd number of electrons per unit cell. Similarly, if a material has an even number of electrons per unit cell, it will act as an insulator due to fully filled energy bands. The above descriptions through band theory are valid for most materials, where interaction between electrons does not play an essential role in the determination of physical properties like conductivity. Hence, the low-temperature physical properties can be understood through a model like free-electron model [START_REF] Slater | The Electronic Structure of Metals[END_REF] in these materials. However, with strong electronelectron interactions, a material with a partially filled band can become an insulator. A concrete example would be V 2 O 3 [START_REF] Imada | Metal-insulator transitions[END_REF], where a transition from metal to an insulator happens due to localization of conducting electrons driven through strong electronelectron Coulomb repulsion at low temperature even with partially filled energy bands. Furthermore, the variation of these electron-electron interactions can further lead the localized character of an electron to an itinerant. This is the case found in many 𝑓 -electron systems [START_REF] Moore | Nature of the 5 𝑓 states in actinide metals[END_REF][START_REF] Fujimori | Band structures of 4f and 5f materials studied by angle-resolved photoelectron spectroscopy[END_REF] with odd or non-integer number of electrons per unit cell, where the application of external pressure can drive a phase transition with localized to itinerant electrons. Again in 𝑓 -electron systems, a new paradigm [START_REF] Knöpfle | The Fermi surface of UPd 2 Al 3[END_REF][START_REF] Zwicknagl | Microscopic description of origin of heavy quasiparticles in UPt 3[END_REF] has emerged where 5 𝑓 electrons can acquire dual character: it can have both itinerant and localized features. From this observation, a natural question about the nature of 𝑓 -electron with the variation of the electron-electron interaction can be raised, which we will be treating in this thesis. Besides form this specific itinerant versus localized 1 CHAPTER 1. GENERAL INTRODUCTION feature, the strong correlation between electrons induces many other phenomena, which we will be presenting below.

Strongly correlated electron systems

The history of SCES started in the 1930s when transition-metal oxide NiO [START_REF] De Boer | Semi-conductors with partially and with completely filled 3d-lattice bands[END_REF] partially filled 𝑑-electron band was found to be an insulator. In the same decade, increasing resistivity was observed in impure gold and silver [START_REF] De Haas | The electrical resistance of gold, copper and lead at low temperatures[END_REF] at low temperatures, which was again unexpected. Later on, Nevill Mott and Rudolf Peierls [START_REF] Mott | Discussion of the paper by de Boer and Verwey[END_REF] pointed out the possible role of electron-electron correlations for this insulating behavior in NiO with an odd number of electrons per unit cell, and in 1949 a theory [START_REF] Mott | The basis of the electron theory of metals, with special reference to the transition metals[END_REF] was established based on very-strong Coulomb repulsion. However, increasing resistivity [START_REF] De Haas | The electrical resistance of gold, copper and lead at low temperatures[END_REF] in some impure metals at low temperature remained unsolved for almost three decades. It was only explained three decades later by Jun Kondo [START_REF] Kondo | Resistance minimum in dilute magnetic alloys[END_REF], naming this effect as the Kondo effect that became a prototype of strongly correlated phenomena. Besides these two historical examples, we could note a series of experimental SCES discoveries, features, and breakthroughs like the first heavy-fermion system [START_REF] Andres | 4 𝑓 -Virtual-Bound-State Formation in CeAl 3 at Low Temperatures[END_REF], first heavy-fermion superconductivity [START_REF] Steglich | Superconductivity in the Presence of Strong Pauli Paramagnetism: CeCu 2 Si 2[END_REF], high temperature superconductivity in cuprates [START_REF] Bednorz | Possible high T c superconductivity in the Ba-La-Cu-O system[END_REF], non-Fermi liquids [START_REF] Stewart | Non-Fermi-liquid behavior in 𝑑-and 𝑓 -electron metals[END_REF] quantum phase transitions [START_REF] Vojta | Quantum phase transitions[END_REF], and ferromagnetic superconductors in 5 𝑓 -electron systems [START_REF] Saxena | Superconductivity on the border of itinerant-electron ferromagnetism in UGe 2[END_REF].

Large diversity of families of systems with remarkable macroscopic emergent phenomena

In this section, we will present various families of SCES with their macroscopic properties. Let's start with 𝑓 -electron systems, where quasiparticle mass can reach hundreds of times of bare electron mass. The first 𝑓 -electron system to be found of this kind was CeAl 3 [START_REF] Andres | 4 𝑓 -Virtual-Bound-State Formation in CeAl 3 at Low Temperatures[END_REF]. Later on, a wide range of 4 𝑓 and 5 𝑓 -electron heavyfermions were discovered. Figure 1.1 shows this wide range of materials and the universal scaling feature when the so-called Kadowaki-Woods ratio between the 𝐴 coefficient of the resistivity 𝜌(𝑇) = 𝜌 0 + 𝐴𝑇 2 and the square linear coefficient of the specific heat 𝛾 = 𝐶 𝑣 /𝑇 is plotted. In addition to heavy-fermion behavior, these systems may present unconventional1 heavy-fermion superconductivity. The first compounds to be found as heavy-fermion superconductors in 4 𝑓 and 5 𝑓 -systems are CeCu 2 Si 2 [START_REF] Steglich | Superconductivity in the Presence of Strong Pauli Paramagnetism: CeCu 2 Si 2[END_REF] and in UBe 13 [START_REF] Ott | UBe 13 : An Unconventional Actinide Superconductor[END_REF] respectively. Also, a few numbers of 5 𝑓 -systems were also found to have ferromagnetic superconductivity [START_REF] Aoki | Review of U-based ferromagnetic superconductors: Comparison between UGe 2 , URhGe, and UCoGe[END_REF] and while might present duality [START_REF] Knöpfle | The Fermi surface of UPd 2 Al 3[END_REF][START_REF] Zwicknagl | Microscopic description of origin of heavy quasiparticles in UPt 3[END_REF][START_REF] Aoki | Review of U-based ferromagnetic superconductors: Comparison between UGe 2 , URhGe, and UCoGe[END_REF]. SCES are also famous for high-temperature superconductivity 𝑇 𝑐 which can be found in cuprates and iron pnictides. Cuprates remains on the top of the podium with 𝑇 𝑐 of 133K [START_REF] Schilling | Superconductivity above 130 k in the hg-ba-ca-cu-o system[END_REF] at 1 atm and 166 K [START_REF] Monteverde | High-pressure effects in fluorinated HgBa 2 Ca-2Cu 3 O 8 + 𝛿[END_REF] at 23 GPa. The highest 𝑇 𝑐 observed in iron pnictides is 55 K [START_REF] Zhi-An | Superconductivity at 55 K in iron-based F-doped layered quaternary compound Sm [O1-xFx[END_REF]. Another example of SCES is vanadates with vanadium sesquioxide V 2 O 3 seen as an archetype of the Mott insulator [START_REF] Imada | Metal-insulator transitions[END_REF]. Indeed, V 2 O 3 is metal at high temperatures, and it becomes an insulator at low temperatures due to strong correlations. SCES also exhibit exotic phases like spin-liquids [START_REF] Gardner | Magnetic pyrochlore oxides[END_REF] and spin-ice [START_REF] Gardner | Magnetic pyrochlore oxides[END_REF][START_REF] Bramwell | Spin ice state in frustrated magnetic pyrochlore materials[END_REF], and can be found in pyrochlores oxides family (R 2 Ti 2 O 7 , where R 3+ is a rare earth atom) and spin ladders systems like Sr 14-𝑥 Ca 𝑥 Cu 24 O 41 [START_REF] Vuletić | The spin-ladder and spin-chain system (La, Y, Sr, Ca) 14Cu24O41: Electronic phases, charge and spin dynamics[END_REF]. Furthermore, the spin ladders systems like Sr 14-𝑥 Ca 𝑥 Cu 24 O 41 also exhibit charge density wave phases along with the superconducting phase. Apart from 𝑓 -electron systems, heavy-fermion behavior can be observed in transition metal oxide such as LiV 2 O 4 [START_REF] Kondo | LiV 2 𝑂 4 : A Heavy Fermion Transition Metal Oxide[END_REF]. As most of SCES, ruthenates like Sr 2 RuO 4 can present superconductivity [START_REF] Maeno | Evaluation of spintriplet superconductivity in Sr2RuO4[END_REF]. SCES also include organic conductors like 𝜅 -(𝐵𝐸𝐷𝑇 -𝑇𝑇𝐹) 2 𝑋 systems [START_REF] Kanoda | Recent progress in NMR studies on organic conductors[END_REF][START_REF] Mckenzie | A strongly correlated electron model for the layered organic superconductors 𝜅-(BEDT-TTF) 2 X[END_REF], or even in cold atoms [START_REF] Zwerger | Mott-Hubbard transition of cold atoms in optical lattices[END_REF][START_REF] Juzeli Ūnas | Generalized Rashba-Dresselhaus spin-orbit coupling for cold atoms[END_REF] systems.

Besides the theoretical study and understanding of SCES, the strong correlation in some materials has a real potential to be used in near-future technologies in electronic devices [START_REF] Rozenberg | Strong electron correlation effects in nonvolatile electronic memory devices[END_REF] like non-volatile memory, also known as Mottronics [START_REF] Tokura | Emergent functions of quantum materials[END_REF]. Colossal magnetoresistance [START_REF] Ramirez | Colossal magnetoresistance[END_REF] and magnetocaloric effect [START_REF] Phan | Review of the magnetocaloric effect in manganite materials[END_REF] found in manganites [START_REF] Phan | Review of the magnetocaloric effect in manganite materials[END_REF][START_REF] Rozenberg | Nonvolatile Memory with Multilevel Switching: A Basic Model[END_REF][START_REF] Haghiri-Gosnet | CMR manganites: physics, thin films and devices[END_REF][START_REF] Yu-Kuai | Colossal magnetoresistance in manganites and related prototype devices[END_REF] can be used in spintronics [START_REF] Haghiri-Gosnet | CMR manganites: physics, thin films and devices[END_REF][START_REF] Yu-Kuai | Colossal magnetoresistance in manganites and related prototype devices[END_REF] and in magnetic refrigeration technology. Complex oxides like doped lanthanum manganites, we can observe multiferroicity [START_REF] Fiebig | Revival of the magnetoelectric effect[END_REF] effect, which has its potential use in devices [START_REF] Fusil | Magnetoelectric devices for spintronics[END_REF][START_REF] Ramesh | Multiferroics: progress and prospects in thin films[END_REF]. Of course, there is high 𝑇 𝑐 superconductivity a wide range of uses like high transmission lines.

Various microscopic mechanisms and interactions

Strong correlations in strongly correlated electron systems come from the interplay and competition between the multiple degrees of freedom of an electron: spin, orbital, lattice, or charge. Coulomb interaction could be the most common interaction that may lead to a variety of spin ordering patterns as well as metal-insulator transition [START_REF] Imada | Metal-insulator transitions[END_REF]. In iridates [START_REF] Zhao | Evidence of an odd-parity hidden order in a spin-orbit coupled correlated iridate[END_REF][START_REF] Kim | Novel 𝐽 eff = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4[END_REF] or cold atoms [START_REF] Juzeli Ūnas | Generalized Rashba-Dresselhaus spin-orbit coupling for cold atoms[END_REF], spin-orbit coupling may contribute orbital-dependent Mott transition. Orbital-dependent physics may appear from Hund's coupling in multi-orbital SCES like in ruthenates [START_REF] Mravlje | Coherence-incoherence crossover and the mass-renormalization puzzles in Sr 2 RuO 4[END_REF][START_REF] Georges | Strong correlations from Hund's coupling[END_REF][START_REF] Dang | Electronic correlations, magnetism, and Hund's rule coupling in the ruthenium perovskites SrRuO 3 and CaRuO 3[END_REF], and iron pnictides [START_REF] Chen | Strong Coupling Theory for Superconducting Iron Pnictides[END_REF][START_REF] Yin | Magnetism and charge dynamics in iron pnictides[END_REF], iron chalcogenides or in 5 𝑓 systems [START_REF] Efremov | Dual nature of 5 𝑓 electrons: Effect of intra-atomic correlations on hopping anisotropies[END_REF]. More importantly, there can be a complex interplay between all these couplings, which might necessitate a theoretical treatment.

Appropriate theoretical methods

Exploration of the physical properties of SCES presents a challenge on theoretical sides. To illustrate this challenge, we can note the failure of density functional theory to describe for Mott insulator with partially 𝑑-shells [START_REF] Engel | Density functional theory[END_REF]. Thus, one needs to choose a method carefully to treat electronic correlations. Ab-intio methods like local density Figure 1.1: Kadowaki-Woods ratio for a wide range of 4 𝑓 and 5 𝑓 -electron heavyfermion materials with normalized à = 𝐴/(𝑁(𝑁 -1)) and γ = 𝛾/(𝑁(𝑁 -1)) with ground-state degeneracy 𝑁. From [START_REF] Tsujii | Universality in Heavy Fermion Systems with General Degeneracy[END_REF] approximation (LDA) + U [START_REF] Anisimov | Band theory and Mott insulators: Hubbard U instead of Stoner I[END_REF][START_REF] Anisimov | First-principles calculations of the electronic structure and spectra of strongly correlated systems: the LDA+ U method[END_REF], self-interaction corrected LDA [START_REF] Perdew | Density-functional theory of the correlation energy in atoms and ions: A simple analytic model and a challenge[END_REF], linear muffintin orbitals [START_REF] Skriver | The LMTO method: muffin-tin orbitals and electronic structure[END_REF][START_REF] Zwicknagl | Quasi-particles in Heavy Fermion systems[END_REF], hybrid functional approach [START_REF] Perdew | Rationale for mixing exact exchange with density functional approximations[END_REF][START_REF] Heyd | Hybrid functionals based on a screened Coulomb potential[END_REF][START_REF] Bylander | Good semiconductor band gaps with a modified local-density approximation[END_REF] and generalized gradient approximation [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] could be employed to calculate more accurately the chemical and electronic structure of a material. However, one might need to know some characteristic parameters value, normally experimental data, for a specific material.

Simpler but effective, model-based approaches could be an alternative to ab-intio since they tend to capture electron-electron interaction more accurately giving better qualitatively results. One could choose any methods among: slave-bosons [START_REF] Barnes | New method for the Anderson model. II. The U=0 limit[END_REF][START_REF] Kotliar | New Functional Integral Approach to Strongly Correlated Fermi Systems: The Gutzwiller Approximation as a Saddle Point[END_REF][START_REF] Frésard | Unified slave boson representation of spin and charge degrees of freedom for strongly correlated Fermi systems[END_REF][START_REF] Lechermann | Rotationally invariant slave-boson formalism and momentum dependence of the quasiparticle weight[END_REF], variational method with Gutzwiller wave function approximation [START_REF] Metzner | Analytic calculation of ground-state properties of correlated fermions with the Gutzwiller wave function[END_REF], mean-field methods like dynamical mean-field theory [START_REF] Georges | Dynamical meanfield theory of strongly correlated fermion systems and the limit of infinite dimensions[END_REF], exact diagonalization [START_REF] Caffarel | Exact diagonalization approach to correlated fermions in infinite dimensions: Mott transition and superconductivity[END_REF], density matrix renormalization group [START_REF] White | Density matrix formulation for quantum renormalization groups[END_REF], quantum Monte Carlo [START_REF] Becca | Quantum Monte Carlo approaches for correlated systems[END_REF] and cluster perturbation theory [START_REF] Sénéchal | Spectral Weight of the Hubbard Model through Cluster Perturbation Theory[END_REF][START_REF] Sénéchal | Cluster perturbation theory for Hubbard models[END_REF]. Additionally, both ab-intio and model-based method can be mixed for more realistic modelling of electronic properties in real materials. For instance, one can combine dynamical mean-field theory with density functional [START_REF] Pavarini | From Infinite Dimensions to Real Materials tech[END_REF][START_REF] Kotliar | Electronic structure calculations with dynamical mean-field theory[END_REF]. DFT+DMFT has been used extensively and successfully to study real materials [START_REF] Kent | Toward a predictive theory of correlated materials[END_REF][START_REF] Paul | Applications of DFT+ DMFT in materials science[END_REF].

Motivations

In this section of this thesis, we will disclose our motivations and objectives.

Strongly correlated 𝑓 -electron systems

In this thesis, we focus on the effect of strong electronic correlations present on 𝑓 -electrons material: the rare earth metals (lanthanides) and the actinides with electronic configurations [Xe]4 𝑓 𝑛 5𝑑 0-1 6𝑠 2 and [Rn] 5 𝑓 𝑛 6𝑑 0-1 7𝑠 2 respectively. The whole series of lanthanides and actinides are formed by successive addition of an electron on the 4 𝑓 or 5 𝑓 shells.

From the figure 1.2, we can observe that the wave functions of 4 𝑓 and 5 𝑓 shells are contracted while compared to the wave functions of other electrons in 𝑠, 𝑝 and 𝑑 shells. Thus, the 4 𝑓 shell is more inner towards the nucleus and spatially localized except in europium, samarium, and ytterbium. Furthermore, the spatial extension of these orbitals shrinks as the atomic number increases, with Ce being more spatially extended for lanthanides. In 4 𝑓 systems, the strong correlation is the result of the interaction between partially filled localized electrons in 4 𝑓 electronic shells and the conduction electrons. In contrast to lanthanides, the spatial extent of 5 𝑓 orbital varies upon atomic number: larger spatial distribution for light actinides [START_REF] Brooks | Electronic structure of NaCl-type compounds of the light actinides. I. UN, UC, and UO[END_REF] whereas for heavy actinides [START_REF] Keller | Chemistry of the heavy actinides and light transactinides[END_REF] spatial extent is similar to those of lanthanides. 3+ and Pu 3+ respectively with both relativistic and nonrelativistic effects. 𝑥-axis in radial distance from the nucleus whereas 𝑃(𝑅) radial probability to find an electron at a distance 𝑟 from the nucleus. From [START_REF] Clark | The chemical complexities of plutonium[END_REF]

Lattice coherence and local-itinerant duality

In 4 𝑓 and 5 𝑓 systems, the localized 𝑓 -electrons are immersed in a sea of conduction electrons with which they may interact. At high temperatures, 𝑓 -electrons remains localized as seen through the observation of Curie Weiss susceptibility. However, at sufficiently low temperatures, strong electronic correlations may arise at the atomic level. Remarkably, a macroscopic coherent state can manifest on periodic lattice despite the short-distance nature of these local correlations. In this thesis, we will study the formation and robustness of this lattice coherence. As a first approach, we conduct this study by diluting the atoms with 𝑓 -electrons. The dilution can be done through isostructural atomic substitution of the atom with 𝑓 -electron by the atoms without 𝑓 -electrons. The second approach for this study consists of varying electronic correlations based up on the application of external parameters like pressure. We address all these questions about the lattice coherence on 4 𝑓 electronic systems in the first part of this thesis. Similarly, the question of duality will be treated on the in uranium alloys in the second part of this thesis.

Part I

Kondo alloys

Chapter 2

An overview of Kondo systems

Kondo systems are the systems where magnetic impurities are randomly distributed in a metallic crystal and interact with conduction electrons. In our case, these metallic impurities are the 4 𝑓 electrons present in rare earth metals. In these systems, the interaction between these localized magnetic impurities and conduction electrons, also known as Kondo effect [START_REF] Hewson | The Kondo problem to heavy fermions[END_REF][START_REF] Fulde | Strongly correlated electrons[END_REF]. This effect can lead to unexpected behaviors [START_REF] Sumiyama | Coherent Kondo State in a Dense Kondo Substance: Ce 𝑥 La 1-𝑥 Cu 6[END_REF] at low temperature, rich phase diagrams, non-Fermi liquid behaviors [START_REF] Stewart | Non-Fermi-liquid behavior in 𝑑-and 𝑓 -electron metals[END_REF], small and large Fermi surfaces [START_REF] Fujimori | Band structures of 4f and 5f materials studied by angle-resolved photoelectron spectroscopy[END_REF], and many other emergent phenomenons [START_REF] Michishita | Relationship between exceptional points and the Kondo effect in 𝑓 -electron materials[END_REF]. In this introductory chapter of part one of this thesis, we will present an overview of Kondo physics following the historical events in the development of Kondo physics while exposing the motivations of this current study.

Introduction to Kondo physics

Kondo effect as a scattering process

The journey of Kondo physics started with the observation of a minimum resistivity in impure gold [START_REF] De Haas | The electrical resistance of gold, copper and lead at low temperatures[END_REF] (see figure 2.2(a)) and in CuFe [START_REF] Meissner | Messungen mit Hilfe von flüssigem Helium XI Widerstand der reinen Metalle in tiefen Temperaturen[END_REF] in the early 1930s at low temperatures. At that time, this came with a big surprise, since normally, the

CHAPTER 2. AN OVERVIEW OF KONDO SYSTEMS

resistivity decreases with decreasing temperature and tends towards saturation at low temperatures. Later, the logarithm increase of resistivity was observed whenever magnetic impurities are present in the host metal. This effect puzzled for thirty years until Jun Kondo [START_REF] Kondo | Resistance minimum in dilute magnetic alloys[END_REF] provided an explanation with the assumption that the effect arises through the interaction spins of the localized and conduction electrons. This interaction leads to the model known as the Kondo single-impurity model

ℋ = k 𝜖 k 𝑐 † k𝜎 𝑐 k𝜎 + 𝐽 𝐾 S 𝐼 s 𝑐 (0) , (2.1)
where 𝑐 † k𝜎 (𝑐 k𝜎 ) describe creation (annihilation) operators of conduction electrons with spin 𝜎 =↑, ↓ and momentum k. 𝜖 k is non-interacting dispersion. S 𝐼 is the impurity spin, and s 𝑐 (0) is the conduction electron spin density at the impurity site with 𝐽 𝐾 > 0 being antiferromagnetic coupling. The resistivity in ordinary metals originates from the scattering processes where the phonon contribution is proportional to 𝑇 5 , dominating at high temperatures and the electron-electron contribution of the order 𝑇 2 , characterizing the Fermi liquid state. In addition to these contributions, the scattering of conduction electrons from the impurity is also considered through the spin-flip process. In the later process, the electron spin can flip together with a simultaneous spin-flip of the impurity. Schematic representation of spin process can be found in the figure 2.1. Within model 2.1 and Born approximations, an expression of resistivity for impurity scattering part 1 is obtained

𝑅 𝑖𝑚𝑝 = 𝑅 0 [1 - 2𝐽 𝐾 𝜌 𝑁 log( 𝑘 𝐵 𝑇 𝑊 ) + . . .] , (2.2)
where 𝑅 0 is temperature-independent residual resistivity from Born approximation and 𝑊 electronic bandwidth. For antiferromagnetic coupling 𝐽 𝐾 > 0 and for 𝑘 𝐵 𝑇 ≫ 𝑊, the resistance will increase logarithmically. However, the expression (2.2) contain the problem that it diverges as 𝑇 → 0. A year after Jun Kondo explanation, Abrikosov extended the previous calculation by calculating all the terms given by [(𝐽 𝐾 𝜌/𝑁) log(𝑘 𝐵 𝑇/𝑊)] 𝑛 which gave rise to even strong divergence at 𝑇 → 0. Through the summation of leading terms, he obtained

𝑅 𝑖𝑚𝑝 = 𝑅 0 [1 + 𝐽 𝐾 𝜌 log( 𝑘 𝐵 𝑇 𝑊 )] 2 . (2.3)
One can remark that the above equation (2.3) diverges at a characteristic temperature which is expressed as

𝑘 𝐵 𝑇 𝐾 ≈ 𝑊 exp -1/𝐽 𝐾 𝜌) . (2.4)
Thus, the characteristic temperature 𝑇 𝐾 determined from the expression (2.4) is the so-called Kondo temperature.

1A demonstration can be found in chapter four in [START_REF] Yamada | Electron correlation in metals[END_REF] Figure 2.2: Two historical example showing Kondo effect. (a) Resistivity of impure gold from 1 K to 5 K where we observe a minimum resistivity around 4 K, extracted from [START_REF] De Haas | The electrical resistance of gold, copper and lead at low temperatures[END_REF]. (b) Resistivity of Ce 𝑥 La 1-𝑥 Cu 6 series with Ce-La substitution. Singleimpurity Kondo effect can be seen at 𝑥 = 0.094 and multi-impurity at 𝑥 = 1.0, extracted from [START_REF] Sumiyama | Coherent Kondo State in a Dense Kondo Substance: Ce 𝑥 La 1-𝑥 Cu 6[END_REF].

In order to study the low energy characteristics, Anderson and his co-workers [START_REF] Anderson | Exact Results in the Kondo Problem: Equivalence to a Classical One-Dimensional Coulomb Gas[END_REF][START_REF] Anderson | A poor man's derivation of scaling laws for the Kondo problem[END_REF][START_REF] Anderson | Exact Results in the Kondo Problem. II. Scaling Theory, Qualitatively Correct Solution, and Some New Results on One-Dimensional Classical Statistical Models[END_REF] used a new approach based on scaling known as poor's man scaling, where highenergy excitations are eliminated gradually. The effect of eliminated states is retained in a set of energy-dependent running couplings 𝐽 𝐾 (𝑊) in order to preserve the lowenergy behavior of the system. They found an expression of Kondo temperature similar to equation (2.4), but with a pre-factor 𝐽 𝐾 𝜌. From this scaling approach, few conclusions can be drawn: physics is governed by scaling independent Kondo temperature 𝑇 𝐾 , and different systems have the same low energy behavior.

The theory developed from the perturbation theory progressively breaks at low temperatures. Similarly, all the analytical methods starting from the hightemperature region break down near 𝑇 𝐾 . With a severe need to understand the low energy characteristics, Wilson developed a non-perturbative numerical normalization group [START_REF] Wilson | The renormalization group: Critical phenomena and the Kondo problem[END_REF] which showed that when the Kondo interacting becomes large and below the Kondo temperature 𝑇 𝐾 , the ground-state is marked by the formation of spin-singlet states between the conduction band electrons and the local impurity spin. In his study, he also found that the ratio of magnetic susceptibility 𝜒 to Sommerfeld coefficient 𝛾 was two, which is twice the value observed for a non-interacting system.

Nozières [START_REF] Nozieres | Fermi-liquid" description of the Kondo problem at low temperatures[END_REF][START_REF] Nozières | Impuretés magnétiques et effet Kondo in[END_REF] also provided the essentials of the ground-state physical properties at the strong coupling regime (when 𝐽 𝐾 → ∞). In his study, the impurity is bound to the conduction electrons and forms a singlet state, and is decoupled from the rest of the system. It acts only as a scattering center for the conduction electrons. Since the coupling is large but not infinite, the conduction electrons can polarize the CHAPTER 2. AN OVERVIEW OF KONDO SYSTEMS singlet, which can again affect another electron. In this way, a "local Fermi liquid" can be formed through local electron-electron interaction. This single-impurity "local Fermi liquid", analogous to Landau's Fermi liquid theory where the resistivity curves behave as 𝜌(𝑇) = 𝜌 0 + 𝐴𝑇 2 , specific heat behaves as 𝐶 = 𝛾𝑇 and magnetic susceptibility behave as 𝜒(𝑇) = 𝜒 0 -𝛼𝑇 2 . This was observed experimentally in a lot of diluted Kondo alloys [START_REF] Ragel | Effects of La dilution on the CePt 2 Si 2 Kondo lattice[END_REF][100][101].

Finally, almost 50 years later, the single-impurity Kondo problem was solved exactly [102,103] by using the Bethe ansatz. The exact solution confirmed the singlet ground state calculated by Wilson and the local Fermi liquid theory elaborated by Nozières.

From single-impurity to Kondo lattice

In the 1970s, the experimental realizations of a new class of systems based on rare earth metals gave rise to new phenomena due to the presence of multiple impurities and the correlations generated by them. Concurrently, a series of theoretical developments were made to explain the physics of multi-impurity systems. Thus, in this section, we will present an overview of the physics of multi-impurity Kondo systems.

Magnetism versus Kondo: Doniach's argument

The complexity in Kondo systems increases with the increasing number of impurities. Indeed, the impurities can interact with each other via Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Doniach [104] argued the scaling behaviors of this RKKY interaction in multi-impurity systems. In his paper, he showed that there are two competing energy scales in Kondo systems proportional to the Kondo coupling 𝐽 𝐾 . The magnetically ordered phase competes with the paramagnetic Kondo phase with two different temperature scales: Kondo energy scales as 𝑇 𝐾 ∝ 1 𝜌 0 𝑒 -1/𝐽 𝐾 𝜌 0 whereas the Ruderman-Kittel-Kasuya-Yoshida state (RKKY) inter-impurity magnetic interaction scales as 𝑇 𝑅𝐾𝐾𝑌 ∝ 𝐽 2 𝐾 𝜌 0 . Indeed, magnetic order is stabilized when the strength of the Kondo interaction, i.e. the corresponding 𝑇 𝐾 , is relatively small. Figure 2.3 resumes schematic view of this competition in a phase diagram. This picture was confirmed through multiple theoretical studies [105][106][107][108][109][110][111].

Later based on Doniach argument, another kind of quantum criticality was proposed [112][113][114][115][116][117] where we observe the breakdown of Kondo effect with change of localized to itinerant behavior of impurity spins. Indeed, a system can undergo a phase transition at zero temperature upon application of external parameters due to quantum fluctuations, known as quantum phase transition [START_REF] Vojta | Quantum phase transitions[END_REF]. In Kondo lattice this can be done through the application of pressure like in CeRu 2 Ge 2 [118,119], external field like in YbRh 2 Si 2 [120] or substitution like in CeCu 6-𝑥 Au 𝑥 [121], is of secondorder nature. This transition at 𝑇 = 0 at the quantum critical point (QCP), separates a magnetically ordered phase from the non-ordered phase. The two separated phases on two sides of QCP compete with each other with similar energy scales, and in its vicinity a non-Fermi liquid behavior [START_REF] Stewart | Non-Fermi-liquid behavior in 𝑑-and 𝑓 -electron metals[END_REF]118,122] can be observed. However, mostly this QCP is hidden by a superconducting dome [START_REF] Steglich | Superconductivity in the Presence of Strong Pauli Paramagnetism: CeCu 2 Si 2[END_REF]. The microscopic nature and mechanism of QPT seem an interesting subject to explore, but it remains out of the scope of this thesis.

Effect of electronic filling: Nozières's exhaustion problem and coherence formation

The physical properties of the impurity scarce Kondo system can be understood in terms of a single-impurity scenario. However, in lattice systems, a coherent macroscopic Fermi liquid ground state can be realized where localized quantum magnetic impurities contribute to the formation of non-local, i.e. Bloch waves, fermionic quasiparticle excitations. A question about the mechanism of formation of this coherent Fermi liquid Kondo ground state can be raised when the concentration of magnetic impurities is larger than the concentration of conduction electrons. This was the question of 'exhaustion' raised by Nozières [START_REF] Nozières | Impuretés magnétiques et effet Kondo in[END_REF]123]. Photoemission [124] results on Kondo lattice showed that the spectral weights and the Kondo resonance at 𝐸 𝐹 were inconsistent with the description of the single-impurity model with the temperature scale 𝑇 𝐾 . Thus, he proposed that there are two energy scales, the Kondo temperature 𝑇 𝐾 and a coherence temperature 𝑇 * lower than 𝑇 𝐾 . This proposition was discussed in many papers, and authors now agree that the ratio of these two energy scales depends only on the band filling 𝑛 𝑐 [125][126][127][128][129]. The formation of the coherence macroscopic phase influences the transport properties. For instance, figure 2.2 shows the resistivity for Ce 𝑥 La 1-𝑥 Cu 6 . For low Ce concentrations, a saturation of maximum resistivity is observed as 𝑇 → 0. For high Ce concentrations, the resistivity gets maximum around coherence temperature 𝑇 * , and then it starts to decrease for lower temperatures. The difference in these two behaviors can be understood from the coherence formation in the lattice. At low temperatures, the screened moments become strong scatters, thus the rise in resistivity is observed at first. Below, coherence temperature 𝑇 * , coherent scattering of the Kondo singlets takes place with the conservation of momentum, this leads to the decrease of resistivity after a maximum of around 𝑇 * . Furthermore, this decrease in resistivity at low temperature shows the characteristic Fermi liquid 𝑇 2 dependence.

In the coherent phase, 𝑓 -electron levels may enter the description of the Fermi surface, which enlarges the Fermi surface. Experimentally, this enlargement of the Fermi surface can be observed from photoemission experiments, for example, the figure 2.5 presents a large Fermi surface of Kondo lattice YbRh 2 Si 2 obtained through ARPES for 𝑇 < 𝑇 𝐾 . Also, at low temperatures, the local density of states at Fermi level 𝜌(𝐸 𝐹 ) presents a distinct large peak also known as . In order to understand the coherence formation and the large Fermi surface in Kondo lattice, we should at first consider the case of large Kondo coupling as in the exact strong-coupling treatment [133] of Kondo lattice. Here, we have 𝑁 𝑓 = 𝑁 𝑥 number of Kondo impurities on 𝑁 sites and 𝑁 𝑐 = 𝑁 𝑛 𝑐 number of conduction electrons. Since, every conduction electron is bound to an impurity spin, which gives us 𝑁 𝑓 -𝑁 𝑐 number of unscreened impurity spins per site. Knowing this, we can deduce that the conduction electrons can only hop from the impurity bound site to the bachelor impurity site. We could now think the other way around: the effective charge carriers are the holes from the site with the unscreened Kondo spin. In this case, we can deduce that the number of quasiparticles would be 2𝑁 𝑓 -(𝑁 𝑓 -𝑛 𝑐 ) = 𝑁 𝑓 + 𝑁 𝑐 , where the two come from spin degeneracy. This also shows that the observation of enlarged CHAPTER 2. AN OVERVIEW OF KONDO SYSTEMS Fermi surface (see figure 2.5) comes from the participation of impurity electron in its formation, as per Luttinger's theorem [134]. A schematic representation is presented in the figure 2.4.

The coherent Fermi liquid phase can also be established from the magnetically ordered ground state in the Kondo lattice. This can be done by increasing the Kondo coupling, applying external pressure. Let's take a concrete example of CeRu 2 Ge 2 for the illustration. CeRu 2 Ge 2 is ferromagnetic with Curie temperature of 8 K, and Kondo temperature of 1.9 K. This low 𝑇 𝐾 implies low Kondo coupling 𝐽 𝐾 . By applying pressure of 76 kbar [118], a complete suppression of magnetically ordered phase is observed, followed by coherent Kondo Fermi liquid phase with enhanced effective masse.

Kondo substitution in Kondo alloys

In Kondo alloys, the number of Kondo impurities can be diminished through the substitution of Kondo atoms with magnetic impurities by non-magnetic atoms. This substitution can break the lattice coherence and can induce change in the Fermi surface structure. In this section, we present different aspects of Kondo substitution related to the lattice coherence and its breakdown.

Strong coupling picture

With a description of the large Fermi surface in Kondo lattice obtained, we could now deplete Kondo atoms. This will reduce the number of Kondo impurities 𝑁 𝑓 while the number of conduction electrons 𝑁 𝑐 remains the same. When 𝑁 𝑓 < 𝑁 𝑐 upon dilution of magnetic impurities, each impurity still captures one conduction electron by forming Kondo singlets. This leaves only 𝑁 𝑐 -𝑁 𝑓 free conduction electrons, which can hope freely only on sites without magnetic impurity. In this case, the Fermi surface contains only 𝑁 𝑐 -𝑁 𝑓 free electrons, thus the Fermi surface is considerably reduced. This scenario is illustrated in the figure 2.6. It has been proposed that the transition between large and small Fermi surfaces happens when

𝑁 𝑓 = 𝑁 𝑐 [136].

Weak coupling limit

The theoretical works predict a fundamental difference between dense and dilute Kondo alloys. But, this strong coupling does not represent the experimental reality where Kondo coupling can not be infinity. Thus, one must study the limit of intermediate to weak coupling limit. A study [137,138] on weak coupling limit was conducted on the Bethe lattice through the adapted version of the statistical DMFT method [139,140] and with matrix DMFT [141] to treat the Kondo substitution with the disorder. In that study, the neighborhood fluctuations and low dimensionality issues were analyzed through the local potential scatterings, a site-dependent effective local energy level. For weak coupling, local potential scattering showed a smooth crossover from the dilute to dense impurity concentration regime. This was in contrast with the strong coupling picture, where a clear separation of dilute and dense regimes was obtained. Furthermore, for intermediate values of impurity concentration 𝑥, the distribution local potential scattering could be found outside the non-interacting electronic bandwidth, showing the possible breakdown of Luttinger's theorem [134].

Experimental realization of lattice coherence breakdown

In order to show the lattice coherence breakdown, let's take the previous example of CeCu 6 (see figure 2.2(b)) where magnetic Ce is substituted by non-magnetic and iso-electronic La atom. In this particular example, one can follow the gradual breakdown of coherence as Ce atoms get depleted by following the resistivity curves. The breakdown of Kondo coherence also marks the change of the Fermi surface from large to small. This is because the singlet forming 𝑓 -electrons does not participate in the Fermi surface. Similarly, the enhanced effective mass becomes far lesser enhanced.

Substitution of Kondo atoms can also generate pressure effects in addition to the dilution of Kondo impurities. Ce atoms with the ionic radius of 101 pm are slightly smaller than La atoms with the ionic radius of 103.2 pm [142]. Thus Ce-La substitution increases the volume of the cell without altering the lattice structure. The increase in cell volume has the same effect as negative pressure, which decreases 𝐽 𝐾 and favors the magnetically ordered phase. A concrete example would be the case of Ce 𝑥 La 1-𝑥 Ru 2 Si 2 [119] series where only 8% of Ce substitution generates a phase transition from coherent Kondo to antiferromagnetic phase.

In addition to cell volume change, the Kondo substitution can generate substitutional disorder. This disorder can also be responsible for the breakdown of Kondo coherence, along with phase transition from coherent Kondo magnetically ordered phase or to a single-impurity local Fermi liquid regime. Non-Fermi liquid behavoir [START_REF] Stewart | Non-Fermi-liquid behavior in 𝑑-and 𝑓 -electron metals[END_REF], a deviation from Fermi liquid properties, can be observed in disordered 𝑓 -electron system with substitution [143]. In disordered Kondo system Ce 𝑥 La 1-𝑥 Ni 2 Ge 2 [101] series, no magnetic ordering was observed upon magnetic dilution, but the coherent Fermi liquid in dense Kondo regime get replaced by the non-Fermi liquid state before local Fermi liquid state at the very dilute regime. The disorder may be one of the possible origins of this non-Fermi liquid behavior, without ignoring the fact the non-Fermi liquid behavior can appear in the vicinity of magnetic phase transition.

Kondo substitution can also break the ground-state magnetic phase present in many Kondo lattices like antiferromagnetism in CeCu 2 Ge 2 [144] or ferromagnetism like in CeRu 2 Ge 2 [118] ground-state. For instance, the substitution in CeCu 2 Ge 2 [100] series found that the antiferromagnetic phase survives up to dilution state with 𝑥 ≈ 0.8 and further dilution 𝑥 ≈ 0.03 led to single-impurity behavior. Similar results were seen on Ce 𝑥 La 1-𝑥 Ni 2 Ge 2 [101] series at very dilute case, even though CeNi 2 Ge 2 has a paramagnetic ground-state. Compiling all the observations, we can deduce the following possible scenarios with substitution: coherent paramagnetic Kondo phase (dense Kondo) leading to single-impurity Kondo regime, coherent paramagnetic Kondo phase to magnetically ordered phase, coherent paramagnetic Kondo phase to non-Fermi liquid phase to single-impurity Kondo regime, magnetically ordered phase to single-impurity Kondo regime.

Motivations and objectives

Crucial differences between dilute and dense regimes were observed through DMFT on Bethe lattice at strong coupling regime [136,141], through finite-size calculations [145], Monte Carlo simulations [146][147][148], numerical renormalization group [149], or local-moment approach [150,151] and also through statistical DMFT by analyzing local potential scatterings [137]. However, these studies lacked a systematic study of the Fermi surface with the depletion of Kondo atoms which we will be addressing in this work. Also, in this thesis, we will consider the possibility of having a magnetically ordered phase in order to verify Doniach's argument with Kondo substitution. For this particular study, we will consider 1D, 2D, and 3D systems. The possible experimental signatures of coherence breakdown between dilute and dense Kondo regimes through photoemission will be also analyzed on 2D system. We also analyze local potential scattering and the possibility of charge ordering. More importantly, each of our studies will be considered with the possibility of having experimental realizations. This part is organized as follows:

The next chapter 3 is dedicated to the methodical and numerical aspect of Kondo alloys. There, we present two matrix DMFT methods: one for the paramagnetic Kondo phase and the other for the magnetically ordered phase. The DMFT method for the magnetically ordered phase is one of the original works of this thesis, which can be seen as an extension of previous DMFT [START_REF] Georges | Dynamical meanfield theory of strongly correlated fermion systems and the limit of infinite dimensions[END_REF] and matrix DMFT [141]. Again, in this chapter, we will present our approach to decouple the Kondo interactions and the numerical algorithm.

Chapter 4 present the results concerning the magnetic phase diagram of Kondo alloys upon substitution using DMFT. The pertinence of our results will be compared with experimental data of cerium-lanthanum substitution and with Doniach's picture.

Chapter 5 will be the core chapter of this part where we search various signatures of lattice coherence breakdown through photoemission. Systematic studies will be done with magnetic impurity concentration 𝑥, electronic filling 𝑛 𝑐 , and the Kondo coupling strength 𝑇 𝐾 . We will analyze the Fermi surfaces, the effective masses, density of states, and the ARPES derived bands along with the disorder effect.

In chapter 6, we present and analyze our results on local potential scattering and charge order. This chapter has two motivations: one to produce the pertinence of our results with previous studies [137,138] and the other to support the results obtained in the chapter 5 for coherence breakdown.

Chapter 7 resumes this study and presents the future perspectives.

Chapter 3

Model, method and approximations

This chapter is dedicated to the theoretical aspects of Kondo alloys. First, we start with the presentation of our model Hamiltonian, followed by the dynamical meanfield theory, which treats the disorder. Thereafter, we derive our self-consistent equations through mean-field approximations on Kondo interaction, and finally, we present our method of their numerical resolution.

The Kondo alloy model

We consider the Kondo alloy model (KAM)

ℋ = 𝑖𝑗𝜎 (𝑡 𝑖𝑗 -𝜇𝛿 𝑖𝑗 )𝑐 † 𝑖𝜎 𝑐 𝑗𝜎 + 𝐽 𝐾 𝑖∈𝒦 S 𝑖 s 𝑖 , (3.1)
where

𝑐 ( †)
𝑖𝜎 corresponds to annihilation (creation) operators for conduction electrons on a site 𝑖 with spin 𝜎 =↑, ↓. 𝐽 𝐾 > 0 is the local Kondo antiferromagnetic interaction between the local spin density of conduction electrons s 𝑖 , and quantum spin 1/2 operators S 𝑖 representing Kondo impurities. The Kondo impurities are distributed randomly with a site concentration 𝑥 on a sub-part Kondo-sites 𝒦 of the periodic lattice with 𝑁 sites. The complementary non-Kondo sites, with concentration 1 -𝑥, will be denoted 𝒩. 𝜇 is the chemical potential fixing the electronic filling 𝑛 𝑐 per site. Despite the disordered nature of Kondo alloys, the nearest-neighbor inter-site electronic hopping energy 𝑡 𝑖𝑗 remains constant, 𝑡 𝑖𝑗 = 𝑡, respectively from the random nature 𝒦 or 𝒩 of sites 𝑖 and 𝑗. 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. Thus, every 𝒦 -site will have exactly one impurity.

Treatment of disorder due to Kondo impurity substitution

The Eq. (3.1) describes the randomness of the distribution of Kondo atoms during Kondo substitutional. Thus, an appropriate method is required to treat the disorder generated due to substitutional randomness. In our case, we have opted dynamical mean-field theory [START_REF] Georges | Dynamical meanfield theory of strongly correlated fermion systems and the limit of infinite dimensions[END_REF] method. However, below we present a short selection of available methods.

An overview of methods

Multiple methods can take account of disorder presence in alloys and depends upon the considered systems. For example, exact diagonalization [152,153] works only for small systems. Density-matrix renormalization group [154] is efficient to obtain the low-energy properties for low-dimensional systems. We can also note real-space variational Gutzwiller wave functions [155], quantum monte carlo [156] methods, and Hartree-Fock based diagonalization [157] can be used to simulate disordered systems. And finally, there is dynamical mean-field theory [START_REF] Georges | Dynamical meanfield theory of strongly correlated fermion systems and the limit of infinite dimensions[END_REF]141] which we will be using in this part of this thesis.

A brief introduction to dynamical mean-field theory (DMFT)

A simple but effective mean-field theory can be employed in order to have a qualitative insight into physical properties. A well-known mean-field approach is the Weiss mean-field theory, which is a static mean-field approach. However, the static mean-field approximations are unable to capture a complete picture of the effect of correlations and also suffer from drawbacks [158]. This leads us to a dynamical mean-field theory (DMFT) where the fluctuation due to correlations are treated dynamically. The conceptual DMFT framework was developed Metzner and Vollhardt [159], Kotliar and Georges [160]. Later DMFT was applied to numerous correlated models describing strongly correlated systems [START_REF] Georges | Dynamical meanfield theory of strongly correlated fermion systems and the limit of infinite dimensions[END_REF]. DMFT makes use of infinite dimension or coordination 𝑑, 𝑍 = ∞ where a model defined in a lattice is reduced to a single site local problem embedded into a dynamical mean-field generated by other fermions. In DMFT, the mean-field is dynamical, and the quantum fluctuations are taken into account at the local level. In SCES, DMFT is a very versatile method that can be used to study the effect of coupling, electronic fillings, densities, and temperature on a system. It has been used extensively and has been successful to study Metal insulator transitions [START_REF] Georges | Dynamical meanfield theory of strongly correlated fermion systems and the limit of infinite dimensions[END_REF]161,162], disorder [149,[163][164][165][166][167], to generate phase diagrams and ARPES properties [START_REF] Kotliar | Electronic structure calculations with dynamical mean-field theory[END_REF]. For a much better accurate understanding of electronic properties in real materials, the DMFT method has also been combined with ab-initio methods [START_REF] Pavarini | From Infinite Dimensions to Real Materials tech[END_REF][START_REF] Kotliar | Electronic structure calculations with dynamical mean-field theory[END_REF]. This com-CHAPTER 3. MODEL, METHOD AND APPROXIMATIONS bined method DFT+DMFT has been used extensively in real materials study [START_REF] Kent | Toward a predictive theory of correlated materials[END_REF][START_REF] Paul | Applications of DFT+ DMFT in materials science[END_REF].

Matrix DMFT/CPA

In this part of this chapter, we will extend the matrix DMFT/CPA formalism initially developed in [START_REF] Georges | Dynamical meanfield theory of strongly correlated fermion systems and the limit of infinite dimensions[END_REF]141] to treat the effect of randomness and disorder in binary Kondo alloys with a possibility to include the magnetically ordered (MO) phases. One of the key points of our approach is that it permits us to investigate disorder in Kondo alloys at finite temperature, whereas is not possible in a static mean-field approach. At first, we will start by detailing the matrix DMFT/CPA formalism for the paramagnetic phase. Thereafter, we explain matrix DMFT formalism for a bipartite system for Néel ordered antiferromagnetic phase.

Paramagnetic Kondo phase

In this section, we detail our matrix DMFT/CPA formalism to treat paramagnetic Kondo phase for Kondo alloys which was initially developed in [141]. In order to capture the alloying effect, we remap the single local site scheme [START_REF] Georges | Dynamical meanfield theory of strongly correlated fermion systems and the limit of infinite dimensions[END_REF] into two local site problems which can be either Kondo site (𝒦 -site) or non-Kondo site (𝒩-site). This gives us the Eq. (3.1) as

ℋ = 𝑖𝑗𝜎 (𝑡 𝑖𝑗 -𝜇𝛿 𝑖𝑗 )𝑐 † 𝑖𝜎 𝑐 𝑗𝜎 + 𝐽 𝐾 𝑖∈𝒦 S 𝑖 s 𝑖 . (3.2)
The term on the right-hand side of the Eq. (3.2) represents local Kondo impurities on a fixed subset 𝒦 (Kondo) of lattice sites, which have been randomly distributed with a site concentration 𝑥. To take account of all the configurations, we introduced the projection operators as P † 𝑖 = x𝑖 ŷ𝑖 and P 𝑖 = x𝑖 ŷ𝑖 , where x𝑖 = 1 -ŷ𝑖 equals to 1 if 𝑖 is an 𝒦 -site site or '0' otherwise. These projection operators make possible to take account of various configuration possibilities between Thus, we rewrite the KAM Eq. (3.2) as

𝐻 = 𝑖𝑗𝜎 𝛾 𝑖𝑗 P † 𝑖 WP 𝑗 𝑐 † 𝑖𝜎 𝑐 𝑗𝜎 + 𝐽 𝐾 𝑖 x𝑖 S 𝑖 s 𝑖 , (3.3)
where W = 𝑡 𝑡 𝑡 𝑡 is transfer matrix. 𝛾 𝑖𝑗 is the structure factor of the underlying periodic lattice such that the non-interacting dispersion is given by 𝜖 k = 𝑖𝑗 𝛾 𝑖𝑗 𝑡 𝑖𝑗 𝑒 𝑖k(R 𝑗 -R 𝑖 ) . Furthermore, we can write the action corresponding to the Eq. (3. In the above equation, 𝑐 † 𝑖𝜎 (𝜏) and 𝑐 𝑗𝜎 (𝜏) are Grassemann variables which follows anti-commutation relation, but they aren't fermionic operators as in 3.3 while 𝜏 represents imaginary time. Equally, we write the partition function related to the action (3.4) as

𝒵 = Π 𝑖 Tr 𝑖 [𝒯 𝜏 𝑒 𝑆 ] , (3.5)
where 𝒯 𝜏 describes chronological order in imaginary time variable. Following DMFT formalism, the next step to map the lattice problem into a two local sites effective problem as in figure 3.1.

Local effective action and cavity method

Here, a local site can be either 𝒦 -site or 𝒩-site which is tagged by the index 𝑎. The action (3.4) The term 𝑆 𝑎 𝑙𝑜𝑐 includes all the interactions including the local site 𝑎 and the cavity term 𝑆 (𝑎) include all the contributions that are not taken account by the term 𝑆 𝑎 𝑙𝑜𝑐 . The partition function is expressed as

𝒵 = 𝑒 𝛽ℱ (𝑎) Tr 𝑎 [𝒯 𝜏 𝑒 𝑆 𝑎 𝑙𝑜𝑐 ],
where 0) ]) is the free energy in the presence of the cavity. The term 𝑆 𝑎 𝑙𝑜𝑐 is rewritten as

ℱ (𝑎) = 1 𝛽 𝑙𝑛(Π 𝑖≠𝑎 Tr 𝑖 [𝑒 𝑆 (
𝑆 𝑎 𝑙𝑜𝑐 = 𝜎 ∫ 𝛽 0 𝑑𝜏𝑐 † 𝑎𝜎 (𝜏)(𝜕 𝜏 -𝜇)𝑐 𝑎𝜎 (𝜏) -𝐽 𝐾 ∫ 𝛽 0 𝑑𝜏 x𝑎 S 𝑎 (𝜏)s 𝑎 (𝜏) + 𝑆 ∆ , (3.8)
where

𝑆 ∆ = 𝑙𝑛⟨𝑒𝑥𝑝(-𝑗𝜎 ∫ 𝛽 0 𝑑𝜏{𝛾 𝑎 𝑗 P † 𝑎 WP 𝑗 𝑐 † 𝑎𝜎 (𝜏)𝑐 𝑗𝜎 (𝜏) + 𝛾 𝑗𝑎 P † 𝑗 WP 𝑎 𝑐 † 𝑗𝜎 (𝜏)𝑐 𝑎𝜎 (𝜏)})⟩ (𝑎)
, with ⟨. . . ⟩ (𝑎) denotes average value in the presence of the cavity.

Limit of infinite coordination number (𝑧 = ∞)

The next step is to evaluate the newly expressed dynamical local bath term 𝑆 ∆ in the limit of large coordination (𝑧) number. In the limit of infinite dimensions or, equivalently, of infinite coordination number 𝑧 → ∞, DMFT becomes exact as a mean-field theory in classical statistical mechanics. In this case, one can neglect spatial fluctuation in the system, considering only on-site dynamical fluctuations.

So at first, we rescale the hopping parameter as t𝑖𝑗 = 𝑡 𝑖𝑗 √ 𝑧 in order to have finite average kinetic energy in this limit [168]. Similarly, the transfer matrix W will be aslo rescaled as W = W/ √ 𝑧. Now, the average value ⟨. . . ⟩ (𝑎) is calculated by cumulant development as

𝑆 ∆ = ∞ 𝑛=1 = 𝐶 𝑛 𝑛! .
The first order cumulant is the average value of the expression (3.8) and since the average value Grassemann variables ⟨𝑐 𝑗𝜎 (𝜏)⟩ = 0 and ⟨𝑐 † 𝑗𝜎 (𝜏)⟩ = 0 due to the disorder nature of the phase, the

𝐶 1 = -𝑗𝜎 ∫ 𝛽 0 𝑑𝜏{P † 𝑎 WP 𝑗 𝑐 † 𝑎𝜎 (𝜏)⟨𝑐 𝑗𝜎 (𝜏)⟩+P † 𝑗 WP 𝑎 ⟨𝑐 † 𝑗𝜎 (
𝜏)⟩𝑐 𝑎𝜎 (𝜏)} = 0 and equally all the impair order terms 𝐶 2𝑛+1 = 0. Furthermore, the pair order terms 𝐶 2𝑛 ∼ 1/𝑧 𝑛-1 when 𝑧 → ∞. Thus, the dynamical bath term can be expressed as

𝑆 ∆ = 1 2 = - ∫ 𝛽 0 𝜕𝜏 ∫ 𝛽 0 𝜕𝜏 ′ 𝑖𝑗𝜎 𝛾 𝑎𝑖 𝛾 𝑗𝑎 𝑐 † 𝑎𝜎 (𝜏)P † 𝑎 W⟨P 𝑖 P † 𝑗 𝑐 𝑖𝜎 (𝜏)𝑐 † 𝑗𝜎 (𝜏 ′ )⟩ (𝑎) WP 𝑎 𝑐 𝑎𝜎 (𝜏 ′ ) = - ∫ 𝛽 0 𝜕𝜏 ∫ 𝛽 0 𝜕𝜏 ′ 𝜎 𝑐 † 𝑎𝜎 (𝜏)P † 𝑎 ∆ 𝜎 𝑎 (𝜏 -𝜏 ′ )P 𝑎 𝑐 𝑎𝜎 (𝜏 ′ ) , (3.9)
where

∆ 𝜎 𝑎 (𝜏 -𝜏 ′ ) = 𝑖𝑗 𝛾 𝑎𝑖 𝛾 𝑗𝑎 W⟨P 𝑖 P † 𝑗 𝑐 𝑖𝜎 (𝜏)𝑐 † 𝑗𝜎 (𝜏 ′ )⟩ (𝑎) W = 𝑖𝑗 WG (𝑎)
𝑖𝑗 W is the dynamical local bath.

Determination of dynamical local bath and local action

Here, we define a Green's function as 𝐺 𝜎 𝑖𝑗 (𝜏 -𝜏 ′ ) = -⟨𝑇 𝜏 𝑐 † 𝑖𝜎 (𝜏)𝑐 𝑗𝜎 (𝜏 ′ )⟩. Thus, the next step of this formalism to establish an expression of dynamical local bath ∆ 𝜎 𝑎 (𝜏 -𝜏 ′ ). To do so, we need to redefine the Green's function in the presence of the cavity to the Green's function without the cavity. For that, we do a diagrammatic expansion of the cavity correlation function along with all the paths and excluding the paths passing through site 𝑎. We also define a local propagator Π 𝜎 𝑖𝑖 (𝜏 -𝜏 ′ ) which contains all the local correlations. With the help of Fourier transform as 𝐺(𝜏 -𝜏 ′ ) = 1 𝛽 𝑖𝜔 𝑛 𝑒 -𝑖𝜔 𝑛 (𝜏-𝜏 ′ ) 𝐺(𝑖𝜔 𝑛 ), all the Green's function as well as local propagators can be expressed in terms of fermionic Matsubara frequencies.

Since, we treat a paramagnetic phase and both 𝐺 𝜎 𝑖𝑗 (𝑖𝜔) and the local propagator Π 𝜎 𝑖𝑖 (𝑖𝜔) are diagonals on spin, we omit the spin index. We also escape 𝑖𝜔 from now in order to simplify the reading. We express the correlation Green function 𝐺 𝜎 𝑖𝑗 with the sum of all the direct paths leading 𝑖 → 𝑗 as

𝐺 𝑖𝑗 = 𝑎𝑙𝑙𝑝𝑎𝑡 ℎ𝑠 Π 𝑖𝑖 𝛾 𝑖𝑖 1 P † 𝑖 WP 𝑖 1 Π 𝑖 1 𝑖 1 𝛾 𝑖 1 𝑖 2 P † 𝑖 1 WP 𝑖 2 . . . Π 𝑖 𝑝 𝑖 𝑝 𝛾 𝑖 𝑝 𝑗 P † 𝑖 𝑝 WP 𝑗 Π 𝑗 𝑗 , (3.10)
We introduced site-dependent Green's function matrix as (3.12) Since we consider only direct paths connecting 𝑖 → 𝑗 and a site can be either 𝒦 CHAPTER 3. MODEL, METHOD AND APPROXIMATIONS or 𝒩-site, the above-average value G 𝑖𝑗 lead to

P
⟨G 𝑖𝑗 ⟩ 𝑑𝑖𝑠 = 𝑎𝑙𝑙𝑝𝑎𝑡 ℎ𝑠 Π 0 𝛾 𝑖𝑖 1 WΠ 0 𝛾 𝑖 1 𝑖 2 W . . . Π 0 𝛾 𝑖 𝑝 𝑗 WΠ 0 , (3.13)
with Π 0 = ⟨P 𝑖 Π 𝑖𝑖 P † 𝑖 ⟩ 𝑑𝑖𝑠 . From the diagrammatic expansion, one can obtain a Dyson like equation as (see appendix B.1)

⟨G 𝑖𝑗 ⟩ 𝑑𝑖𝑠 = Π 0 (𝛿 𝑖𝑗 + 𝑙 𝛾 𝑖𝑙 W⟨G 𝑙 𝑗 ⟩ 𝑑𝑖𝑠 ) .
(3.14)

Local Green's function

We introduce disordered-averaged local Green's matrix as

G 𝑙𝑜𝑐 = ⟨G 𝑖𝑖 ⟩ 𝑑𝑖𝑠 = 𝑥𝐺 𝒦 0 0 (1 -𝑥)𝐺 𝒩 , (3.15)
where 𝐺 𝒦 and 𝐺 𝒩 are the local Green's functions for site 𝒦 and 𝒩-site respectively. Further, we define the Green's function in reciprocal k space through Fourier transform as G k = k ⟨G 𝑖𝑗 ⟩ 𝑑𝑖𝑠 𝑒 𝑖k(R 𝑗 -R 𝑖 ) . Using the Fourier transform and the Eq. (3.14), we obtain

G -1 k = Π -1 0 -E k , (3.16)
where 𝐸 k is dispersion matrix as

E k = 𝜖 k 𝜖 k 𝜖 k 𝜖 k . (3.17)
Furthermore, the transformation of Green's function form imaginary time 𝜏 to Matsubara frequency (𝑖𝜔 𝑛 ) is obtained through Fourier transformation as 𝐺(𝜏-𝜏 ′ ) = 1 𝛽 𝑖𝜔 𝑛 𝑒 -𝑖𝜔 𝑛 (𝜏-𝜏 ′ ) 𝐺(𝑖𝜔 𝑛 ). Thus, we can now express the local Green's function matrix in terms of Green's function in reciprocal k space as 3.18) where 𝑖𝜔 𝑛 = 𝑖𝜋𝑇(2𝑛+1) is fermionic Matsubara frequency, while 𝑇 is the temperature of the system. .20) Inserting the Eq. (3.20) into the equation of dynamical local bath in (3.9)

G 𝑙𝑜𝑐 (𝑖𝜔) = k G k (𝑖𝜔) , (
G 𝑙𝑜𝑐 (𝑖𝜔) = k Π -1 0 (𝑖𝜔) -E k . ( 3
⟨P 𝑖 𝐺 (𝑎) 𝑖𝑗 P † 𝑗 ⟩ 𝑑𝑖𝑠 = ⟨G (𝑎) 𝑖𝑗 ⟩ 𝑑𝑖𝑠 = ⟨G 𝑖𝑗 ⟩ 𝑑𝑖𝑠 -⟨G 𝑖𝑎 ⟩G -1 𝑙𝑜𝑐 ⟨G 𝑎 𝑗 ⟩ 𝑑𝑖𝑠 . ( 3
∆(𝑖𝜔 𝑛 ) = 𝑖𝑗 𝛾 𝑎𝑖 𝛾 𝑗𝑎 W⟨G (𝑎) 𝑖𝑗 ⟩ 𝑑𝑖𝑠 W -( 𝑖 𝛾 𝑖𝑎 W⟨G 𝑎𝑖 ⟩)G -1 𝑙𝑜𝑐 ( 𝑗 𝛾 𝑗𝑎 W⟨G 𝑎 𝑗 ⟩) . (3.21)
From now, we can follow the standard DMFT formalism, and after some straightforward algebraic steps, we obtain

∆(𝑖𝜔) = [[Π 0 (𝑖𝜔)] -1 -[G 𝑙𝑜𝑐 (𝑖𝜔)] -1 ] .
(3.22)

Self-consistent equations

At the end, the our self-consistent equation for local Green's matrix becomes

G 𝑙𝑜𝑐 (𝑖𝜔) = k [(𝑖𝜔 + 𝜇)I -K(𝑖𝜔) + G 𝑙𝑜𝑐 (𝑖𝜔) -1 -E k ] -1 (3.23) = k [(𝑖𝜔 + 𝜇) 1 0 0 1 - 𝜅 𝒦 𝜅 𝒦 𝒩 𝜅 𝒩𝒦 𝜅 𝒩 + 𝑥𝐺 𝒦 0 0 (1 -𝑥)𝐺 𝒩 -1 - 𝜖 k 𝜖 k 𝜖 k 𝜖 k ] -1 , (3.24)
where, K(𝑖𝜔) = Σ(𝑖𝜔) + [G 𝑙𝑜𝑐 (𝑖𝜔)] -1 . Σ(𝑖𝜔) is self-energy matrix, is related to local propagator Π 0 (𝑖𝜔) = 𝑖𝜔 + 𝜇 -Σ(𝑖𝜔). After some lines of algebraic operations, we get our final results is: where 2 and knowing Δ 𝒦 (𝑖𝜔) = 𝑖𝑤 + 𝜇 -𝜅 𝒦 and Δ 𝒩 (𝑖𝜔) = 𝑖𝑤 + 𝜇 -𝜅 𝒩 , our three self-consistent equations are :

𝑑𝑒𝑡(𝑖𝜔) = (𝑖𝜔 + 𝜇 -𝜅 𝒦 (𝑖𝜔) + 1 𝑥𝐺 𝒦 (𝑖𝜔) -𝜖 k )(𝑖𝜔 + 𝜇 - 𝜅 𝒩 (𝑖𝜔) + 1 (1-𝑥)𝐺 𝒩 (𝑖𝜔) -𝜖 k ) -(𝜅 𝒦 𝒩 (𝑖𝜔) + 𝜖 k )
𝑥𝐺 𝒦 (𝑖𝜔) = k 1 𝑑𝑒𝑡(𝑖𝜔) (Δ 𝒩 (𝑖𝜔) + 1 (1 -𝑥)𝐺 𝒩 (𝑖𝜔) -𝜖 k ) , (3.25) 0 = k (𝜅 𝒦 𝒩 + 𝜖 k ) 𝑑𝑒𝑡(𝑖𝜔) , (3.26) (1 -𝑥)𝐺 𝒩 (𝑖𝜔) = k 1 𝑑𝑒𝑡(𝑖𝜔) (Δ 𝒦 (𝑖𝜔) + 1 𝑥𝐺 𝒦 (𝑖𝜔) -𝜖 k ) . ( 3
𝑆 𝒦 = - ∫ 𝛽 0 𝜕𝜏 ∫ 𝛽 0 𝜎 𝜕𝜏 ′ 𝑐 † 𝒦 𝜎 (𝜏)((𝜕 𝑡 -𝜇)𝛿(𝜏 -𝜏 ′ ) + Δ 𝜎 𝒦 (𝜏 -𝜏 ′ ))𝑐 𝒦 𝜎 (𝜏 ′ ) , -𝐽 𝐾 ∫ 𝛽 0 𝑑𝜏S 𝑎 (𝜏)s 𝑎 (𝜏) (3.29) 𝑆 𝒩 = - ∫ 𝛽 0 𝜕𝜏 ∫ 𝛽 0 𝜎 𝜕𝜏 ′ 𝑐 † 𝒩 𝜎 (𝜏)((𝜕 𝑡 -𝜇)𝛿(𝜏 -𝜏 ′ ) + Δ 𝜎 𝒩 (𝜏 -𝜏 ′ ))𝑐 𝒩 𝜎 (𝜏 ′ ) .
(3.30)

Antiferromagnetic phase

Matrix DMFT/CPA formalism detailed in the previous section (3.2.3) is designed to treat only the paramagnetic Kondo phase. Thus, to take account of the magnetically ordered phase, we have generalized the previous approach. To do so, we consider a bipartite lattice where a unit cell is composed of two lattice sites (see figure 3.2). Thus, we have remapped two local sites schemes to four local site problems (see figure 3.3) belonging to two sub-lattices 𝐴 or 𝐵 on a bipartite lattice. In this new mapping scheme, a local site can be either Kondo (𝒦 ) or non-Kondo (𝒩) marked by index 𝑎, and it can either belong to sublattice 𝐴 or 𝐵 which is marked by index 𝛼. So all together, we have four local site problems (see figure 3.3), which makes it more complex than it was initially. But further, we show that these four local sites problem can be reduced into two local sites problems, using the symmetries of the phases considered.

Local Green's function

The formalism here is similar to that in section (3.2.3) up to average over the disorder of Green's functions. The details of the calculations can be found in the appendix B.1. We rewrite the Dyson-like equation in k-space .31) Due to the bipartite nature of lattice, we define the local propagator of a site in sublattice A as Π 𝐴 and a site in sublattice B as Π 𝐵 . By considering the bipartite nature of the lattice and developing the expression ( 3.31) over the Reduced Brillouin zone (RBZ), we get a (4 × 4) matrix equation (3.32) where

G 𝜎 kk ′ = Π 𝜎 k ′ -k + k ′′ Π 𝜎 k ′′ -k 𝛾 k ′′ G 𝜎 k ′′ k ′ . ( 3
Ḡ𝜎 kk ′ = G 𝜎 kk ′ G 𝜎 kk ′ +Q G 𝜎 k+Qk ′ G 𝜎 k+Qk ′ +Q =        Π + 𝜎 Π - 𝜎 Π - 𝜎 Π + 𝜎 -1 - E k 0 0 E k+Q        -1 𝛿 kk ′ ,
Π + 𝜎 = Π 𝜎 𝒩 +Π 𝜎 𝐵 2 and Π - 𝜎 = Π 𝜎 𝒩 -Π 𝜎 𝐵 2
and kk ′ ∈ RBZ and Q is order vector of considered phase. Ḡ𝜎 kk ′ is a tensor with it's each element is a (2 × 2) matrix. The details of the calculations to obtain the Eq. (3.32) can be found in the appendix B.2.

Similarly, we have two disordered-averaged local Green's matrix for two sublattices 𝐴 and 𝐵 as G 𝜎 𝑙𝑜𝑐,𝐴 and G 𝜎 𝑙𝑜𝑐,𝐵 respectively. From these two (2 × 2) local Green's matrices, we define a (4 × 4) local Green's matrices as

Ḡ𝜎 𝑙𝑜𝑐 = G 𝜎 𝑙𝑜𝑐,𝐴 0 0 G 𝜎 𝑙𝑜𝑐,𝐵 . (3.33)
Here, the unit cell is two times larger than the original, thus the Brillouin zone is reduced by two folds. So expanding the over k to reduced Brillouin zone (RBZ), we get

G 𝜎 𝑙𝑜𝑐,𝐴 = 1 𝑁 k∈𝑅𝐵𝑍 {G 𝜎 kk + G 𝜎 kk+Q + G 𝜎 k+Qk + G 𝜎 k+Qk+Q } , (3.34)
G 𝜎 𝑙𝑜𝑐,𝐵 = 1 𝑁 k∈𝑅𝐵𝑍 {G 𝜎 kk -G 𝜎 kk+Q -G 𝜎 k+Qk + G 𝜎 k+Qk+Q } . (3.35) (3.36)
Expressing the (4 × 4) local Green's matrix with the help of the equations (3.34) and (3.35) and (3.32), and with some algebraic calculation (see appendix B.4), we get

Ḡ𝜎 𝑙𝑜𝑐 = 2 𝑁 k∈𝑅𝐵𝑍 [[Π 𝜎 𝐴 ] -1 -E k Π 𝜎 𝐵 E k ] -1 0 0 [[Π 𝜎 𝐵 ] -1 -E k Π 𝐴 E k ] -1 .
(3.37)

Finally, we deduce the expression for the local Green's matrix as

G 𝜎 𝑙𝑜𝑐,𝛼 (𝑖𝜔 𝑛 ) = 2 𝑁 k∈𝑅𝐵𝑍 [Π 𝜎 𝛼 (𝑖𝜔 𝑛 )] -1 -E k Π 𝜎 ᾱ(𝑖 𝜔 𝑛 )E k ] -1 . (3.38)

Determination of dynamical local bath

Similarly, as in paramagnetic case, the dynamical local bath is for bipartite lattice is expressed as

∆ 𝜎 = 𝑖𝑗 𝛾 𝑖𝛼 𝛾 𝛼𝑗 W⟨G (𝛼) 𝑖𝑗 ⟩ 𝑑𝑖𝑠 W . ( 3
𝑖𝑗 ⟩ 𝑑𝑖𝑠 = ⟨G 𝑖𝑗 ⟩ 𝑑𝑖𝑠 -⟨G 𝑖𝛼 ⟩G -1
𝑙𝑜𝑐 ⟨G 𝛼𝑗 ⟩ 𝑑𝑖𝑠 which relates the disorder-averaged Green's function to the disorder-averaged Green's function in the presence of the cavity and putting int the expression (3.40) and with some algebraic manipulations (see appendix...), we get

∆ = [Π 𝜎 𝐴 ] -1 0 0 [Π 𝜎 𝐵 ] -1 - [G 𝜎 𝑙𝑜𝑐,𝐴 ] -1 0 0 [G 𝜎 𝑙𝑜𝑐,𝐵 ] -1 .
(3.41)

The above Eq. (3.41) is further simplified to

∆ 𝜎 = [Π 𝜎 𝛼 ] -1 -[G 𝜎 𝑙𝑜𝑐,𝛼 ] -1 . (3.42)
The detailed step-by-step calculations can be found in the appendix B.5.

Self-consistent equations

In order to study the disordered binary Kondo alloy problem, we need to solve the self-consistent equations numerically. The selfconsistent equations are determined from the matrix Eq. (3.38) and are presented below

𝑥𝐺 𝜎 𝒦 𝛼 = 2 𝑁 k∈𝑅𝐵𝑍 1 𝐷 𝜎 k Δ 𝜎 𝒩 𝛼 + ((1 -𝑥)𝐺 𝜎 𝒩 𝛼 ) -1 -𝐻 𝜎 k ᾱ , (3.43) 0 = 2 𝑁 k∈𝑅𝐵𝑍 1 𝐷 𝜎 k 𝜅 𝜎 𝒦 𝒩 𝛼 + 𝐻 𝜎 k ᾱ , (3.44
)

(1 -𝑥)𝐺 𝜎 𝒩 𝛼 = 2 𝑁 k∈𝑅𝐵𝑍 1 𝐷 𝜎 k Δ 𝜎 𝒦 𝛼 + (𝑥𝐺 𝜎 𝒦 𝛼 ) -1 -𝐻 𝜎 k ᾱ , (3.45)
where

𝐻 𝜎 k ᾱ = 𝜖 2 k (𝑑𝑒𝑡) 𝜎 ᾱ (Δ 𝜎 𝒦 ᾱ + Δ 𝜎 𝒩 ᾱ + (𝑥𝐺 𝜎 𝒦 ᾱ) -1 + ((1 -𝑥)𝐺 𝜎 𝒩 ᾱ) -1 + 2𝜅 𝒦 𝒩 ᾱ) and 𝐷 𝜎 k = (Δ 𝜎 𝒦 𝛼 + (𝑥𝐺 𝜎 𝒦 𝛼 ) -1 -𝐻 𝜎 k ᾱ)(Δ 𝜎 𝒩 𝛼 + ((1 -𝑥)𝐺 𝜎 𝒩 𝛼 ) -1 -𝐻 𝜎 k ᾱ) -(𝜅 𝜎 𝒦 𝒩 ,𝛼 + 𝐻 𝜎 k ᾱ) 2 with 𝜅 𝜎 𝒦 𝒩 𝛼 = - 1 -2𝑥 + 𝑥(𝑖𝜔 + 𝜇 -Σ 𝜎 𝒦 𝛼 )𝐺 𝜎 𝒦 𝛼 -(1 -𝑥)(𝑖𝜔 + 𝜇)𝐺 𝜎 𝒩 𝛼 𝑥𝐺 𝜎 𝒦 𝛼 -(1 -𝑥)𝐺 𝜎 𝒩 𝛼 . (3.46)

Treatment of local Kondo interaction

Within DMFT formalism, one have to use a local impurity solver to resolve DMFT/CPA self-consistent equations as in section (3.2.3) and(3.2.3.3) representing many-body problem defined by the local effective action on a Kondo site, Eq. 3.29.

An overview of methods

In order to treat an interaction, one needs to choose an adapted impurity solver to that interaction keeping in mind the goal of the study. Each decoupling can have its own advantages like in the continuous-time quantum Monte Carlo method (CTQMC) [169] which is particularly applicable for multiband and time-dependent correlations. CTQMC suffers from potential sign problems and slow convergence.

One can also employ density-matrix renormalization group [170,171] for low dimensional quantum problems. Dynamical cluster approximation [172] is a technique that includes short-ranged dynamical correlations in addition to the local dynamics of the dynamical mean-field approximation while preserving causality. One can also choose the numerical renormalization group approach [173] for quantum impurity systems. There are approximate solvers like non-crossing approximation [174] which are fast but not exact.

Here, we employ mean-field approximations as local impurity solvers to describe both purely magnetically ordered and pure Kondo paramagnetic phases. Indeed, we want to focus on the physical observable like the ARPES and PES signatures of these different phases along with effective masses of quasi-particles that may emerge from the KAM. The aim is to investigate the effect of Kondo alloying and the effect of disorder, focusing on the change in Fermi surface structure via ARPES signatures, decoherence effects, and discontinuity of effective masses. We thus use the Weiss mean-field approximation for MO phases while Kondo mean-field approximation for pure paramagnetic Kondo phase since it was previously successfully used before [105,113,126,138]. Below, we present the two different mean-field approaches.

Magnetically ordered phases: Weiss mean-field approximation

Weiss mean-field channel is used to solve the action (?? while assuming a magnetically ordered phase. We restrict our study to pure ordered phase ignoring the possibility of mixed phases where the 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 ↔ 𝐵 symmetries of these phases as in table 4.1, 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 𝛼 =A, and we omit the 𝛼 index.

CHAPTER 3. MODEL, METHOD AND APPROXIMATIONS

The Kondo interaction is decoupled using the standard Weiss mean-field approximation as S•s ≈ ⟨𝑆 𝑧 ⟩𝑠 𝑧 +𝑆 𝑧 ⟨𝑠 𝑧 ⟩ -⟨𝑆 𝑧 ⟩⟨𝑠 𝑧 ⟩ which leads us to the local magnetizations

𝑚 𝑓 ≡ ⟨𝑆 𝑧 ⟩, and 𝑚 𝑐 ≡ ⟨𝑠 𝑧 ⟩ = 1 2 𝜎 𝜎 𝑧 ⟨𝑐 †
𝜎 𝑐 𝜎 ⟩ as order parameters on a Kondo site belonging to the sublattice 𝐴. The local effective action (3.29) becomes quadratic, and the local Green function on a 𝒦 -site can be expressed explicitly in terms of the bath and order parameters as

𝐺 𝜎 𝒦 (𝑖𝜔) = 1 𝑖𝜔 + 𝜇 -Δ 𝜎 (𝑖𝜔) -𝜎𝐽 𝐾 𝑚 𝑓 2 . (3.47)
Finally, to complete the general DMFT/CPA self-consistent equations described in section 3.2.3.3 which relate the local Green functions and the dynamical bath, the chemical potential 𝜇 and the order parameters 𝑚 𝑓 and 𝑚 𝑐 are determined by solving the following self-consistent equations:

𝑛 𝑐 = 𝑥 1 𝛽 𝑖𝜔 𝐺 𝒦 𝜎 (𝑖𝜔) + (1 -𝑥) 1 𝛽 𝑖𝜔 𝐺 𝒩 𝜎 (𝑖𝜔) , (3.48
)

𝑚 𝑐 = 1 2𝛽 𝑖𝜔,𝜎 𝜎𝐺 𝒦 𝜎 (𝑖𝜔) , (3.49
)

𝑚 𝑓 = - 1 2 tanh( 𝛽𝑚 𝑐 𝐽 𝐾 2
) .

(3.50)

These equations (3.48, 3.49, 3.50) are solved self-consistently using the DMFT in order to find the self-consistent solutions for 𝜇, 𝑚 𝑓 and 𝑚 𝑐 .

Paramagnetic Kondo phases: "large N" slave-bosons meanfield approximation

Like above, the Kondo interaction is decoupled using mean-field to solve the local effective action on a K-site, which in given by Eq. (3.29) assuming a paramagnetic Kondo correlated state. We neglect again the possibility that magnetic order might co-exist with Kondo local strong correlations. Consequently, hereafter, in this section, we omit the 𝛼 sublattice index. We follow the standard mean-field approximation as introduced by Lacroix and Cyrot in ( [105]), which is analogous to the "large N" expansion or slave boson approximation developed by Coleman in ( [175]) and Read and Newns in ( [176]).

At first, we start by representing Kondo spin operator within Abrikosov's fermionic representation

S 𝜎𝜎 ′ = 𝑓 † 𝜎 𝑓 𝜎 ′ -𝛿 𝜎𝜎 ′ /2.
Thereafter, the Kondo interaction in Eq. (3.29) CHAPTER 3. MODEL, METHOD AND APPROXIMATIONS is mapped and decoupled as

S • s → 1 2 𝜎𝜎 ′ 𝑐 † 𝒦 𝜎 𝑐 𝒦 𝜎 ′ 𝑓 † 𝜎 ′ 𝑓 𝜎 (3.51) ≈ 1 2 𝜎𝜎 ′ ⟨𝑐 † 𝒦 𝜎 𝑓 𝜎 ⟩ 𝑓 † 𝜎 ′ 𝑐 𝒦 𝜎 ′ + ⟨ 𝑓 † 𝜎 ′ 𝑐 𝒦 𝜎 ′ ⟩𝑐 † 𝒦 𝜎 𝑓 𝜎 -⟨ 𝑓 † 𝜎 ′ 𝑐 𝒦 𝜎 ′ ⟩⟨𝑐 † 𝒦 𝜎 𝑓 𝜎 ⟩ . (3.52)
The mean-field description as in Eq. (3.52) results in an emergent effective hybridization between the conduction electrons and the Abrikosov fermions, 𝑟 = 𝐽 𝐾 2

𝜎 ⟨𝑐 † 𝒦 𝜎 𝑓 𝜎 ⟩, which can be identified as an order parameter for the Kondo phase. An additional constraint 𝜎 𝑓 † 𝜎 𝑓 𝜎 = 1 restricts the number of Abrikosov fermions to one, which are imposed by introducing a Lagrange parameter 𝜆(𝜏). With the meanfield approximation, 𝜆 is assumed to be constant and determined self-consistently in order to satisfy the 𝑓 occupancy constraint on average.

Since here we are considering the paramagnetic Kondo phase, for the sake of clarity, hereafter we skip the spin index 𝜎. The local effective action (3.29) becomes quadratic and the local Green function on a 𝒦 -site can be expressed explicitly in terms of the bath and order parameters as

𝐺 𝒦 (𝑖𝜔) = 1 𝑖𝜔 + 𝜇 -Δ 𝜎 (𝑖𝜔) -𝑟 2 𝑖𝜔+𝜆 . (3.53)
For a non-Kondo site, the Green function can be obtained directly from the electronic bath Δ

𝐺 𝑐 𝒩 = 1 𝑖𝜔 + 𝜇 -Δ(𝑖𝜔) . (3.54)
The Kondo self-energy involved in Eq. (3.53) thus has a pole singularity

Σ 𝐾 (𝑖𝜔) = 𝑟 2
𝑖𝜔+𝜆 which captures several relevant aspects of Kondo physics. Finally, to complete the general DMFT/CPA self-consistent equations described in section 3.2.3 which relate the local Green functions and the dynamical bath, the chemical potential 𝜇, the order parameter 𝑟, and the Lagrange multiplier 𝜆 are determined by solving the following self-consistent equations:

𝑛 𝑐 = 𝑥 2 𝛽 𝑖𝜔 𝐺 𝒦 (𝑖𝜔) + (1 -𝑥) 2 𝛽 𝑖𝜔 𝐺 𝒩 (𝑖𝜔) , (3.55) 𝑟 𝐽 𝐾 = - 𝑟 𝛽 𝑖𝜔 𝐺 𝒦 (𝑖𝜔) 𝑖𝜔 + 𝜆 , (3.56) 1 = 2 𝛽 𝑖𝜔 𝑟 2 𝐺 𝒦 (𝑖𝜔) (𝑖𝜔 + 𝜆) 2 .
(3.57) CHAPTER 3. MODEL, METHOD AND APPROXIMATIONS agnetic phase. It describes the low temperature magnetic entanglement between conduction electrons and Kondo impurities spins. The Kondo crossover temperature is replaced by a transition at Kondo temperature 𝑇 𝐾 , defined as the temperature where 𝑟 continuously vanishes. It can be computed from 𝐽 𝐾 by solving the equation 1

𝐽 𝐾 = -1 𝛽 𝑖𝜔 𝐺 0 (𝑖𝜔)
𝑖𝜔 , where 𝐺 0 (𝑖𝜔) is the non-interacting Green's function and is independent on 𝑥, as discussed in [141]. The equations (3.55, 3.56, 3.57) are solved self-consistently to find the self-consistent solutions for 𝜇, 𝜆 and 𝑟.

Numerical approach and algorithm scheme

Two different DMFT loops were used to solve numerically paramagnetic Kondo phase and magnetically ordered phases as depicted on figures 3.4a and 3.4b respectively. DMFT full self-consistency was obtained using imaginary Matsubara fermionic frequencies to determine the effective dynamical electronic bath and the mean-field order parameters. Due to the constraints over computational power and time-limit, we introduced a cut-off frequency that was taken much larger than the bandwidth of the system, and the finite temperature introduced was taken at least 80 times smaller than 𝑇 𝐾 . Thus, the solutions described here correspond to the possible ground states. We describe our numerical approach in the following steps below: i) We start with our solution obtained for 𝑇 = 0 as ansatz at 𝑥 = 1 or with the solution obtained when 𝐽 𝐾 = 0. ii) 𝐺 𝜎 𝒦 ,𝛼 and 𝐺 𝜎 𝒩 ,𝛼 are actualized from the Eq. (3.42). iii) From the actualized 𝐺 𝜎 𝒦 ,𝛼 and 𝐺 𝜎 𝒩 ,𝛼 , we compute new dynamical local bath (∆ 𝜎 𝛼 (𝑖𝜔)) from Eq. (3.38). We iterate until convergence for the order parameters. iv) We evaluate our new MF orders parameter form the set of equations (3.55,3.56,3.57) or (3.48,3.49,3.50) for a given phase.

Ansatz

Chapter 4

Magnetic phase diagrams of Kondo alloys

Kondo alloys present very rich phase diagrams, and these phase diagrams depend upon various parameters. In this chapter, we analyze experimental and theoretical phase diagrams of Kondo alloys considering three parameters: the Kondo coupling between magnetic impurities and conduction electrons 𝐽 𝐾 , the electronic filling 𝑛 𝑐 , and the concentration of magnetic impurities 𝑥. At first, we start by presenting the state-of-the-art on both experimental and theoretical aspects of phase diagrams of Kondo alloys. Later, we present and discuss our results on the ground state phase diagrams obtained for Kondo lattice (𝑥 = 1.0) and with substitution.

A state of the art

Experimental phase diagrams

Decades of experiments in Kondo alloys revealed that various types of ground states could be stabilized with a large diversity of unconventional quantum phases and behaviors [177]. The unconventional phase includes unconventional superconductivity [178,179] like in heavy-fermion compound CeCu 2 Si 2 [START_REF] Steglich | Superconductivity in the Presence of Strong Pauli Paramagnetism: CeCu 2 Si 2[END_REF]180], heavyfermion paramagnetic phase, metallic spin-liquid [181] and also non-Fermi liquid phases [START_REF] Stewart | Non-Fermi-liquid behavior in 𝑑-and 𝑓 -electron metals[END_REF]. They also exhibit more conventional magnetically ordered phases such as antiferromagnetic phases like in CeCu 2 Ge 2 [100,182,183] or ferromagnetic phase like in CeRu 2 Ge 2 [118]. Furthermore, a phase transition can be obtained by the application of pressure or atomic substitution from these conventional or unconventional phases. Below, we present an experimental overview of these parameters on various Kondo alloys.

Mechanical pressure as tuning parameter

In chapter 2, we have presented Doniach's argument explaining the stabilization of magnetically ordered phases at low Kondo coupling. This Doniach's argument was verified experimentally for a multitude of Kondo lattices where the Kondo coupling 𝐽 𝐾 was controlled by applying pressure. The applied pressure can be either external mechanical pressure, negative pressure through hydrogenation, or negative or positive chemical pressure through atomic substitution. The application of external pressure usually reduces the volume of lattice, which consequently increases the Kondo interaction. Thus, this increase in Kondo interaction can yield a phase transition from a magnetic phase to a paramagnetic phase. For instance, application of pressure in Ce-based Kondo alloys like in CeRu 2 Ge 2 [118] or in CeAu 2 Si 2 [184] changed the magnetically ordered ground state present at atmospheric pressure into heavy-fermion paramagnetic phase. Meanwhile, the scaling of the Kondo temperature [118] was also in good agreement with Doniach's argument. Other than the positive external pressure, a negative pressure also can be applied from hydrogenation. For example, the hydrogenation in CeRuSi [185] led to the transition from paramagnetic heavy-fermion phase to antiferromagnetic phase.

Application of pressure is not limited to magnetic to non-magnetic transition but can also reveal unconventional phases. Indeed, pressure-induced superconductivity was found in Ce based compounds like CeRh 2 Si 2 [186], CePtSi 2 [187], CeRhIn 5 [188] or CeNiGe 3 [189] and unusual nonmagnetic ordered state in CeCoSi [190]. Pressure can also induce breakdown of Fermi liquid properties like in CeRu 2 Ge 2 [118].

Substitution as tuning parameter

In Kondo alloys, one can substitute either the rare-earth atom, transition metal, or metalloid atom. Each of these substitutions can act as a tuning parameter which we will be presenting below.

Transition metal or metalloid substitution

In most cases, Cerium-based Kondo alloys are of type CeT 𝑎 X 𝑏 where T is a transition metal like Ru and X is a metalloid such as Si or Ge. Apart from rare-earth atoms, in these alloys, we can substitute either a transition metal atom or metalloid atom with another atom of the same group of the periodic table. This substitution can modify the Kondo coupling and the electronic filing at the same time, even though the electronic is difficult to evaluate experimentally. Thus, these substitutions can act as a tuning parameter. An example of a magnetic phase diagram through metalloid substitution is presented in the figure 4.1 where Si atoms are substituted by Ge atoms (extracted from [119]). This Si-Ge substitution generated a cascade of magnetic phase transitions from ferromagnetic to two different antiferromagnetic phases and paramagnetic phase at the end. A cascade of phase transitions is expected with

Rare earth metal substitution

In Kondo systems, the concentration of Kondo impurities 𝑥 is another relevant parameter besides the electronic filling 𝑛 𝑐 and pressure. Continuous tuning of 𝑥 can be realized experimentally in Kondo alloys heavy-fermions by isostructural substitution of a magnetic rare-earth atom with a non-magnetic one. This induces remarkable changes in macroscopic physical properties, and the dilution of magnetic impurities from the Kondo lattice can yield a phase transition. As La atom is slightly larger than Ce atom, Ce-La isostructural substitution enlarges the crystal volume, thus reducing the Kondo coupling. One can imagine that this reduction of Kondo coupling can lead from paramagnetic phase to magnetic ordered phase transition. Indeed, this was observed for Ce 𝑥 La 1-𝑥 Ru 2 Si 2 [119] where the dilution of magnetic impurities led to a phase transition from paramagnetic to magnetically ordered phase. This is not always true since in Ce 𝑥 La 1-𝑥 Cu 2 Ge 2 [100] series a phase transition from antiferromagnetic (AF) to paramagnetic (PM) was observed. Another Ce-La substitution in the series Ce 𝑥 La 1-𝑥 Cu 6 [START_REF] Sumiyama | Coherent Kondo State in a Dense Kondo Substance: Ce 𝑥 La 1-𝑥 Cu 6[END_REF]195] revealed an evolution from coherent dense Kondo lattice regime to dilute Kondo regime whereas in the series Ce 𝑥 La 1-𝑥 FePO [196] coexistence of spin glass with dilute Kondo regime is observed. Other types of rare-earth substitutions can also unexpected result like Fermi-surface instabilities in unconventional superconductors like Ce 𝑥 Yb 1-𝑥 CoIn 5 [197] and Nd 2-𝑥 Ce 𝑥 CuO 4 [198].

The Fermi liquid properties were seen from resistivity measurements for Ce 𝑥 La 1-𝑥 Pt 2 Si 2 [START_REF] Ragel | Effects of La dilution on the CePt 2 Si 2 Kondo lattice[END_REF] and Ce 𝑥 La 1-𝑥 Cu 2 Ge 2 [100] series. But in multiple cases, Ce-La substitution led to non-Fermi liquid behavior [122]. Such as for Ce 𝑥 La 1-𝑥 Ni 2 Ge 2 [101] and Ce 𝑥 La 1-𝑥 PtIn [199] series, substitution led to dense-dilute transition along with non-Fermi liquid behavior. In some Ce-La substitution, Fermi liquid properties were not consistent over different physical quantities. This is the case for Ce-La substitution in Ce 𝑥 La 1-𝑥 Ni 9 Ge 4 [200,201] where non-Fermi liquid behavior was observed in specific heat measurements while Fermi liquid behavior in magnetic susceptibility. This raises an additional question about the origin of these non-Fermi liquid behaviors upon dilution. These handfuls of examples clearly show that Ce-La substitution can considerably alter the behavior of physical quantities, which also motivates our study of binary Kondo alloys of this thesis.

Phase diagrams through various theoretical approaches

Another pertinent parameter besides pressure is the electronic filling (𝑛 𝑐 ). Various theoretical studies revealed Doniach-like phase diagrams [105,106,202,203]. Multiple theoretical techniques were employed in order to explore the effect of 𝑛 𝑐 for Kondo lattice model (KLM) like for one-dimensional systems by exact diagonalizationn [129], by density matrix renormalization group [204][205][206], by a unitary transformation involving a bosonization of delocalized conduction electrons [207], for two-dimensional systems by DMFT with numerical renormalization group [111,208,209], cellular dynamical mean-field theory and variational Monte Carlo [210] and by dynamical cluster approach [211] and for three-dimensional systems by mean-field [105]. Some the theoretical studies [109,212,213] in Kondo lattice model also pointed out that magnetic order can coexist with Kondo effect.

Important attention was also focused on the issue of coherence in the paramagnetic phase of dense Kondo systems. The robustness of a coherent Kondo state has then been investigated in the framework of the Kondo lattice model by several complementary theoretical approaches [123, 125-129, 133, 214-217].

Symmetries considerations for numerical calculations

In our study of Kondo alloys, we consider only pure paramagnetic Kondo phase (K), pure ferromagnetic phase (F) or antiferromagnetic (AFII) phase with purely staggered Néel order with ordering vector Q = (𝜋, 𝜋). Using the DMFT method, the lattice problem has been mapped onto four single-site effective problems that account for the possible local correlations, 𝑎 = 𝒩 or 𝒦 , on sublattice 𝛼 =A or B. For different reasons, in each case, we can restrict the problem to two effective sites. In table 4.1, we analyze the symmetry properties of the various phases that we consider. Sites A and B are equivalent for Kondo decoupled F and K phases, and there we recover the two effective sites dynamical approach as developed in [141]. As AF phase with Néel order obeys the transformations (A→B, 𝜎 → σ), the basis and the Green's function remains invariant through this transformation. Thus, the correlation functions for a site in sublattice A with spin 𝜎 are equivalent to the correlation functions for a site in sublattice B with spin σ. Therefore, evaluating Green's functions for a site in sublattice A or B with spin 𝜎 and σ is sufficient to study the AFII phase. So again, we bring back two effective sites dynamical approach instead of four effective sites.

Transformations invariance K F AFII A → B 𝜎 → σ yes no yes A → B 𝜎 → 𝜎 yes yes no A → A 𝜎 → σ yes no no

Results: ground-state phase diagram

In this section, we present and discuss the ground state phase diagrams obtained for periodic Kondo lattice as well as for Kondo alloys for electronic filling 𝑛 𝑐 = 0.30, 0.70 and 0.90. In each case, the phase diagrams were obtained by comparing the energies of each considered phase. The expressions to compute the energies of each phase are detailed in the appendix A. At First, we studied the phase diagrams for periodic Kondo lattice for 1D 'chain', 2D 'square', and 3D 'cubic' lattices. We reproduced the existing results, which validated our method. Thereafter, we studied the phase diagrams of Kondo alloys by varying the Kondo atom concentration 𝑥.

4.

3.1 Tuning of 𝑛 𝑐 and 𝑇 𝐾 for periodic Kondo lattice

Q 𝐴𝐹𝐼 0 Q 𝐴𝐹𝐼𝐼 0 Q 𝐴𝐹𝐼𝐼𝐼 0 1D 𝜋 2D (𝜋, 0) (𝜋, 𝜋) 3D (0, 0, 𝜋) (𝜋, 𝜋, 0) (𝜋, 𝜋, 𝜋)
Table 4.2: commensurate AF phases with and their respective wave ordering vectors.

In order to validate our DMFT approach, we calculated the solutions for paramagnetic Kondo phase (K), ferromagnetic phase (F), and commensurate antiferromagnetic phases for periodic 1D 'chain', 2D 'square' and 3D 'cubic' Kondo lattices without using DMFT algorithm at 𝑇 = 0 with varying 𝑛 𝑐 . We then studied the same problem for the 2D 'square' lattice using the DMFT algorithm. We came up with the same results. However, we didn't calculate the solutions below 𝑛 𝑐 < 0.25 with DMFT because the algorithm becomes more time-consuming. We have equally omitted to evaluate AF I in our DMFT calculations, however, it doesn't change the qualitative results.

Figure 4.2 shows the phase diagrams obtained for 1D 'chain', 2D 'square' and 3D 'cubic' lattices at T = 0 with K, F and AF phases with several commensurate antiferromagnetic phases (see table 4.2). The ground-state phase diagrams are consistent with Doniach's Phase diagram [104]. Three distinct phases can be seen for 1D lattice with magnetic phase dominates at small 𝐽 𝐾 𝑊 and Kondo phase is observed at large 𝐽 𝐾 𝑊 . At half-filling, the transition between AF to Kondo phase occurs around 𝐽 𝐾 𝑊 ≃ 0.40, and it gradually decreases with 𝑛 𝑐 . With decreasing 𝑛 𝑐 at 𝑛 𝑐 ≈ 0.62, transition from AF to F is observed. Further reduction of 𝑛 𝑐 saturates the ferromagnetic phase, and it is stabilized for low electronic filing. The phase diagram obtained for 2D square lattice is remarkably similar to that obtained by Lacroix and Bernhard [109]. At half-filling a transition between K and AFII is obtained at around 𝐽 𝐾 𝑊 ≃ 0.20 at a similar value obtained in [218]. The transition from AFII to F is observed at 𝑛 𝑐 = 0.57, similar as in [106,109]. The transition between AFII and AF I is observed at around 𝑛 𝑐 ≃ 0.68 whereas AF I → F is observed at 𝑛 𝑐 ≃ 0.28. In [111], a stripped magnetism was observed at 0.32 < 𝑛 𝑐 < 0.65 which is quite similar to the region(0.28 < 𝑛 𝑐 < 0.68) where our AF I is stabilized. Similar observations for 3D 'cubic' lattice were observed. When the number of conducting electrons 𝑛 𝑐 is low, saturated F phase is observed up-to 𝐽 𝐾 𝑊 ≃ 1.0 which is followed by Kondo phase afterwards. At half filling, the transition AFIII → K, occurs at In summary, a cascade of antiferromagnetic phases to ferromagnetic phases is observed as we move from higher 𝑛 𝑐 to lower 𝑛 𝑐 in each case for low 𝐽 𝐾 /𝑊, and paramagnetic Kondo phase is observed for higher 𝐽 𝐾 /𝑊. Since all three phases diagrams have the same form, we will concentrate only on the 2D square lattice to study the effect of dilution of magnetic impurities.

Tuning of 𝑥 and 𝑇 𝐾 at fixed 𝑛 𝑐 for Kondo alloys

We used the DMFT algorithm to study binary Kondo alloys for the concentrations 𝑥 ∈ [0.01, 1] for different 𝑛 𝑐 with different couplings 𝐽 𝐾 considering pure K, F, and AFII phases. We present in the figure 4.3 the ground-state phase diagrams obtained for 𝑛 𝑐 = 0.90 and 0.30. Here, the strength of the Kondo interaction is represented by the ratio between the corresponding Kondo temperature 𝑇 𝐾 and the non-interacting electronic bandwidth 𝑊. All three-phase diagrams present similar forms and are Doniach-like: at low Kondo temperature (𝑇 𝐾 ) MO phase is found to be the ground state, and this MO phase is suppressed by the paramagnetic Kondo phase at higher 𝑇 𝐾 .

Discussion: comparison with experimental data

The above phase diagrams are compatible with the phase diagrams of some alloys as Ce 𝑥 La 1-𝑥 Ru 2 Si 2 [119] where substitution of Ce with La give rises to MO, Ce 𝑥 La 1-𝑥 Cu 2 Ge 2 [100] where antiferromagnetic phase was seen up-to very low concentration of Ce, and Ce 𝑥 La 1-𝑥 Pt 2 Si 2 [START_REF] Ragel | Effects of La dilution on the CePt 2 Si 2 Kondo lattice[END_REF] paramagnetic Kondo phase persists with substitution. In the real isostructural Ce-La substitution series, experiments show that the strength of the Kondo interaction decreases when decreasing Cerium concentration. The opposite monotonic variation of 𝑇 𝐾 with 𝑥 is observed with Ytterbium-based Kondo alloys. This is due to the effective pressure effect that is related, with the difference of lattice parameters between Ce-(or Yb-) and La-based (or Lu-) compounds. Therefore, in order to provide a scenario for real Kondo alloys, using the (𝑥-𝑇 𝐾 ) phase diagrams depicted in figure 4.3, one has to characterize Ce-La substitution with a non-constant but rather monotonous line. HF materials. Here, we focus only on 122-family, but the scenario isn't restricted to these compounds, and we expect that it can be much more general. Figure 4.4 is the generalized phase diagram obtained through our DMFT calculations where we have placed a few Ce-based 122-family HF materials. For several of Kondo lattice (𝑥 = 1) 122 compounds, the Doniach argument is consistent with the occurrence of magnetically ordered ground state when the strength of the Kondo coupling 𝑇 𝐾 becomes smaller than about 10 Kelvin. We can thus depict various Kondo alloys on a single schematic phase diagram, where the energy scale is arbitrarily fixed (see figure 4.4). For Ce 𝑥 La 1-𝑥 Pt 2 Si 2 , the paramagnetic phase persists with substitution, with 𝑇 𝐾 decreasing from 70 K (𝑥 = 1) to 2.6 ± 0.6 K (𝑥 = 0.10) [START_REF] Ragel | Effects of La dilution on the CePt 2 Si 2 Kondo lattice[END_REF]. For Ce 𝑥 La 1-𝑥 Ni 2 Ge 2 , the ground state also remains paramagnetic, but the Kondo interaction is smaller, from 30 K for 𝑥 = 1 to 1 K for 𝑥 = 0.01 [101]. Invoking the present scenario, the vicinity of a magnetically ordered phase at intermediate Ce concentration might explain the non-Fermi liquid behavior that has been reported for this Kondo alloy [101]. For Ce 𝑥 La 1-𝑥 Ru 2 Si 2 , the Kondo lattice (𝑥 = 1) is paramagnetic with 𝑇 𝐾 = 24 K, but an antiferromagnetic ground state is stabilized below the critical concentration 𝑥 = 0.91 [119] and down to the smallest concentrations for which this alloy was synthesized. The Kondo lattice CePd 2 Si 2 , which is characterized by a smaller 𝑇 𝐾 =9 K, has an antiferromagnetic ground state with a Néel temperature 𝑇 𝑁 =9.9 K. For Ce 𝑥 La 1-𝑥 Pd 2 Si 2 , the antiferromagnetic phase is observed when 𝑥 > 0.75, but a paramagnetic Kondo ground state is obtained at lower Cerium concentrations, with 𝑇 𝐾 decreasing down to 2.8 K for 𝑥 = 0.20 [220]. Another example of 122 Ce-La substituted compound has a similar behavior as CePd 2 Si 2 : the Kondo alloy Ce 𝑥 La 1-𝑥 Cu 2 Ge 2 was indeed reported to be antiferromagnetic with 𝑇 𝑁 = 4.1 K and 𝑇 𝐾 = 4 K for 𝑥 = 1, and a surprising persistence of antiferromagnetic order down to 𝑥 = 0.10 [100]. Our present scenario might also explain this persistence. Of course, the quantitative energy scale used for the schematic figure 4.4 was chosen arbitrarily to mimic various 122 Ce-based compounds in a coherent scenario. However, we expect that the qualitative properties observed experimentally should remain universal beyond the specific cases that are analyzed here.

Chapter 5

Photo-emission properties of Kondo alloys

In this chapter, we present our results on the photo-emission properties of Kondo alloys upon the dilution of Kondo impurities for a large range of Kondo coupling and three electronic fillings 𝑛 𝑐 = 0.30, 0.70 and 0.90. For this, we will analyze the photo-emission spectrum in terms of Fermi surfaces, band-structure and density of states, and effective mass focusing on the paramagnetic Kondo phase. This analysis will further complete the phase diagram presented in chapter 5.

A state of the art

The angle-resolved photoemission spectroscopy (ARPES) technique is based on the photoelectric effect described a century ago. In this technique, a photon of sufficient energy is projected to the crystal, which ejects an electron from the material, following the absorption of an X-ray photon. By measuring the kinetic energy and angle distributions of the emitted photoelectrons, the technique can map the electronic band structure and Fermi surfaces of the system. ARPES is used to characterize materials because it gives access directly to the Fermi surface topology and band structure. On Kondo alloys, it was first Park et al. [221] conducted ARPES experiments on multiple Ce-based Kondo lattice systems, since then this probe was employed extensively to study the electronic structure of correlated electronic systems [START_REF] Fujimori | Band structures of 4f and 5f materials studied by angle-resolved photoelectron spectroscopy[END_REF]. These experimental technics have already been proven to be very useful to investigate the physics of 𝑓 -electron systems, including Kondo alloys [START_REF] Fujimori | Band structures of 4f and 5f materials studied by angle-resolved photoelectron spectroscopy[END_REF]222].

Enlargement of Fermi surface due to the coherent participation of [135,229,230] showed large Fermi surfaces due to the coherent participation of 4 𝑓 electrons. ARPES can also reveal anisotropic properties present in some Kondo alloys. For example, soft x-ray ARPES revealed anisotropic Fermi surface structures: large along Γ -𝑋 direction for CeRu 2 Si 2 [231] and smaller along with other directions. Meanwhile, LaRu 2 Si 2 showed a smaller Fermi surface as predicted through model calculations (see figure 5.1). Additionally, photoemission spectroscopy experiments also revealed limitations of the single impurity models for describing dense Kondo systems [124,232,233]. However, a complete picture of the evolution of large Fermi surface for dense Kondo to small Fermi surface in dilute Kondo upon dilution remains still an open question.

ARPES can give access directly also to the band structure, and angle-integrated photoemission spectroscopy gives access to the density of states of a crystal. In Kondo lattice,ARPES experiment [223] on CeRu 2 Si 2 and URu 2 Si 2 indicated 𝑓 -𝑑 band mixing at low temperatures and exclusions of 𝑓 -electrons in the Fermi surface above 𝑇 𝐾 . While in CeCoGe 1.2 Si 0.8 [234] the heavy hybridized conduction band as well as dispersive Kondo resonance peaks were observed directly. In another ARPES experiment, similar band structures were obtained on both Kondo lattice CeRh 2 Si 2 and in Ce-La substituted Kondo alloy Ce 0.84 La 0.16 Ru 2 Si 2 [235]. This similarity was astonishing because below the critical concentration 𝑥 = 0.07, the antiferromagnetic phase appears. Similarly in the band structure before and after the critical concentration indicates that the change in band structure might be continuous across 𝑥 𝑐 . ARPES in CeRh 2 Si 2 [236] also revealed the surface and bulk hybridization in antiferromagnetic Kondo lattice.

Important works were also done in theoretical part to determine the electronic structures of Kondo alloys either by ab-initio methods like DFT, linear muffin-tin orbital [START_REF] Zwicknagl | Quasi-particles in Heavy Fermion systems[END_REF]237], or DMFT/DMFT+ab-initio calculations [START_REF] Kotliar | Electronic structure calculations with dynamical mean-field theory[END_REF]. Fermi surface obtained through ab-initio calculations were consistent with ARPES derived Fermi surfaces [227,229,230,238]. Combined DMFT and DFT method was employed to study temperature dependent Fermi surface in multiple cases [239][240][241][242].

Theoretical approach for disordered systems

Some physical quantities like the density of states, spectral function, and effective masses, showing the electronic structure of a system can be extracted directly from local self-energy. Since the system that we are studying is disordered and random, one must include all the processes of creation and annihilation of a fermion that goes from one type of site to another type of site. In our case, we have four different processes: Kondo site to Kondo site, Kondo site to non-Kondo site, non-Kondo site to Kondo site and non-Kondo site to non-Kondo site. Below, we obtain a disordered-averaged one-body Green's function from these four different processes. This disordered-averaged one-body Green's function will be used to obtain photoe-

Disorder averaged one-body Green's function

Here, we will derive the expression for one-body Green's function for paramagnetic Kondo phase. In pure Kondo phase the sub-lattices A and B are equivalent and doesn't depend on spin 𝜎, the sites 𝛼 and ᾱ are equivalent which gives makes the local propagator equivalent for each site

[Π 𝛼 (𝑖𝜔 𝑛 )] -1 = [Π ᾱ(𝑖 𝜔 𝑛 )] -1 = [Π 0 (𝑖𝜔 𝑛 )] -1
and similarly the dynamical local bath for the sites 𝛼 and ᾱ are equivalent, hence

Δ 𝛼 (𝑖𝜔 𝑛 ) = Δ ᾱ(𝑖 𝜔 𝑛 ) = Δ(𝑖𝜔 𝑛 ). We recover the equation for G 𝜎 𝑙𝑜𝑐 (𝑖𝜔 𝑛 ) as in [141] G 𝑙𝑜𝑐 (𝑖𝜔 𝑛 ) = k G(k, 𝑖𝜔 𝑛 ) = k [[Π 0𝜎 (𝑖𝜔 𝑛 )] -1 -W k ] -1 , (5.1) with [Π 0𝜎 (𝑖𝜔 𝑛 )] -1 = (𝑖𝜔 𝑛 + 𝜇)I -Σ 𝜎 (𝑖𝜔 𝑛 )
, where I is 2 × 2 identity matrix. Following the steps as in [141], disordered-averaged K-dependent 2 × 2 Green's function matrix G(k, 𝑖𝜔 𝑛 ) can be expressed as

G(k, 𝑖𝜔 𝑛 ) = 𝑖𝜔 𝑛 +𝜇-Σ 𝒦 (𝑖𝜔 𝑛 )-(1-𝑥)Δ(𝑖𝜔 𝑛 ) 𝑥 -𝜖 k Δ(𝑖𝜔 𝑛 ) -𝜖 k Δ(𝑖𝜔 𝑛 ) -𝜖 k 𝑖𝜔 𝑛 +𝜇-𝑥Δ(𝑖𝜔 𝑛 ) 1-𝑥 -𝜖 k -1 , (5.2)
where, self-energy Kondo Σ 𝒦 (𝑖𝜔 𝑛 ) = 𝑟 2 𝑖𝜔 𝑛 +𝜆 . Inverting this (2 × 2) matrix, we get the matrix elements of G(k, 𝑖𝜔 𝑛 ). We sum up all the four matrix elements of the matrix (5.2) defining the four different processes to get our one-body Green's function as

𝒢(k, 𝑖𝜔 𝑛 ) = 𝐺 𝒦 𝒦 (k, 𝑖𝜔 𝑛 ) + 𝐺 𝒦 𝒩 (k, 𝑖𝜔 𝑛 ) + 𝐺 𝒩𝒦 (k, 𝑖𝜔 𝑛 ) + 𝐺 𝒩𝒩 (k, 𝑖𝜔 𝑛 ) , (5.3)
where 𝐺 𝒦 𝒦 (k, 𝑖𝜔 𝑛 ), 𝐺 𝒦 𝒩 (k, 𝑖𝜔 𝑛 ), 𝐺 𝒩𝒦 (k, 𝑖𝜔 𝑛 ) and 𝐺 𝒩𝒩 (k, 𝑖𝜔 𝑛 ) are the matrix elements of the matrix (5.2). With a some straight forward algebraic manipulations, we obtain an expression analogous to one-body interacting Green's function as

𝒢(k, 𝑖𝜔 𝑛 ) = 𝑖𝜔 𝑛 + 𝜇 -Σ 𝐴𝑙𝑙𝑜 𝑦 (𝑖𝜔 𝑛 ) -𝜖 k -1
.

(5.4)

where the self-energy is expressed with the help of Kondo self-energy Σ k (𝑖𝜔 𝑛 ) = 𝑟 2 /(𝑖𝜔 𝑛 + 𝜆) alloy as

Σ 𝑎𝑙𝑙𝑜 𝑦 (𝑖𝜔 𝑛 ) = 𝑥 Σ -1 𝒦 (𝑖𝜔 𝑛 ) -(1 -𝑥)𝐺 𝒩(𝑖𝜔 𝑛 )
.

(5.5)

We can easily verify that this one-body Green's function (Eq. 5.4) reproduces standard results at several limits. At extreme dilute limit (𝑥 = 0), we obtain non-interaction Green's function 𝐺 0 (k, 𝑖𝜔 𝑛 ) = 1/(𝑖𝜔 𝑛 + 𝜇 -𝜖 k ) whereas when 𝑥 = 1, we obtain

𝐺(k, 𝑖𝜔 𝑛 ) = 1/(𝑖𝜔 𝑛 + 𝜇 -Σ 𝒦 (𝑖𝜔 𝑛 ) -𝜖 k
). Thus, one-body Green's function take account all the processes relates both 𝒦 and 𝒩 sites. All the observable properties related to our system is confined into Σ 𝑎𝑙𝑙𝑜 𝑦 . So now, we present below the definitions and expressions of some quantities that we have analyzed in this chapter.

Spectral function and Fermi surface

In non-interacting system, the spectral function is defined as

𝒜 0 (k, 𝜔) = 𝛿(𝜔 -𝜖 k ) , (5.6)
with 𝛿 being delta Dirac function. Similarly, in this study, we define the spectral function at a frequency 𝜔 and a wave-vector k from the disorder-averaged one-body Green's function as

𝒜(k, 𝜔) = - 1 Π 𝐼𝑚(𝒢(k, 𝜔 + 𝑖𝜂 + )) . (5.7)
where 𝜂 is an infinitesimal positive number. The Fermi surface for a system is obtained from the equation 5.7 by evaluating it at Fermi level 𝜔 = 0 over first the Brillouin zone.

Local density of states

Another quantity that we can obtain form one-particle Green function is density of states (d.o.s). The d.o.s is obtained by a summation of spectral function (Eq. 5.6) over all k.

𝜌(𝜔) = 1 𝑉 ∫ 𝑘 𝒜(k, 𝜔)𝑑k , (5.8)
where 𝑉 is the volume of Brillouin zone. This quantity can be obtained from angle integrated photoemission spectroscopy.

Local self-energy and effective mass

In this chapter, we are also interested in the effective mass of quasiparticles. This quantity can be obtained through self-energy. At low-energy, the Taylor expansion of self-energy can be written in as

Σ(𝜔) = Σ(0) + 𝜔 𝜕Σ(𝜔) 𝜕𝜔 𝜔=0 + 𝑂(𝜔 2 ) (5.9) = Σ(0) + (1 -𝑚 * )𝜔 + 𝑂(𝜔 2 ) , (5.10)
where 𝑚 ★ is effective mass and is defined as

𝑚 ★ 𝑚 = 1 - 𝜕ℜ(Σ(𝜔)) 𝜕𝜔 𝜔=0 .
(5.11)

Results: Spectral function and electronic density of states

In this section, we analyze the spectral function 𝒜(k, 𝜔), which is related ARPES signal and the local electronic density of states 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 dispersion in spectral function along with gap-less, pseudogapped, or gapped local electronic density of states. We solved the self-consistent equations (3.25, 3.26 and 3.27) 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 5.3) and the relatively small 𝑇 𝐾 regime (see figure 5.4), typically separated by a threshold corresponding to 𝑇 𝐾 around 𝑊/10. In both cases, the dilute Kondo limit (𝑥 ≪ 1) and the dense Kondo limit (𝑥 = 1) generate usual results. The dilute Kondo limit (𝑥 ≪ 1) reproduces the noninteracting 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.

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 5.3. 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 structures 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 [133,[243][244][245]. The corresponding "Coulomb repulsion" 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 [START_REF] Imada | Metal-insulator transitions[END_REF], 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. When decreasing 𝑥, this third band (see 𝑥 = 0.30 in the figure 5.3) looks like the 𝑥 = 0.01 band but distorted. This is consistent with previous argument about the origin of the third band since in the both cases, most of the sites are non-Kondo. Thus in the extreme dilute case, the d.o.s of non-interacting 2D square lattice is recovered while the lower and the upper Kondo-related bands disappear. 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.

Evidence for a new critical concentration at low 𝑇 𝐾

We now focus on the output of spectral function obtained for the relatively small Kondo coupling case. Figure 5.4 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 branches merge to form a single branch structure along with the disappearance of the pseudogap. This occurs at a critical concentration 𝑥 ★ which depends on the strength of the Kondo interaction. We find that 𝑥 ★ ≪ 𝑛 𝑐 at very small 𝑇 𝐾 , and 𝑥 ★ → 𝑛 𝑐 when 𝑇 𝐾 approaches around 𝑊/10.

Results: Fermi surface

In this section we analyze the experimental signatures in the Fermi surfaces obtained from the spectral function 𝒜(k, 𝑖𝜔 = 𝑖0 + ). Figure 5.5 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 [START_REF] Fujimori | Band structures of 4f and 5f materials studied by angle-resolved photoelectron spectroscopy[END_REF]135,[222][223][224][225][226][227][228][229][230]. It can be well understood in terms of the Luttinger theorem, which stipulates that all fermionic degrees of freedom participate in the formation of the Fermi liquid ground state.

Fermi surface in the Kondo phases: 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 [136]. In this case, the spectral function corresponds to relatively long lifetime quasiparticles with a welldefined Fermi surface (excepted at the transition). Figure 5.6 shows clearly this discontinuity of Fermi surface at 𝑇 𝐾 /𝑊 = 0.175. In this case, 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 along 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 spectral functions (see section 5.3) at 𝑥 ★ characterizing the merging of two spectral function branches. However, the Fermi surfaces at figure 5.5 do not present clear signatures of any characterized feature at or around this critical concentration 𝑥 ★ . This is not surprising since the electronic excitation spectrum is not accessible from the Fermi surfaces.

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 5.7 (for 𝑛 𝑐 = 0.30, where the small Kondo coupling ground state is ferromagnetic) and in figure 5.8 (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 as in [246] . 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 to 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 coexist with Kondo effect. For such states, we expect coherence breakdown signatures that might be concomitant with Fermi surface reconstructions resulting from magnetic order.

Results: emergence of two transitions in the paramagnetic Kondo phases

The phase diagram of the KAM is depicted in figure 5.9. It was obtained by comparing the energies of each phase considered: paramagnetic Kondo, antiferromagnetic and ferromagnetic phases. From these 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 The transition observed in the Fermi surfaces (see section 5.4) at 𝑥 = 𝑛 𝑐 for sufficiently large values of Kondo coupling is characterized by a discontinuity of Σ 𝐴𝑙𝑙𝑜 𝑦 (𝜔 = 0) (solid line in figure 5.9). 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 [136] 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 ≳ 𝑊 10 . Therefore, in order to observe a signature of coherence breakdown at 𝑥 = 𝑛 𝑐 from ARPES experiments on 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. 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.

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 (dotted line in figure 5.9, see also figure 5.10).

Whereas, the imaginary part, Σ ′′ 𝐴𝑙𝑙𝑜 𝑦 (0), which is relatively small at strong cou- pling, becomes large around this crossover at smaller coupling (see color/blackwhite gradient in figure 5.9).

In agreement with the spectral function analysis of section 5.3 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 ★ , the effective mass 𝑚 ★ vanishes continuously. This transition separates the very dilute regime (𝑥 < 𝑥 ★ ) with 𝑚 ★ > 0 from an intermediate regime (𝑥 ★ < 𝑥 < 𝑛 𝑐 ) characterized by 𝑚 ★ < 0 (see inset of figure 5.10 and dashed line in figure 5.9). From our numerical data, we found a power law relation 𝑇 𝐾 /𝑊 ∝ (𝑥 ★ ) 𝛾 with an exponent 𝛾 < 1 and the exponent itself depends on the filling 𝑛 𝑐 .

We found 𝛾 = 0.71, 0.78 and 0.83 for 𝑛 𝑐 = 0.30, 0.70 and 0.90 respectively as shown in the figure 5.12. This transition produces signatures in spectral function (see section 5.3) where two branches merge to one branch at 𝑥 ★ (see figure 5.4). Meanwhile, Fermi surface structures (see figure 5.5) do not show a clear signature of 𝑥 ★ . Keeping in mind that the dispersion relation is 𝜔 + 𝜇 -Σ 𝐴𝑙𝑙𝑜 𝑦 (𝜔) = 𝜖 k , the frequency dependence of the real part 𝜔 -Σ ′ 𝐴𝑙𝑙𝑜 𝑦 (𝜔) + Σ ′ 𝐴𝑙𝑙𝑜 𝑦 (0) is represented in figure 5.13. We can thus interpret the transition at 𝑥 = 𝑥 ★ 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 ★ ) to locally non-monotonous (at 𝑥 > 𝑥 ★ ). 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 5.3. This intermediate state is thus a precursor to the coherent state which is realized around 𝑥 ≈ 1. It is very interesting to see that such a pre-coherent state may start being formed at a relatively small concentration 𝑥 ★ .

Imaginary part of self-energy Σ ′′ (0)/𝑇 𝐾 is presented in the figure 5.11. We quickly remark that Σ ′′ (0)/𝑇 𝐾 < 0, which shows that our calculation respects the causality. We could also observe that the Σ ′′ (0)/𝑇 𝐾 increases considerably with increasing 𝑇 𝐾 while getting maximum value at 𝑥 = 𝑛 𝑐 . This shows us that finite lifetime of quasiparticle is minimum at 𝑥 = 𝑛 𝑐 . However, two very important features could make the transition at 𝑥 ★ observable experimentally: first, the imaginary part of Σ ′′ 𝐴𝑙𝑙𝑜 𝑦 (0) remains relatively small around 𝑥 ★ (see color/black-white gradient in figure 5.9 and figure 5.11). We can thus expect that the excitations are long lifetime quasiparticles that could be revealed by photo-emission. Secondly, 𝑥 ★ 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 a 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 oneelectron excitation spectrum, we may expect signatures in other sorts of experiments that could be realized under pressure, e.g., Raman spectroscopy. 𝐴𝑙𝑙𝑜 𝑦 (0) for 𝑛 𝑐 = 0.70 and 𝑇 𝐾 = 0.019. Top: for 𝑥 in the vicinity of the critical point 𝑥 ★ = 0.08 which is characterized by 𝑚 ★ = 0, we observe the emergence of a non-monotonicity at low energy. This leads to the gradual formation of a multiplebranches dispersion for 𝑥 > 𝑥 ★ . Center: for 𝑥 in the vicinity of 𝑛 𝑐 , the maximum is realized at 𝜔 < 0 for 𝑥 < 𝑛 𝑐 and at 𝜔 > 0 for 𝑥 > 𝑛 𝑐 resulting in a second change of sign of 𝑚 ★ . Bottom: for 𝑥 close to 1, we observe signatures of the singularity

Σ 𝐾 (𝜔) = 𝑟 2
𝜔+𝜆 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.

Discussion

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 ) [136]. This Lifshitz-like transition may be observed in ARPES experiments through the analysis 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 enlarged with 𝑥, in connection with the 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 ★ < 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, the negative effective mass does not mean instability, but it is rather a Fermi liquid phase. 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 ★ .

We also analyze the effects of decoherence resulting from disorder, which may spoil the 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 the Fermi surface at 𝑥 = 𝑛 𝑐 for small coupling. We still expect relatively well-defined quasiparticles around 𝑥 = 𝑥 ★ even though contrary to the transition at 𝑥 = 𝑛 𝑐 , the transition at 𝑥 = 𝑥 ★ cannot be observed directly from the Fermi surface structures. We considered an infinite coordination number, the fluctuations in neighboring atoms configurations [138] in our calculations are neglected. This might induce fluctuations of the local effective Kondo hybridization [165,166,248], resulting in additional broadening of spectral function. 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 on Kondo alloys. Since we predict that 𝑥 ★ 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 ).

Chapter 6

Local potential scattering and charge inhomogeneity in Kondo alloys

In this chapter, we will present our results on local potential scattering and charge inhomogeneities in order to characterize the dilute and dense Kondo regimes. Similar to the previous chapter 5, here we will analyze different quantities with large Kondo coupling and also with Kondo coupling for various electronic filling upon the dilution of Kondo impurities. This will permit us to further characterize the critical concentrations at 𝑥 = 𝑛 𝑐 and 𝑥 = 𝑥 * seen through the photo-emission spectrum.

Introduction

Local potential scattering

We define local potential scattering (LPS) 𝑆 𝑎 (𝑖𝜔) as

𝐺 𝑐𝑐 𝑎 (𝑖𝜔) = 𝐺 0 (𝑖𝜔 + 𝑆 𝑎 (𝑖𝜔)) , (6.1)
where the index 𝑎 denotes either 𝒦 -site or 𝒩-site, 𝐺 0 (𝑖𝜔) = 1/(𝑖𝜔 -𝜖 k ) is noninteracting Green's function and 𝐺 𝑐𝑐 𝑎 (𝑖𝜔) is local Green's function for conduction electrons as expressed in the equations (3.53 -3.54). 𝑆 𝑎 (𝑖𝜔) is a site dependent complex quantity with its real part can be seen as an effective energy level of conduction electron on a given site with respect to the non-interacting bandwidth. By comparing this real part with the chemical potential 𝜇 0 (𝑛 𝑐 ) for non-interacting electrons with 𝑛 𝑐 electronic filling as defined by the Eq. ( 6.2), we can deduce hole-dopant or particle-dopant nature of the quasiparticles.

𝑛 𝑐 = ∫ 𝜇 0 (𝑛 𝑐 ) -𝑊/2 𝜌 0 (𝜔)𝑑𝜔 (6.2)
Furthermore, the imaginary part of 𝑆 𝑎 (𝑖𝜔) is linked with the quasiparticle's INHOMOGENEITY IN KONDO ALLOYS lifetime. Thus, one can extract the information about localized or itinerant behavior of 𝑓 -electrons depending upon the position of the 𝑆 𝑎 (𝑖𝜔) to the non-interacting bandwidth.

In this chapter, we will only concentrate on static part of LPS at zero frequency. Thus, we define

𝑆 𝑎 (0) = 𝑆 ′ 𝑎 + 𝑖𝑆 ′′ 𝑎 , (6.3)
where 𝑆 ′ 𝑎 and 𝑆 ′′ 𝑎 are real and imaginary part of 𝑆 𝑎 (0). Local potential scattering was previously studied for Bethe lattice in [138] in the Kondo alloy model using two complementary DMFT and stat-DMFT approaches while taking account of structural disorder fluctuations due to impurity substitution. In their study, they found that the real part of the LPS can shift outside the noninteracting electronic bandwidth while depleting Kondo sites. They also found a clear signature of Kondo lattice coherence breakdown 𝑥 = 𝑛 𝑐 marking emergence of two Fermi liquid phases: a coherent Fermi liquid for 𝑥 > 𝑛 𝑐 and a local Fermi liquid 𝑥 < 𝑛 𝑐 at strong 𝐽 𝐾 .

Charge inhomogeneity

It was first in 1934, Wigner [249] introduced the concept of charge ordering, the long-range ordered pattern of electron density, in the gas of electrons. Following its first introduction, numerous systems [250] have been found showing charge ordering. In strongly correlated systems, charge ordering is found in transition metal compounds like in manganates [251,252]. Again in SCES, charge ordering is found to cause multiferroicity [253] in some groups of strongly correlated systems. Regarding 𝑓 -electron systems, the charge ordering is less common but can occur in the systems where the intersite Coulomb interaction may be strong enough to lead to a 4f-charge disproportionate like in Yb 4 As 3 [254,255].

Charge order was studied previously in Kondo lattice system [208] at quarter filling using DMFT where the paramagnetic charge-ordered state was found for small coupling strengths for a bipartite Bethe lattice, and it vanishes as a first-order phase transition for strong coupling. In this study, the charge order was found to be an insulator. Further, charge order was also studied in a two-dimensional Kondo Lattice Model [210] using two complementary approaches: variational Monte Carlo method [START_REF] Becca | Quantum Monte Carlo approaches for correlated systems[END_REF] for the ground state and cellular dynamical mean-field theory [256]. In their study, they found charge order as an insulator at quarter filling.

In this part of this chapter, inspired from previous studies of charge order, we will analyze charge inhomogeneities present between Kondo and non-Kondo sites. Thus, we define the charge occupation for each type of site as

𝑛 𝑎 = 1 𝛽 𝑖𝜔 𝐺 𝑐𝑐 𝑎 (𝑖𝜔) , (6.4

Large 𝑇 𝐾 case

At first, we focus on a large Kondo coupling scenario with 𝑇 𝐾 /𝑊 = 0.169 for 𝑛 𝑐 = 0.70 where evidence of Lifshitz-like transition was observed through the analysis of Fermi surface structures (see section 5.4). Figure 6.1 represents the results for the LPS at Fermi level for large coupling. An obvious limit to analyze should be the Kondo lattice limit (𝑥 ≃ 1). At Kondo lattice limit (𝑥 ≃ 1), the real part 𝑆 ′ 𝒩 is situated outside the electronic bandwidth with finite positive imaginary part, and the real part 𝑆 ′ 𝒦 is inside the electronic bandwidth with the negligible but non-negative imaginary part. One should notice that 𝑆 ′ 𝒦 is almost constant for the concentrations 𝑥 > 𝑛 𝑐 . This shows the extent of coherence of coherent dense regime up to the concentration 𝑥 > 𝑛 𝑐 as in [138] for Bethe lattice at finite 𝐽 𝐾 . For 𝑥 = 1.00, 𝑆 ′ equals to the chemical potential of non-interacting electrons 𝜇 0 (𝑛 𝑐 + 1), with 𝑛 𝑐 + 1 electronic filling. This corresponds to the large Fermi surface verifying the Luttinger 'theorem' [134] where all the fermionic degree of freedoms participates in the formation of the Fermi surface. Upon dilution, at 𝑥 = 𝑛 𝑐 , both quantities 𝑆 𝒦 and 𝑆 𝒩 presents discontinuities. INHOMOGENEITY IN KONDO ALLOYS At 𝑥 < 𝑛 𝑐 , 𝑆 ′ 𝒦 crosses out the non-interacting bandwidth through the upper limit while 𝑆 ′ 𝒩 enters inside the non-interacting bandwidth through lower limit. Interestingly, 𝑆 ′′ 𝒦 acquire a finite value whereas 𝑆 ′′ 𝒩 cancels out. For 𝑥 ≪ 𝑛 𝑐 , 𝑆 ′ 𝒩 coincide with the chemical potential 𝜇 0 (𝑛 𝑐 ) corresponding to a small Fermi surface.

In contrast to the previous calculations in [138] for the Bethe lattice, here we did not observe the change of sign in the imaginary part of LPS for either 𝒦 -site or for 𝒩-site.

Low 𝑇 𝐾 case

From our previous analysis of spectral function in the low Kondo coupling case, we had identified a new critical concentration 𝑥 * where the cancellation of effective mass happens. Here, we present and analyze results for the same value of 𝑇 𝐾 /𝑊 = 0.019 and for 𝑛 𝑐 = 0.70. Figure 6.2 present the LPS for 𝒦 -site and 𝒩-site for low 𝑇 𝐾 case.

As soon as we compare the figure 6.2 for low 𝑇 𝐾 case with the figure 6.1 for large 𝑇 𝐾 , we find two clear differences: i) the discontinuity at 𝑥 = 𝑛 𝑐 for both 𝑆 ′ 𝒦 and 𝑆 ′ 𝒩 disappears and ii) 𝑆 ′ 𝒩 is always situated inside the electronic bandwidth. Like for the large 𝑇 𝐾 , at 𝑥 = 1.00, 𝑆 ′ 𝒦 corresponds to 𝜇 0 (𝑛 𝑐 + 1) showing that the Luttinger theorem is still verified. Even though the discontinuity at 𝑥 = 𝑛 𝑐 disappears, 𝑆 ′ 𝒦 crosses the upper limit shortly after 𝑥 = 𝑛 𝑐 marking the extent of coherence of Kondo lattice. Thus, this disappearance of discontinuity at 𝑥 = 𝑛 𝑐 translates the transformation of dilute-dense transition into a crossover. Upon further dilution of magnetic impurities at 𝑥 ≪ 𝑛 𝑐 , 𝑆 ′ 𝒩 corresponds to the chemical potential 𝜇 0 (𝑛 𝑐 ) and also 𝑆 ′′ 𝒩 cancels out. LPS clearly presents evidence of Liftshitz-like transition, but it was unable to present any signatures of the transition at 𝑥 = 𝑥 ★ .

Results: Charge inhomogeneity

In this section, we present our results on the average charge occupation and its average distribution over the randomly distributed 𝒦 -sites and 𝒩-sites to study further the critical concentrations: 𝑥 = 𝑛 𝑐 and 𝑥 ★ in paramagnetic Kondo phase. We also analyze the distribution of charge between the Kondo sites and non-Kondo sites concerning the impurity concentration 𝑥 and electronic filling 𝑛 𝑐 . It permits us to study the distribution of charge inhomogeneity with the alloying effect.

Let us first start by analyzing the infinite 𝐽 𝐾 → ∞ case.

Discussion

In this chapter, we have analyzed local potential scattering and charge inhomogeneity with substitution to find new signatures of the transitions at 𝑥 = 𝑥 * and 𝑥 = 𝑛 𝑐 . At first, we can observe that for large Kondo coupling at 𝑥 ≳ 𝑛 𝑐 , 𝑆 ′ 𝒩 is situated outside and below of non-interacting electronic bandwidth, showing that 𝒩-sites are empty. However, 𝑆 ′ 𝒦 is found inside the non-electronic bandwidth, with 𝑆 ′ 𝒦 corresponding to 𝜇 0 (𝑛 𝑐 + 1) at 𝑥 = 1.00. This verifies the Luttinger theorem and shows that the occupied Kondo sites behave like particle doping for the concentrations 𝑥 ≳ 𝑛 𝑐 even for the low 𝑇 𝐾 . Furthermore, 𝑆 ′′ 𝒦 is very small, indicating the quasiparticles are welldefined with a long lifetime on Kondo sites. This result is concordant with the large Fermi surface and self-energy found in the previous chapter 5 in the dense regime. Upon dilution for large Kondo coupling, at 𝑥 ≃ 𝑛 𝑐 , 𝑆 ′ 𝒦 shifts outside above while 𝑆 ′ 𝒦 enters inside the non-interacting bandwidth. This shows that all the 𝒦 -sites are fully occupied and don't contribute to the transport properties. On the other hand, at the very dilute case, 𝑆 ′ 𝒩 situated inside the non-interacting bandwidth corresponds to 𝜇 0 (𝑛 𝑐 ), confirming the small Fermi surface. In this case, the quasiparticles in 𝒩-sites are well-defined since 𝑆 ′′ 𝒦 ≃ 0. However, as expected, from the charge occupation, INHOMOGENEITY IN KONDO ALLOYS the probability of occupation of non-Kondo sites increases with decreasing 𝑇 𝐾 even for the dense regime. The transition between dilute and dense Kondo regimes at large 𝑇 𝐾 happens at 𝑥 = 𝑛 𝑐 is marked by a discontinuity in local potential scattering and a sudden rise in occupations of non-Kondo sites. We again confirm that this transition at 𝑥 = 𝑛 𝑐 becomes a mere crossover for intermediate to low 𝑇 𝐾 , since the discontinuities in local potential scattering disappear and the kinks in charge occupations become smoother. In this study, either local potential scattering or charge occupations gave any signatures of the transition at 𝑥 = 𝑥 ★ or negative effective mass of quasiparticles. This can be understood since the transition at 𝑥 = 𝑥 ★ is related to the electronic excitation spectrum, thus not accessible at zero energy.

Chapter 7

Conclusion and Perspectives

In this part of the thesis, we have thoroughly studied 𝑓 -electron substitution in Kondo alloys which permitted us to answer multiple pending questions related to lattice coherence and its breakdown.

For this study, we used DMFT to treat the disorder in combination with the mean-field coupling of the Kondo interaction. Firstly, the study of the paramagnetic phase was carried out by using matrix DMFT, which was already developed previously. However, the study of magnetically ordered phases for the binary alloy with substitution needed its generalization. Thus, we generalized this formalism in order to incorporate magnetically ordered phases with the disorder. On the purely theoretical side, this generalization is an original work of this thesis. Using the above methods, we constructed at first the phase diagram of the Kondo lattice (𝑥 = 1.00) for the 1D chain, 2D square, and 3D cubic lattices. For this particular study, we considered a wide range of electronic fillings 𝑛 𝑐 along with low to strong Kondo couplings. The results obtained for all three lattices were consistent with previous studies proving the pertinence of our numerical approach. Thereafter, we diluted the impurity concentration for only 2D square system by tuning 𝑥 for 𝑛 𝑐 = 0.30, 0.70, and 0.90. We obtained a Doniach-like phase diagram with Kondo substitution marked by magnetically ordered phases dominating at low 𝐽 𝐾 .

Even though this study is theoretical, the experimental relevance of our results was not left behind. Our magnetic phase diagrams of Kondo alloys were compared with the experimental data of Ce-La substitution for various cerium-based heavyfermions. Our comparisons were consistent, and it provided how the Kondo alloys with the magnetically ordered ground-state like in CeCu 2 Ge 2 [100] or the paramagnetic Kondo ground-state like in CePt 2 Si 2 [START_REF] Ragel | Effects of La dilution on the CePt 2 Si 2 Kondo lattice[END_REF] can resist the dilution of magnetic impurities.

After the generalization of Doniach's argument to Kondo substitution, our study focuses on the study of the breakdown of the coherent Kondo phase. For this, we analyzed photoemission spectra, local potential scattering, and charge inhomogeneity under dilution while varying Kondo coupling. Predicted results [136] were obtained for large Kondo coupling on the dense (𝑥 ∼ 1.00) and dilute (𝑥 ∼ 0.01), with large and small Fermi surfaces respectively. Furthermore, the analysis of local potential scattering on Kondo-sites (𝑆 ′ 𝒦 ) confirms the verification of the Luttinger theorem [134] at the dense regime for all the values of Kondo couplings. The transition between the dilute and dense regimes happens around 𝑥 = 𝑛 𝑐 for Kondo strength 𝑇 𝐾 /𝑊 ≳ 0.10. This transition is characterized by Lifshitz-like transition and the shifting of 𝑆 ′ 𝒦 outside and above the non-interacting bandwidth, revealing that the transport properties are assured only from the movement of electrons on the 𝒩sites. As 𝑇 𝐾 /𝑊 decreases, the probability of charge occupation at 𝒩-sites increases.

Thus, for 𝑇 𝐾 /𝑊 ≲ 0.10, a crossover happens between the dense and dilute regimes.

Surprisingly for intermediate to low Kondo strength, the crossover at 𝑥 ≃ 𝑛 𝑐 is followed by a vast region where quasiparticles acquire negative-effective masses with two-branches electronic dispersion. In this negative-mass region, 𝑆 ′ 𝒦 also shifts outside and above the non-interacting bandwidth, as seen for large 𝑇 𝐾 /𝑊. At the end of this negative effective mass region, we found the presence of a critical concentration 𝑥 * where the quasiparticle effective mass cancels out. This critical concentration 𝑥 * was characterized by the merging of two the spectral function bands. This result was completely unexpected. Compiling all the information, we constructed more detailed versions of the phase diagram for Kondo alloys for various 𝑛 𝑐 .

In the continuation of this work, one could explore the optical conductivity in CPA formalism [257][258][259][260][261][262][263][264] to find the signatures of the transitions at 𝑥 * and 𝑥 = 𝑛 𝑐 . Indeed, optical conductivity can provide additional information about particle-hole nature of quasiparticle, spectral weight [265] which is also liked with electron mass and also possible non-Fermi liquid behavior [266]. From a fundamental point of view, this work also suggests the possible emergence of exceptional points at 𝑥 ★ where two dispersive branches merge and 𝑚 ★ vanishes. Recently, the presence of exceptional points along with the Kondo effect was proposed on multiple occasions [START_REF] Michishita | Relationship between exceptional points and the Kondo effect in 𝑓 -electron materials[END_REF]267] by theoretical means. In appendix D, we propose an origin of negative mass. This has to be further developed and analyzed in the context of the exceptional point, which makes it another promising perspective of this thesis.

Our extension of matrix DMFT formalism only considers two commensurate magnetically ordered phases: Néel ordered antiferromagnetic phase and ferromagnetic phase. So, in future works, this work can be extended to include other commensurate and non-commensurate magnetically ordered phases. We have also omitted the possibility of having coexisting phases, which has been a subject of study in multiple theoretical studied [109,111,213,268] primarily for Kondo lattices. In future works, the effect of disorder and alloying could be studied on the coexistence of multiple phases.

Experimental consistencies of transition between dilute-dense paramagnetic Kondo phases still remain to be explored. An obvious choice would experiment like ARPES or dHvA can give direct access to the electronic structure and thus can provide di-rect evidence of our predicted transitions. There might be a need to apply external pressure to cross the transition temperature of 𝑥 = 𝑥 * and 𝑥 = 𝑛 𝑐 . Knowing that these experimental probes are not fully compatible with the applied external pressure, an alternative option would be Compton scattering. It can be conducted with respect to various experimental conditions like external pressure, atomic substitution, temperature, or magnetic field. More recently, Compton scattering experiments were successfully conducted on Kondo lattices CeRu 2 Si 2 [269] and YbRh 2 Si 2 [270] in order to explore the Fermi surfaces related to localized versus itinerant behavior of 𝑓 -electrons.

Part II 5f electrons in Uranium alloys

Chapter 8 Introduction

In continuation of our study of strongly correlated 𝑓 -electron materials, in this part of this thesis, we proceed to 5 𝑓 systems. In 5 𝑓 systems, due to their non-integer partially filled shells, multiple energy scales compete with each other: the exchange bandwidth, the 5 𝑓 bandwidth, the spin-orbit interaction, and intra-atomic 𝑓 -𝑓 Coulomb interaction. The interplay and competition between these interactions can give rise to a very complex phase diagram at low temperature with conventional phases like ferromagnetic [START_REF] Aoki | Review of U-based ferromagnetic superconductors: Comparison between UGe 2 , URhGe, and UCoGe[END_REF], antiferromagnetic [178], or conventional superconductivity, unconventional heavy-fermion superconductivity [START_REF] Aoki | Review of U-based ferromagnetic superconductors: Comparison between UGe 2 , URhGe, and UCoGe[END_REF]178] or even some sometimes enigmatic states [271]. This plethora of phases with very different microscopic mechanisms makes them interesting to study. Upon the 5 𝑓 -shell filling, the 𝑓 -electrons can be either localized, itinerant, or sometimes ambiguously can acquire both localized and non-localized characteristics [START_REF] Zwicknagl | The dual nature of 5f electrons and the origin of heavy fermions in U compounds[END_REF]250,273]. Let us focus, at first, only on the localized versus itinerant behavior in actinides. This can be visualized by comparing the Wigner-Seitz radius of actinides with transition metals and rare-earth metals. Figure 8.1 presents this comparison. With the increase in 5 𝑓 -electrons count, at first, the volume of actinides decreases similarly as transition metals, and then a sharp rise is noticed around Pu. After Pu, the Wigner-Seitz radius evolves similarly to rare-earth atoms. These two different tendencies mark both the itinerant and localized behavior of 5 𝑓 electrons, and the frontier between them lies near Pu. Additionally, clear evidence of these two different tendencies are also observed through various experiments where transuranium (primarily Np, Pu, and Am) compounds were found to have analogous features as the rare-earth compounds [274][275][276][277][278] and the 𝑓 -electrons in light actinides have the tendency to be itinerant. Similarly, we can observe that uranium lie between these two tendencies, and thus 5 𝑓 states can also be seen as intermediate states between 3𝑑 and 4 𝑓 states.

Nature of 5 𝑓 electrons in uranium based compounds

Hill criterion

In 1970, H. Hill [280] proposed a criterion based on 𝑓 -atom spacing to explain the formation of ground-state phase with localized or itinerant 𝑓 -electrons. Figure 8.2 shows the critical temperatures of various uranium-based compounds according to uranium-uranium atomic distance. From this figure, we can observe that phases with itinerant electrons occur at a short uranium-uranium distance, whereas the phases with localized electrons occur at a large uranium-uranium distance. Overall, this Hill limit between itinerancy and localization lies around 3.5 Å. To understand this criterion, we can look out to the 5 𝑓 wave function, which is more spatially extended than that of 4 𝑓 wave function (see figure 1.2). Thus, 5 𝑓 wave function can overlap with their neighboring sites to form coherent Bloch states within the Hill limit.

The itinerant scenario promoted by Hill criterion for UB 2 with U-U distance of 3.123 Å smaller than Hill limit was confirmed through angle-resolved photoelectron spectroscopy [281] where 5 𝑓 states participate in the formation of Fermi surfaces. On the other hand, the localized 5 𝑓 states were found on UPd 3 [282,283] with large U-U separation.

Heavy fermions

From the figure 8.2, we can observe that some compounds like UPt 3 , UBe 13 do not obey the Hill criterion even with a large U-U distance. More interestingly, they can gain quasiparticle mass hundreds of times of bare electron mass [START_REF] Ott | UBe 13 : An Unconventional Actinide Superconductor[END_REF]284,285]. Furthermore, the discovery of heavy-fermion with metallic low-temperature behavior [284,286] with narrow 5 𝑓 bands in these compounds came as a surprise. Until now, ten uranium-based heavy fermions are found: UBe 13 , UPt 3 , UPd 2 Al 3 , UNi 2 Al 3 , URu 2 Si 2 , UGe 2 , UIr, UCoGe, URhGe, and UTe 2 . One can be tempted to relate the heavy fermionic behavior of 4 𝑓 compounds with those of 5 𝑓 compounds. However, here we deal with intermediate valent compounds [287] with no distinct valence peaks [288], thus the Kondo mechanism treated in the first part of this thesis is excluded. Thus, this raises the question about the microscopic mechanism that leads to heavy fermions.

Microscopic mechanism: dualism of 𝑓 -electron

In this section, we present the dual nature of 5 𝑓 -electrons as a microscopic mechanism that can lead to the heavy fermionic behavior in uranium-based compounds. In our case, a part of 5 𝑓 electrons is itinerant, whereas the remaining remains localized. The itinerant part comes from the hopping mediated through the hybridization with conduction states. Thus, the itinerant part participates in the formation of the Fermi surface. On the other hand, the localized part gets scatters off giving effective mass enhancement, similarly as for Pr metal where heavy-mass of conduction electrons results from virtual crystal-field excitations of localized 4 𝑓 2 electrons [289]. This dual model [290] showed some faithful results on effective masses and dHvA frequencies on UPt 3 [START_REF] Zwicknagl | Microscopic description of origin of heavy quasiparticles in UPt 3[END_REF] and UPd 2 Al 3 [291] when compared with the experiment results [292,293]. Furthermore, the dual model applied to UPt 2 Si 2 [294] was consistent with the experimental observation of probable field-induced first-order Lifshitz-type transition.

Apart from these successful applications, a clear sign of dual nature of 𝑓 electrons for UPd 2 Al 3 , UPt 3 was observed by identifying the high-resolution photoemission spectra [287, 295] with a localized system UPd 3 and an itinerant system UB 2 . More recently, two different types of X-ray experiments were employed to UM 2 Si 2 (M=Pd, Ni, Ru, Fe), which showed both localized and itinerant 5 𝑓 states with different levels of itineracy for each compound depending upon their 5 𝑓 band-filling. The co-existence of superconductivity along with ferromagnetism found in UCoGe and URhGe [296] is also compatible with this vision of the dual character of 𝑓 -electrons.Similarly, Lifshitz transitions were observed in UCoGe [297] indicates further the dual picture of 5 𝑓 states. Furthermore, in these heavy fermionic systems, 𝑓 -electron count was found to be somewhere between 2 and 3 [287, 298-300], supports our dual picture.

Phenomenological modeling of duality

Duality was a direct or indirect subject of study on multiple studies through various approaches. We could note particularly, the case when orbital-selective localization: electrons get Mott localized in particular orbitals while other remains delocalized on other orbitals. Several mechanisms were proposed that can lead to the orbitalselective localization: crystal-field splitting of two bands of equal bandwidth [301] where orbital-selective localization can occur under doping, crystal field splitting of multi-band systems with unequal kinetic energy of electrons, band hopping anisotropies [302], the Hund's rule coupling enhances the orbital differentiation [START_REF] Georges | Strong correlations from Hund's coupling[END_REF]303].

The above examples concern primarily 3𝑑 and 4𝑑 systems with large crystalfield, however the crystal-field effect on 5 𝑓 systems is rather small or irrelevant when compared to other interactions. Thus, here we propose that the competition between intra-atomic correlations and hopping leads to orbital-selective localization. More precisely, intra-atomic correlations resulting from the anisotropic part of the Coulomb repulsion may considerably enhance pre-existing hopping anisotropies where the states with sub-dominant hopping channels get localized.

Model Hamiltonian

In this section, we present our model Hamiltonian. Due to the large nuclear charge in uranium, the spin-orbit interaction 1 is of the order of 1 eV. This large spin-orbit interaction makes the electrons couple with their individual angular momentum 𝑗 𝑗coupling instead of 𝐿𝑆-coupling. With 𝑙 = 3 being orbital angular momentum and 𝑠 = 1 2 being the spin and large spin-orbit, we use total angular momentum 𝑙 ⊗ 𝑠 = (𝑗 =

2 ) ⊕ (𝑗 = 7 2 ) basis as single-particle states instead of 𝑙 = 3, magnetic quantum number 𝑚 and spin 𝑠, noting that both basis are equivalent and transformation between two basis are unitary. Figure 8.3 show a schematic view is of this spin-orbit coupling. Since the spin-orbit splitting between 𝑗 = 5/2 and 𝑗 = 7/2 is around 1 eV, we do not consider the states with 𝑗 = 7/2 [304]. Thus, we write our Hamiltonian on 𝑗 𝑗 𝑧 basis set, with 𝑗 𝑧 being azimuthal quantum number 8.1) where 𝑐 † 𝑎 𝑗 𝑧 (𝑐 𝑎 𝑗 𝑧 ) denotes creator(annihilator) operator which creates(annihilate) a 5felectron on site 𝑎 with angular momentum 𝑗 = 5/2 and 𝑧-components 𝑗 𝑧 = -5/2...5/2, 𝑡 𝑗 𝑧 is the nearest site hopping in orbital index 𝑗 𝑧 . First term describes the Kinetic energy operator whereas the 𝐻 𝐶𝑜𝑢𝑙. describes the local Coulomb repulsion which is identical as in [START_REF] Efremov | Dual nature of 5 𝑓 electrons: Effect of intra-atomic correlations on hopping anisotropies[END_REF] will be defined more precisely in the chapter 9.

ℋ = <𝑎,𝑏>,𝑗 𝑧 𝑡 𝑗 𝑧 𝑐 † 𝑎 𝑗 𝑧 𝑐 𝑏 𝑗 𝑧 + 𝐻 𝐶𝑜𝑢𝑙. , (

Approaches

We intend to solve the Hamiltonian (8.1), but full microscopic treatment becomes very tedious since the Hilbert space increases exponentially with the number of uranium sites. However, the treatment becomes possible for the small clusters like in [START_REF] Efremov | Dual nature of 5 𝑓 electrons: Effect of intra-atomic correlations on hopping anisotropies[END_REF], where this model Hamiltonian was solved for two, three, and four site clusters which captured the essential physics. [START_REF] Efremov | Dual nature of 5 𝑓 electrons: Effect of intra-atomic correlations on hopping anisotropies[END_REF]305] Similarly, approaches with approximations and simplifications can also be applied. For example, cluster perturbation theory was employed for linear chain [250,306] to calculate 5 𝑓 spectral functions. In their study, both dispersive quasiparticle peaks near Fermi energy and incoherent local excitations were seen at the low-energy part of the spectra, suggesting the dual nature of 5 𝑓 electrons. Even though small cluster calculations give essential qualitative physics related to duality, the physical properties in the thermodynamic limit remain to be explored. Thus, in this present work, we adopt a slave-boson mean-field approach to find the ground-state phase diagram.

An outline to slave-bosons approaches

In this section, we will give details on various slave-bosons approaches with their advantages and drawbacks. The slave-boson representations were first introduced by and Schwinger [308] where spins operators may be represented by Bose operators. But, it was Barnes [START_REF] Barnes | New method for the Anderson model. II. The U=0 limit[END_REF]309] who first introduced a representation involving both auxiliary bosonic and fermionic operators for Anderson model [310]. This particular method was further generalized to cure its defaults. Thus, there is not only one possible slave boson representation, but rather multiple versions of them exist [START_REF] Kotliar | New Functional Integral Approach to Strongly Correlated Fermi Systems: The Gutzwiller Approximation as a Saddle Point[END_REF][START_REF] Lechermann | Rotationally invariant slave-boson formalism and momentum dependence of the quasiparticle weight[END_REF]309,311] which are more or less connected to each other. Below, we present two versions of slave-bosons representations in order to present a general idea behind the slave-bosons approaches.

Barnes' representation

In this slave-bosons approach [START_REF] Barnes | New method for the Anderson model. II. The U=0 limit[END_REF]309], the idea is to decompose the local electronic excitations into spins and charge degree of freedom by using bosonic and fermionic operators. With this idea, Barnes mapped the four physical states of an electron by using two bosonic and one fermionic operators. The table 8.1 resumes the mapping.

Initial physical states

Barnes's slave boson representation 𝑖𝜎 obey Fermi anti-commutation relations. One could notice that the empty site |0⟩ is created from a vacuum state rather than assumed as a pre-existing state. The use of auxiliary operators to represent spin and charge components enlarges the Fock space than that of the original physical states. In order to eliminate the non-physical states and to ensure the physical meaning of results, the enlarged Fock space should be span back as original. This can be done by enforcing by constraint below

|0 𝑖 ⟩ ↦ → |𝑣𝑎𝑐⟩ | ↑ 𝑖 ⟩ ↦ → 𝑐 † 𝑖↑ |𝑣𝑎𝑐⟩ | ↓ 𝑖 ⟩ ↦ → 𝑐 † 𝑖↓ |𝑣𝑎𝑐⟩ | ↑↓ 𝑖 ⟩ ↦ → 𝑐 † 𝑖↑ 𝑐 † 𝑖↓ |𝑣𝑎𝑐⟩ |0 𝑖 ⟩ ↦ → 𝑒 † 𝑖 |𝑣𝑎𝑐⟩ | ↑ 𝑖 ⟩ ↦ → 𝑓 † 𝑖↑ |𝑣𝑎𝑐⟩ | ↓ 𝑖 ⟩ ↦ → 𝑓 † 𝑖↓ |𝑣𝑎𝑐⟩ | ↑↓ 𝑖 ⟩ ↦ → 𝑑 † 𝑖 |𝑣𝑎𝑐⟩
𝑒 † 𝑖 𝑒 𝑖 + 𝑑 † 𝑖 𝑑 𝑖 + 𝜎 𝑓 † 𝑖𝜎 𝑓 𝑖𝜎 = 1 . (8.2)
The constraint (8.2) is a completeness equation. The physical second quantized operators can be written in terms of projection operators as (8.3) where 𝑧 𝜎 = 1 or -1 for 𝜎 =↑ or ↓ respectively. With the help of auxiliary bosons and fermions and the equation ( 8.3), the physical creator (annihilator) operators can be expressed by as (8.4) This representation is used in multiple cases primarily when 𝑈 → ∞ [312, 313] because the double occupancy operators drop out. From the above equation (8.4), we can clearly observe that this slave boson representation mixes fermionic operators with bosonic operators. Charges are expressed in terms of bosons, whereas spin degrees are represented by fermion operators. The bosonic operators do not carry spin. Unequal treatment of spin and charge components might lead to unnecessary errors in any approximations [START_REF] Frésard | Unified slave boson representation of spin and charge degrees of freedom for strongly correlated Fermi systems[END_REF]. This disadvantage paves a path to a new slave boson representation where spin and charge degrees of freedom may be expressed in terms of bosons.

𝑐 𝑖𝜎 = |0 𝑖 ⟩⟨𝜎 𝑖 | + 𝑧 𝜎 | σ𝑖 ⟩⟨↑↓ 𝑖 | ,
𝑐 𝑖𝜎 = 𝑒 † 𝑖 𝑓 𝑖𝜎 + 𝑧 𝜎 𝑑 𝑖 𝑓 † 𝑖𝜎 .

Kotliar and Ruckenstein's representation

To overcome the separate bosonic and fermionic representation of charge and spin degree of freedom, Kotliar and Ruckenstien [72] introduced a new representation with two additional bosons linked with the spin degree of freedom. Thus, both the spin and charge degree of freedom of physical electron operators are represented by slave-boson. To do so, they introduced slave bosons to each of four states per site and a spin doublet of fermion operators for singly occupied sites. The table 8.2 resumes the mapping.

Initial physical states KR's slave boson representation

|0 𝑖 ⟩ ↦ → |𝑣𝑎𝑐⟩ | ↑ 𝑖 ⟩ ↦ → 𝑐 † 𝑖↑ |𝑣𝑎𝑐⟩ | ↓ 𝑖 ⟩ ↦ → 𝑐 † 𝑖↓ |𝑣𝑎𝑐⟩ | ↑↓ 𝑖 ⟩ ↦ → 𝑐 † 𝑖↑ 𝑐 † 𝑖↓ |𝑣𝑎𝑐⟩ |0 𝑖 ⟩ ↦ → 𝑒 † 𝑖 |𝑣𝑎𝑐⟩ | ↑ 𝑖 ⟩ ↦ → 𝑝 † 𝑖↑ 𝑓 † 𝑖↑ |𝑣𝑎𝑐⟩ | ↓ 𝑖 ⟩ ↦ → 𝑝 † 𝑖↓ 𝑓 † 𝑖↓ |𝑣𝑎𝑐⟩ | ↑↓ 𝑖 ⟩ ↦ → 𝑑 † 𝑖 𝑓 † 𝑖↑ 𝑓 † 𝑖↓ |𝑣𝑎𝑐⟩ Table 8.

2: Table resuming Kotliar and Ruckenstien's slave bosons representation

Here, 𝑒 † 𝑖 , 𝑝 † 𝑖𝜎 and 𝑑 † 𝑖 are bosonic operators associated with an empty site, a site with one electron with spin 𝜎 and doubly occupied site respectively, whereas 𝑓 † 𝑖𝜎 are auxiliary fermionic operators. Like in Barnes's approach (see section 8.4.1), the introduction of auxiliary operators enlarges the initial Hilbert space. In order to eliminate nonphysical states, constraints (8.5) and (8.6) should be enforced.

𝑒 † 𝑖 𝑒 𝑖 + 𝜎 𝑝 † 𝑖𝜎 𝑝 𝑖𝜎 + 𝑑 † 𝑖 𝑑 𝑖 = 1 , (8.5
)

𝑓 † 𝑖𝜎 𝑓 𝑖𝜎 -(𝑝 † 𝑖𝜎 𝑝 𝑖𝜎 + 𝑑 † 𝑖 𝑑 𝑖 ) = 0 . (8.6)
The constraint (8.5) is a completeness relation that translates either a site is empty (|0 𝑖 ⟩), singly occupied, or double occupied and the number of bosons per site is one. The constraint (8.6) translates that when a site is occupied it is either singly or double occupied. It also translates that the counting of 𝑓 † 𝑖𝜎 𝑓 𝑖𝜎 or 𝑝 † 𝑖𝜎 𝑝 𝑖𝜎 + 𝑑 † 𝑖 𝑑 𝑖 is equivalent. In this representation, the physical electron operators are expressed as (8.8) where σ is the opposite sign of 𝜎 and the relation z𝑖𝜎 = 𝑒 † 𝑖 𝑝 𝑖𝜎 + 𝑝 † 𝑖 σ 𝑑 𝑖 is the new renormalization factor for auxiliary fermions 𝑓 𝑖𝜎 .

𝑐 𝑖𝜎 = 𝑓 𝑖𝜎 (𝑒 † 𝑖 𝑝 𝑖𝜎 + 𝑝 † 𝑖 σ 𝑑 𝑖 ) (8.7) = z𝑖𝜎 𝑓 𝑖𝜎 ,
From the renormalization factor, one can define the quasiparticle weight as 𝑍 𝜎 = ⟨z † 𝑖𝜎 z𝑖𝜎 ⟩. Within mean-field approximation as 𝑒 𝑖 → 𝑒, 𝑝 𝑖𝜎 → 𝑝 𝜎 , and 𝑑 𝑖 → 𝑑, the quasiparticle weight is rewritten as 𝑍 𝜎 = (𝑒𝑝 𝜎 + 𝑝 σ 𝑑) 2 . One of the example of the use of this approach is that it can capture Mott metal-insulator transition. To do so, we can consider half-filled one-band Hubbard model in paramagnetic phase. 𝑍 𝜎 plays the role of mass renormalization and of quasiparticle residue. In this simple case, a localization of an electron can be obtained when quasiparticle weight 𝑍 𝜎 → 0 vanishes.

Different from previous approach, the mixing of fermionic and bosonic operators in the completeness relation is gone and thus faithfully describes the electrons moving on site 𝑖 to 𝑗. Even though KR's representation corrects the previous representation, the spin degree of freedom is not rotationally invariant. The transverse components of the spin operator may not be simply represented in the terms of auxiliary operators since 𝑆 𝑥,𝑦 is neither related to 1 2

𝜎𝜎 ′ 𝑓 † 𝜎 𝜏 𝑥(𝑦) 𝜎𝜎 ′ 𝑓 𝜎 ′ nor to 1 2 𝜎𝜎 ′ 𝑝 † 𝜎 𝜏 𝑥(𝑦)
𝜎𝜎 ′ 𝑝 𝜎 ′ . Thus, the spin representation |𝜎⟩ is not spin rotation invariant making it dependent on the choice of quantization axis in spin space. This shortcoming lead to the development of spin-rotation-invariant slave boson approach [START_REF] Frésard | Unified slave boson representation of spin and charge degrees of freedom for strongly correlated Fermi systems[END_REF].

Li, Wölfe and Hirschfeld's representation

Li, Wölfe, and Hirschfeld's slave boson representation (LWH) [START_REF] Frésard | Unified slave boson representation of spin and charge degrees of freedom for strongly correlated Fermi systems[END_REF]311] corrects the spin non-invariance problem that was stated in previous sections. In their approach, Li, Wölfe and Hirschfeld represented the operator product 𝑝 † 𝑖𝜎 𝑓 † 𝑖𝜎 of KR's mapping (8.2) as a composite particle of spin 1 2 . The LWH mapping is resumed in the table 8.3.

Initial physical states

LWH's slave boson representation (8.9)

|0 𝑖 ⟩ ↦ → |𝑣𝑎𝑐⟩ | ↑ 𝑖 ⟩ ↦ → 𝑐 † 𝑖↑ |𝑣𝑎𝑐⟩ | ↓ 𝑖 ⟩ ↦ → 𝑐 † 𝑖↓ |𝑣𝑎𝑐⟩ | ↑↓ 𝑖 ⟩ ↦ → 𝑐 † 𝑖↑ 𝑐 † 𝑖↓ |𝑣𝑎𝑐⟩ |0 𝑖 ⟩ ↦ → 𝑒 † 𝑖 |𝑣𝑎𝑐⟩ | ↑ 𝑖 ⟩ ↦ → 1 √ 2 {𝑝 † 𝑖↑↓ 𝑓 † 𝑖↓ + 𝑝 † 𝑖↑↑ 𝑓 † 𝑖↑ }|𝑣𝑎𝑐⟩ | ↓ 𝑖 ⟩ ↦ → 1 √ 2 {𝑝 † 𝑖↓↓ 𝑓 † 𝑖↓ + 𝑝 † 𝑖↓↑ 𝑓 † 𝑖↑ }|𝑣𝑎𝑐⟩ | ↑↓ 𝑖 ⟩ ↦ → 𝑑 † 𝑖 𝑓 † 𝑖↑ 𝑓 † 𝑖↓ |𝑣𝑎𝑐⟩
𝑒 𝑖 and 𝑑 𝑖 are associated with empty site and doubly occupied site whereas 𝑝 𝑖𝜎𝜎 ′ is associated with singly occupied. The first index 𝜎 in single-particle boson is associate with local state and the second index 𝜎 ′ is associated with the auxiliary fermion quasi-particle degree of freedom. Also, the bosons 𝑝 𝑖𝜎𝜎 ′ permits to connect the lowenergy quasiparticule excitations to its identical high-energy local counterpart as well as other local configurations such as a state with opposite spin configuration. Thus, this general structure renders spin rotationally invariant. Further, 𝑝 𝑖𝜎𝜎 ′ can be represented as

𝑝 𝑖𝜎𝜎 ′ = 1 2 𝜇=0,𝑥,𝑦,𝑧 𝑝 𝜇 𝜏 𝜇 𝜎𝜎 ′ , (8.10)
where 𝜏 𝜇 are Pauli matrices and 𝜏 0 it the unit matrix.

CHAPTER 8. INTRODUCTION

Knowing that the spin values of auxiliary pseudo-fermions 𝑓 𝑖𝜎 should be 1 2 , the possible spin values for 𝑝 𝑖𝜎𝜎 ′ bosons are 𝑆 = 0 or 𝑆 = 1. Accordingly, a scalar field spin singlet 𝑝 0 and a vector field spin triplet p 𝑖 = (𝑝 𝑖𝑥 , 𝑝 𝑖 𝑦 , 𝑝 𝑖𝑧 ) is defined for 𝑆 = 0 and 𝑆 = 1 respectively. The 𝑆 = 0 boson 𝑝 𝑖0 represent the charge degree of freedom of the spinor states whereas the 𝑆 = 1 boson p 𝑖 describes spin degree of freedom. However, one should note that these fields do not automatically represent the electron charge and spin operators because the density operators involve the square of the matrix p 𝑖 of the equation (8.9).

To recover the initial physical space, the auxiliary operators need to satisfy the following constraints

𝑒 † 𝑖 𝑒 𝑖 + 𝑑 † 𝑖 𝑑 𝑖 + 𝜎𝜎 ′ 𝑝 † 𝑖𝜎𝜎 ′ 𝑝 𝑖𝜎𝜎 ′ = 1 , (8.11
)

𝑓 † 𝑖𝜎 ′ 𝑓 𝑖𝜎 - 𝜎 1 𝑝 † 𝑖𝜎 1 𝜎 ′ 𝑝 𝑖𝜎𝜎 1 -𝛿 𝜎𝜎 ′ 𝑑 † 𝑖 𝑑 𝑖 = 0 , (8.12
)

𝑝 † 𝑖0 p 𝑖 + p † 𝑖 𝑝 𝑖0 -𝑖p † 𝑖 × p 𝑖 = 𝜎𝜎 ′ 𝑓 † 𝑖𝜎 𝜏 𝜎 ′ 𝜎 𝑓 𝑖𝜎 ′ . (8.13)
Enforcement of the completeness constraint (8.11) ensure that a site is either empty, singly occupied, or doubly occupied, whereas enforcement of the constraint (8.12) ensure the number of electrons matches the number of 𝑝 bosons and 𝑑 bosons. The constraint (8.13) ensures that the spin of the physical electrons matches the spin of the bosons. With this at hand, the electron operators may be written as

𝑐 † 𝑖𝜎 = 𝜎 ′ (𝑝 † 𝑖𝜎𝜎 ′ 𝑓 † 𝑖𝜎 ′ 𝑒 𝑖 + 𝜎𝜎 ′ 𝑑 † 𝑖 𝑓 𝑖𝜎 ′ 𝑝 𝑖 σ′ σ) = 𝜎 ′ 𝑅 † 𝑖𝜎𝜎 ′ 𝑓 † 𝑖𝜎 ′ , (8.14)
𝑐 𝑖𝜎 = 𝜎 ′ (𝑒 † 𝑓 𝑖𝜎 ′ 𝑝 𝑖𝜎 ′ 𝜎 + 𝜎𝜎 ′ 𝑝 † 𝑖 σ σ′ 𝑓 𝑖𝜎 ′ 𝑑 𝑖 ) = 𝜎 ′ 𝑓 𝑖𝜎 ′ 𝑅 𝑖𝜎 ′ 𝜎 , (8.15)
where

𝑅 𝑖𝜎 ′ 𝜎 = 𝑒 † 𝑖 𝑝 𝑖𝜎 ′ 𝜎 + 𝜎𝜎 ′ 𝑝 𝑖 σ′ σ 𝑑 𝑖 .
The operator 𝑅 𝑖𝜎 ′ 𝜎 describes the sum of processes: from a singly occupied site to an empty site and from a doubly occupied site to a singly occupied site with time-reserved spin. Furthermore, this formalism makes the foundation for rotationally-invariant slave-boson, which can be applied to the general multi-orbital systems [START_REF] Lechermann | Rotationally invariant slave-boson formalism and momentum dependence of the quasiparticle weight[END_REF] and we will be using the latter formalism throughout this thesis.

Other variations of slave-bosons techniques

In this section, we present a non-exhaustive selection of other variations of slavebosons. Each of the slave-boson variations was developed for specific purposes or as an extension of previous. We could start from the slave-boson developed by Coleman [314] for the mixed-valent impurity problem, which is inspired by Barnes' approach [309]. In his approach, the Hubbard operators [315] were replaced by a product of boson and fermion operators in order to avoid notorious algebraic calculations. Slave-boson techniques are also generalized to study superconductivity [316][317][318][319] or even Kondo cloud [320]. Further, Li, Wölfe, and Hirschfeld's representation in the section 8.4.3 has been generalized to multi-band models by Lechermann,Georges,Kotliar,and Parcollet [74] and also by Bünemann [321] where, e.g., the effects of multiple orbitals, orbital degeneracy, and the Hund's rule can be studied. In the past, multi-orbital RISB method was applied to study the strongly correlated real systems like UO 2 [322], iron chalcogenides [323] or Nd 1-𝑥 Sr 𝑥 NiO 2 [324] with good agreement with experimental data.

Motivations and objectives

The dual model presented by Zwicknagl et al. [10,325] (see section 8.3.1) gave some promising reproducing dHvA frequencies and core-level photoemission spectra [326], suggesting orbital-selective partial localization due to anisotropic bandwidths. Indeed, small cluster calculations [START_REF] Efremov | Dual nature of 5 𝑓 electrons: Effect of intra-atomic correlations on hopping anisotropies[END_REF] showed the partial localization (see figure 8.4) of 𝑓 -electrons, but their calculations suffered problems at non-interacting limit. Moreover, a full thermodynamic calculation is still missing. Thus, in this thesis, we consider all the 𝑓 -electron configurations: 𝑓 0 , 𝑓 1 , 𝑓 2 , 𝑓 3 , 𝑓 4 , 𝑓 5 and 𝑓 6 with non-integer valency, as suggested by multiple experiments. The local correlations will be treated through the rotationally-invariant slave-boson approach [START_REF] Lechermann | Rotationally invariant slave-boson formalism and momentum dependence of the quasiparticle weight[END_REF], and we will consider 𝑗 𝑗-coupling scheme while ignoring crystal field effects. This work is the continuation of the previous work conducted by , where he conducted the necessary analytical calculations. Here, we do not intend to perform 𝑎𝑏 -𝑖𝑛𝑡𝑖𝑜 calculation, but we plan to provide a possible microscopic description of the duality of 𝑓 -electron. Thus, in this thesis, we analyze orbitalselective Mott transitions and construct a phase diagram with partially localized phases, considering paramagnetic and ferromagnetic phases. Further, we will also analyze orbital-dependant electronic filling, quasiparticle weight, magnetization, and 𝑓 -electron configuration in terms of bandwidth anisotropies. Below, we present the organization of this part of the thesis.

Chapter 9 is dedicated to the theoretical details of our study. We will start by detailing the model Hamiltonian. Thereafter, we will present the rotationallyinvariant slave-boson approach and the mean-field approximations within. Finally, we will present our system of equations to be solved numerically.

Chapter 10 presents the numerical aspects of our study. We will give details about our local and global minimization schemes along with different routines used.

Chapter 11 is dedicated to the presentation of our results. At first, we present the result for the isotropic case, and thereafter, the effect of anisotropies in orbitaldependent electronic bandwidth will be studied in terms of orbital-selective Mott localization. Then, we will build and analyze our phase diagram with partially localized phases.

Chapter 12 present the conclusions of our study along with future prespectives.

Chapter 9

Model and rotationally-invariant slave bosons

In this chapter, we present the model, method, and approximations that we have used to study the duality of 5 𝑓 electrons. At first, we start by detailing the model Hamiltonian, thereafter the method rotationally invariant slave-boson approach (RISB), and the mean-field approximations along with the self-consistent equations.

Model Hamiltonian

Our model Hamiltonian to describe 5 𝑓 electrons is

ℋ = <𝑎,𝑏>,𝑗 𝑧 {𝑡 𝑗 𝑧 -𝜇𝛿 𝑎𝑏 }𝑐 † 𝑎 𝑗 𝑧 𝑐 𝑏 𝑗 𝑧 + 𝐻 𝐶𝑜𝑢𝑙. , (9.1)
where 𝑐 † 𝑎 𝑗 𝑧 (𝑐 𝑎 𝑗 𝑧 ) denotes creator(annihilator) operator which creates(annihilate) a 5felectron on site 𝑎 with angular momentum 𝑗 = 5/2 and 𝑧-components 𝑗 𝑧 = -5/2...5/2 and 𝜇 is the chemical potential fixing the 𝑓 -electron occupation 𝑛 𝑓 . We assume that the nearest site hopping 𝑡 𝑗 𝑧 is diagonal in the orbital index 𝑗 𝑧 . The first term of the Hamiltonian describes the kinetic, whereas the second term 𝐻 𝐶𝑜𝑢𝑙. reflects the Coulomb interaction. This Coulomb interaction is further expressed as

𝐻 𝐶𝑜𝑢𝑙. = 1 2 𝑎 𝑗 𝑧 1 ,𝑗 𝑧 2 ,𝑗 𝑧 3 ,𝑗 𝑧 4 ⟨𝑗 𝑧 1 𝑗 𝑧 2 | Û | 𝑗 𝑧 3 𝑗 𝑧 4 ⟩𝑐 † 𝑎 𝑗 𝑧1 𝑐 † 𝑎 𝑗 𝑧2 𝑐 𝑎 𝑗 𝑧3 𝑐 𝑎 𝑗 𝑧4 , (9.2)
with ⟨𝑗 𝑧 1 𝑗 𝑧 2 | Û | 𝑗 𝑧 3 𝑗 𝑧 4 ⟩ are Coulomb matrix elements which are evaluated as

⟨𝑗 𝑧1 𝑗 𝑧2 | Û | 𝑗 𝑧3 𝑗 𝑧4 ⟩ = 𝛿 𝑗 𝑧1 +𝑗 𝑧2 ,𝑗 𝑧3 +𝑗 𝑧4 𝐽 ⟨ 5 2 𝑗 𝑧1 5 2 𝑗 𝑧2 |𝐽𝐽 𝑧 ⟩𝑈 𝐽 ⟨𝐽𝐽 𝑧 | 5 2 𝑗 𝑧2 ⟩ = 𝐽 𝑈 𝐽 𝐶 𝐽𝐽 𝑧 5/2,𝑗 𝑧 1 ;5/2,𝑗 𝑧 2 𝐶 𝐽𝐽 𝑧 5/2,𝑗 𝑧 3 ;5/2,𝑗 𝑧 4 . (9.3)
where 𝐶 ... ... are the Clebsh-Gordon coefficients and 𝐽 denotes the total angular momentum and 𝐽 𝑧 = 𝑗 𝑧1 + 𝑗 𝑧2 = 𝑗 𝑧3 + 𝑗 𝑧4 .

Method: rotationally-invariant slave-bosons

In order to study the model Hamiltonian (9.1) and to treat the intra-atomic correlations, we use the rotationally invariant slave bosons approach (RISB) as introduced by Lechermann et al. [START_REF] Lechermann | Rotationally invariant slave-boson formalism and momentum dependence of the quasiparticle weight[END_REF]. An introduction to slave-boson approaches in the section 8.4 of chapter 8, while below we present a short overview of the RISB formalism and the equations related to it. We would like to point out that this work continues the theoretical and numerical work conducted by Duc-Ahn Le [327] based on the previous work.

Expanded Hilbert space and basis set

By taking account of quasiparticles orbital degree of freedom within RISB formalism, we introduce the auxiliary fermionic operators 𝑓 † 𝑎 𝑗 𝑧 , 𝑓 𝑎 𝑗 𝑧 associated each orbital 𝑗 𝑧 and bosonic operators 𝜙 † 𝐴,𝑛 , 𝜙 𝐴,𝑛 where 𝐴 is a basis of local Hilbert space and 𝑛 label Fock states. These newly introduced operators are used to replace the operators 𝑐 † 𝑗 𝑧 , 𝑐 𝑗 𝑧 and connect the initial physical Hilbert space with the auxiliary particle Hilbert space. To do so a basis set for physical Hilbert space as well as for auxiliary Hilbert space needs to be defined.

Initial physical states for

𝑀 = 0, 1 RISB representation |0⟩ ↦ → |𝑣𝑎𝑐⟩ |𝜂 𝑗 𝑧 ⟩ ↦ → 𝑐 † 𝑗 𝑧 |𝑣𝑎𝑐⟩ |0⟩ ↦ → 𝜙 † |𝑣𝑎𝑐⟩ |𝜂 𝑗 𝑧 ⟩ ↦ → 𝑗 ′ 𝑧 𝜙 † 𝑗 𝑧 𝑗 ′ 𝑧 𝑓 † 𝑗 𝑧 |𝑣𝑎𝑐⟩
= |𝜂 -5/2 𝜂 -3/2 𝜂 -1/2 𝜂 1/2 𝜂 3/2 𝜂 5/2 ⟩
represent 𝑀-particle states such as 𝑀 = 𝑗 𝑧 𝜂 𝑗 𝑧 . The choice of using multiplet-Fock basis is because the local interactions are diagonal in the multiplet basis, and it also simplifies the constraints. For now, we ignore the site-index to simply the notation and a multiplet in multiple-Fock basis is defined as

|Γ⟩ ≡ 1 √ 𝐷 Γ 𝑛 𝜙 † Γ𝑛 |𝑣𝑎𝑐⟩ ⊗ |𝑛⟩ , (9.4)
where 𝐷 Γ denotes the dimension of the sub-space of the Hilbert space with particle number identical to that of Γ. The relation (9.4)

Local constraints

The slave-boson mapping of enlarges the Hilbert phases, which needs to be reduced in order eliminate non-physical states. Thus, the local constraints need to be enforced. In our case, the constraints [START_REF] Lechermann | Rotationally invariant slave-boson formalism and momentum dependence of the quasiparticle weight[END_REF] are

Γ𝑛 𝜙 † Γ𝑛 𝜙 Γ𝑛 = 1 , (9.5)
Γ𝑛𝑛 ′ ⟨𝑛| 𝑓 † 𝑗 𝑧 𝑓 𝑗 𝑧 |𝑛 ′ ⟩𝜙 † Γ𝑛 ′ 𝜙 Γ𝑛 = 𝑓 † 𝑗 𝑧 𝑓 𝑗 𝑧 . (9.6)
The constraint (9.5) is the completeness equation ensuring that the physical states are single boson states as Eq. (8.11). The constraint (9.6) ensures the conservation of quasiparticle number by considering the correct slave-bosons with the same total particle charge sector in the physical and quasiparticle Hilbert space.

Representation of physical electron operators and RISB Hamiltonian

The action of creation operator in enlarged Hilbert space should be same as in physical Hilbert space. This lead us to the equation

⟨Γ|𝑐 𝑗 𝑧 † |Γ ′ ⟩ = ⟨Γ|𝑐 † 𝑗 𝑧 |Γ ′ ⟩ . (9.7)
The 𝑐 𝑗 𝑧 operators are expressed in term of auxiliary bosons and fermions as

𝑐 𝑗 𝑧 = 𝑅 † 𝑗 𝑧 [Φ] 𝑓 𝑗 𝑧 , (9.8)
where Φ is slave-boson matrix and the operator 𝑅 † 𝑗 𝑧 [Φ] is defined as

𝑅 † 𝑗 𝑧 [Φ] = γ𝑗 𝑧 [Φ] n𝑗 𝑧 [Φ](1 -n𝑗 𝑧 [Φ]) , (9.9)
with .11) By using the Eq. ( 9.8) and reintroducing the site-index, the Hamiltonian (9.1) is written in terms of auxiliary bosons as

γ𝑗 𝑧 [Φ] = ΓΓ ′ ,𝑛𝑛 ′ ⟨Γ|𝑐 † 𝑗 𝑧 |Γ ′ ⟩⟨𝑛| 𝑓 † 𝑗 𝑧 |𝑛 ′ ⟩𝜙 † Γ𝑛 𝜙 Γ ′ 𝑛 ′ , (9.10)
n𝑗 𝑧 = Γ𝑛 ⟨𝑛| 𝑓 † 𝑗 𝑧 𝑓 𝑗 𝑧 |𝑛⟩𝜙 † Γ𝑛 𝜙 Γ𝑛 . ( 9
𝐻 = - <𝑎,𝑏>,𝑗 𝑧 𝑡 𝑗 𝑧 𝑅 𝑗 𝑧 [Φ 𝑎 ]𝑅 † 𝑗 𝑧 [Φ 𝑏 ] 𝑓 † 𝑎 𝑗 𝑧 𝑓 𝑏 𝑗 𝑧 + 𝑎,Γ𝑛 𝐸 Γ 𝜙 † 𝑎,Γ𝑛 𝜙 𝑎,Γ𝑛 , (9.12)
where 𝐸 Γ is the eigenvalues of the local interaction Û as Û |Γ⟩ = 𝐸 Γ |Γ⟩.

Basis transformations

The basis set for the physical Hilbert space can be chosen arbitrarily in the RISB formalism due to its rotational invariance. Thus, a linear transformation can be applied to change a basis set to another set. For instance, the basis transformation from multiplet-Fock basis to Fock-Fock basis can be done by applying a linear transformation as

𝜙 † 𝑛 ′ 𝑛 = Γ ⟨Γ|𝑛 ′ ⟩𝜙 † Γ𝑛 . (9.13)
Similarly, a linear transformation to change into multiplet-multiplet basis can be done as

𝜙 † ΓΓ ′ = 𝑛 ⟨𝑛|Γ ′ ⟩𝜙 † Γ𝑛 . (9.14)
We can combine the above two linear transformation (9.13) and (9.14) in order to find the relation between the Fock-Fock bosons and the multiplet-multiplet bosons

𝜙 † ΓΓ ′ = 𝑛𝑛 ′ ⟨Γ|𝑛 ′ ⟩⟨𝑛|Γ ′ ⟩𝜙 † 𝑛 ′ 𝑛 . (9.15)
Whether on a Fock-Fock basis or multiplet-multiplet basis, the first index refers to a state in the initial physical Hilbert space, whereas the second index refers to a state in the auxiliary Hilbert space.

Approximations

We have rewritten the effective Hamiltonian (9.1) within RISB formalism. Now, we can further apply mean-field approximations, considering a constant density of states to obtain self-consistent equations.

Mean-field approximations for auxiliary fields

The mean-field approximations are

• The slave-bosonic operators are replaced by their expected valuers as 𝜙 † 𝑎,Γ𝑛 , 𝜙 𝑎,Γ𝑛 → 𝜑 Γ𝑛 .

• The constraints (9.5-9.6) are applied through enforcing Lagrange parameters 𝜆 and 𝜆 𝑗 𝑧 respectively.

With these approximations in hand and by invoking Fourier transformation, the RISB Hamiltonian (9.12) is rewritten as

ℋ 𝑀𝐹 = k,𝑗 𝑧 𝜖 k,𝑗 𝑧 𝑍 𝑗 𝑧 [Φ] 𝑓 † k,𝑗 𝑧 𝑓 k,𝑗 𝑧 - k,𝑗 𝑧 𝜆 𝑗 𝑧 𝑓 † k,𝑗 𝑧 𝑓 k,𝑗 𝑧 + 𝒩 Γ𝑛 𝐸 Γ 𝜑 2 Γ𝑛 + 𝜆𝒩 Γ𝑛 𝜑 2 Γ𝑛 -1 + 𝒩 𝑗 𝑧 𝜆 𝑗 𝑧 𝑛 𝑗 𝑧 [Φ] + 𝜇𝒩 𝑗 𝑧 𝑛 𝑗 𝑧 [Φ] -𝑛 𝑓 ), (9.16)
where, 𝜖 k𝑗 𝑧 are the energy levels associated with the nearest neighbor hopping integrals 𝑡 𝑗 𝑧 , 𝒩 is the total number of sites and 𝜇 is the chemical potential fixing number of 𝑓 -electrons 𝑛 𝑓 . In the above Eq. ( 9.16), 𝑍[Φ] is the quasiparticle weight which is defined as

𝑍 𝑗 𝑧 [Φ] = 𝑅 2 [Φ] = 𝛾 2 𝑗 𝑧 [Φ] 𝑛 𝑗 𝑧 [Φ](1 -𝑛 𝑗 𝑧 [Φ]) , (9.17)
where 𝛾 𝑗 𝑧 and 𝑛 𝑗 𝑧 are explicit functions of the slave-bosons 𝜑 Γ𝑛 and are defined similarly as the operators γ𝑗 𝑧 and n𝑗 𝑧 as in the equations (9.10 -9.11) but with meanfield approximations. In order to obtain the relations for mean-field parameters 𝜆, and 𝜇, we minimize the free energy of the system ℱ = -1

𝛽 𝑙𝑛𝑇𝑟[𝑒 -𝛽𝐻 𝑀𝐹 ], where 𝛽 = 1 𝑘 𝐵 𝑇 with 𝑘 𝐵 being Boltzmann constant and 𝑇 is the temperature of the system. Thus, we write our saddle-point equations as

Γ𝑛 𝜑 2 Γ𝑛 = 1 , (9.18)
𝑗 𝑧 𝑛 𝑗 𝑧 [Φ] = 𝑛 𝑓 , (9.19)
𝑛 𝑗 𝑧 [Φ] = 1 𝒩 k ⟨ 𝑓 † k𝑗 𝑧 𝑓 k𝑗 𝑧 ⟩ , (9.20)

2(𝐸

Γ + 𝜆)𝜑 Γ𝑛 = - 𝑗 𝑧 (𝜆 𝑗 𝑧 + 𝜇) 𝜕𝑛 𝑗 𝑧 [Φ] 𝜕𝜑 Γ𝑛 - 1 𝒩 𝑗 𝑧 𝜕𝑍 𝑗 𝑧 [Φ] 𝜕𝜑 Γ𝑛 k 𝜖 k𝑗 𝑧 ⟨ 𝑓 † k𝑗 𝑧 𝑓 k𝑗 𝑧 ⟩ , (9.21)
with

⟨ 𝑓 † k𝑗 𝑧 𝑓 k𝑗 𝑧 ⟩ = 𝑛 𝐹 (𝜖 k 𝑍 𝑗 𝑧 [Φ] -𝜆 𝑗 𝑧 ) , (9.22)
where 𝑛 𝐹 is Fermi distribution. From the expressions of 𝑍 𝑗 𝑧 [Φ], 𝛾 𝑗 𝑧 [Φ], and 𝑛 𝑗 𝑧 [Φ],

the partial derivation

𝜕𝑍 𝑗𝑧 [Φ]
𝜕𝜑 Γ𝑛 is expressed as

𝜕𝑍 𝑗 𝑧 [Φ] 𝜕𝜑 Γ𝑛 = 𝑍 𝑗 𝑧 [Φ] 2𝑛 𝑗 𝑧 [Φ] -1 𝑛 𝑗 𝑧 [Φ](1 -𝑛 𝑗 𝑧 [Φ]) 𝜕𝑛 𝑗 𝑧 [Φ] 𝜕𝜑 Γ𝑛 + 2 𝛾 𝑗 𝑧 [Φ] 𝑛 𝑗 𝑧 [Φ](1 -𝑛 𝑗 𝑧 [Φ]) 𝜕𝛾 𝑗 𝑧 [Φ] 𝜕𝜑 Γ𝑛 , (9.23)
with,

𝜕𝛾 𝑗 𝑧 [Φ] 𝜕𝜑 Γ𝑛 = Γ ′ 𝑛 ′ ⟨Γ|𝑐 † 𝑗 𝑧 |Γ ′ ⟩⟨𝑛| 𝑓 † 𝑗 𝑧 |𝑛 ′ ⟩𝜑 Γ ′ 𝑛 ′ , (9.24)
𝜕𝑛 𝑗 𝑧 [Φ] 𝜕𝜑 Γ𝑛 = 2𝜑 Γ𝑛 ⟨𝑛| 𝑓 † 𝑗 𝑧 𝑓 𝑗 𝑧 |𝑛⟩ . (9.25)

Constant density of states

For our numerical calculations, within the mean-field equations (9.18-9.21), we consider an orbital-dependent constant density of states as

𝜌 𝑗 𝑧 (𝜔) = 1 𝑊 𝑗 𝑧 𝜃 𝑊 𝑗 𝑧 2 -|𝜔| , (9.26)
where, 𝑊 𝑗 𝑧 and 𝜃 are orbital-dependent electronic bandwidth and Heaviside function respectively.

Self-consistent equations system and parameters

In this section, we present our self-consistent equations. To do so, we consider the constant density of states provided by the Eq. ( 9.26) along with the average occupation ⟨ 𝑓 † k𝑗 𝑧 𝑓 k𝑗 𝑧 ⟩ from the Eq. ( 9.22), at zero temperature the Eq. ( 9.20) can be expressed as

𝑛 𝑗 𝑧 [Φ] = ∫ 𝜌 𝑗 𝑧 (𝜔)𝑛 𝐹 (𝜔𝑍 𝑗 𝑧 [Φ] -𝜆 𝑗 𝑧 )𝑑𝜔 = 𝜆 𝑗 𝑧 𝑊 𝑗 𝑧 𝑍 𝑗 𝑧 + 1 2 . (9.27)
In the similar way, one can treat the Eq. (9.21). Hence, we resume our final set of self-consistent equations as

Γ𝑛 𝜑 2 Γ𝑛 = 1, (9.28)
𝑗 𝑧 𝑛 𝑗 𝑧 [Φ] = 𝑛 𝑓 , (9.29)
𝑛 𝑗 𝑧 [Φ] = 𝜆 𝑗 𝑧 𝑊 𝑗 𝑧 𝑍 𝑗 𝑧 + 1 2 , (9.30)
[𝐸 Γ + 𝜆 + 𝜇 𝑗 𝑧 ⟨𝑛|𝑐 † 𝑗 𝑧 𝑐 𝑗 𝑧 |𝑛⟩]𝜑 Γ𝑛 = 𝑗 𝑧 𝑊 𝑗 𝑧 2 𝛾 𝑗 𝑧 [Φ] 𝜕𝛾 𝑗 𝑧 [Φ] 𝜕𝜑 Γ𝑛 . (9.31)
Similarly, the energy of the system pre site is obtained from the Eq. ( 9.16) as

⟨ℋ 𝑀𝐹 ⟩
𝑁 , which gives the final equation as

⟨ℋ 𝑀𝐹 ⟩ 𝑁 = 𝐸[Φ] = - 𝑗 𝑧 𝑊 𝑗 𝑧 2 𝑍 𝑗 𝑧 [Φ]𝑛 𝑗 𝑧 [Φ](1 -𝑛 𝑗 𝑧 [Φ]) + Γ𝑛 𝐸 Γ 𝜑 2 Γ𝑛 . (9.32)
The equations (9.28-9.31) makes a set of self-consistent equations to minimize the energy (9.32). Within RISB formalism, a density matrix needs to be constructed

𝑛 𝑗 𝑧 = Γ𝑛 𝜑 2 Γ𝑛 ⟨𝑛| 𝑓 † 𝑗 𝑧 𝑓 𝑗 𝑧 |𝑛⟩.
This construction of this density matrix can lead to a large amount of auxiliary slave-bosons operators. However, the number of slave bosons could be reduced using symmetries of local interactions. For instance, the particle number conservation of the local Hamiltonian and diagonal nature of interaction Û on a multiplet basis leads to cancellation of all the slave-bosons 𝜑 Γ𝑛 with ⟨𝑛|Γ⟩ = 0. Thus in our case, slave-bosons numbers can be reduced considerably to 116 and can be found in the appendix E.

Chapter 10

Numerical approach: algorithm and technical challenges

Numerical resolution of RISB equations can be very tedious and challenging. The difficulties can be purely numerical or algorithmic, or even both. Thus, each step of numerical calculation should be carefully chosen in order to get the correct solution.

Assuming that all the technical difficulties are fixed, the resolution of RISB equations can be very time-consuming. This makes the optimization of numerical codes very necessary whenever possible. In our case, we chose Python due to its large libraries in optimization routines. Furthermore, the python functions were compiled by using Numba [328] whenever possible for faster executions. Finally, for our study, we have used different types of optimization routines mixed together.

Reduction of Hilbert space and slave-bosons

In the previous chapter, we have seen that we are dealing with 116 slave-bosons present in six different particle charge sectors (see appendix E). Ideally, we should consider all the 116 𝜑 Γ𝑛 spanning through the Hilbert space of all six particle sectors. But, this leads to 116 equations to be solved self-consistently. However, depending upon the 𝑊/𝑈 4 , solving all 116 equations along with two physical constraints is not always necessary since the ground state might not be composed of all 𝑓 -electron configurations. To demonstrate at first, we consider the atomic limit 𝑊 = 0. From the previous dual model calculations and from Hund's, the ground-state at 𝑊/𝑈 4 = 0 is composed of 𝑓 2 and 𝑓 3 configurations with 𝐽 = 4 and 𝐽 = 9/2 respectively. So, in this case, we can consider only the slave-bosons representing 𝑓 2 and 𝑓 3 configurations. This assumption will reduce the number of 𝜑 Γ𝑛 to 36 instead of 116. Now, with a slight increase in 𝑊, electrons can hop as 𝑓 3 ↔ 𝑓 4 . Thus, the slave-bosons representing 3 or 4 electrons particle sectors can become non-zeros. Figure 10.1 schematizes this process where we do a preliminary optimization within a 𝜆 and 𝜇 grid. This preliminary grid calculation permits us to have an initial guess parameter .1: Schematic view of intial optimizing process to get guess parameters while considering various 𝑓 -configurations using the Eqs. 9. 28-9.31. Whenever the physical constraints are satisfied, the scheme ejects an output with 𝜆, 𝜇 and slaveboson vector Φ.

𝐺𝑢𝑒 𝑠𝑠 𝑓 2 , 𝑓 3 𝜆, 𝜇, Φ 𝐺𝑢𝑒 𝑠𝑠 𝑓 2 , 𝑓 3 , 𝑓 4 𝜆, 𝜇, Φ 𝐺𝑢𝑒 𝑠𝑠 𝑓 0 , 𝑓 1 ... 𝑓
without any phase consideration. However, one need not do initial calculation over 𝜆 and 𝜇-grid for each value of 𝑊, but rather when there is a jump in mean-field parameters or a phase transition is suspected.

Minimization of Energy

Using a single optimization technique can lead to a local minimum or the worstcase scenario to nonphysical results. In our work, we have used various classes of optimization routines to assure the convergence of RISB mean-field equations to their global minimum. Below, we present the python routines that we have considered and their advantages and drawbacks, and how we have used them in our code.

Choice of optimization algorithms

During our whole numerical optimization process, the choice of algorithm and the method of iteration was a crucial step. We found that gradient-based methods were not so effective for either local minimization or global minimization of energy. So we have used gradient-free methods throughout our study. To solve the auto-consistent equations (9.28-9.31), we have used the derivative-free Powell method from NLopt library [329] along with Scipy's [330] least-squares. For local minimization of energy was either conducted by using Pymoo's [331] heuristic pattern search and/or Scipy's trust region method was used. Furthermore, global minimization was conducted with Scipy's stochastic basin hopping or with differential evolution.

Global and local optimizations

The methods described in section 10.2.1 were used to find local or global minima. Figure 10.2 shows the schematic view of an overall view of local minimizations. During our calculations, we found that an iterative method where a previous solution is injected to calculate a new solution for a new set of parameters was not efficient.However, it can be used in some situations, for example, non-interacting case 𝑈 = 0. The local optimization iteration process can be defined in the following steps. i) At first, a set of non-zeros 𝜑 𝐺𝑢𝑒 𝑠𝑠 Γ𝑛 slave-bosons are determined as guess parameters by using the method in section 10.1.

ii) We actualize the mean-field parameters 𝜆 and 𝜇 for given hopping with 𝜑 𝐺𝑢𝑒 𝑠𝑠 Γ𝑛 as input. In the meantime, 𝜑 𝐺𝑢𝑒 𝑠𝑠 Γ𝑛 is actualized into 𝜑 Γ𝑛 .

iii) A local minimization is conducted with routines stated above in the section 10.2.1. During our numerical calculations, we found that the slave-boson matrix Φ for a given particular solution with partially localized orbitals is not always unique. Thus, in order to assure the ground-state phase diagram, a systematic global optimization must be conducted after the local minimization. Figure 10.3 presents our complete algorithm scheme, which includes the global optimization step.

Ansatz

Numerical error tolerance on constraints

We have altogether two physical constraints: the completeness equation Γ𝑛 𝜑 2 Γ𝑛 = 1 and the conservation of fermionic particle number 𝑗 𝑧 𝑛 𝑗 𝑧 [Φ] = 𝑛 𝑓 . The convergence of a solution is defined upon the tolerance of these constraints. From the equation 9.32, we can also deduce that the numerical error on the energy is proportional to the number of non-zeros slave-bosons. Since, we can have up to 116 non-zeros bosons, this addition can significantly impact the differentiation between two close phases. During our minimization of energy, we found that the minimums of energies can be as close as 0.1% to each other relatively. Thus, all our solutions were calculated with a minimum tolerance of 10 -6 .

Chapter 11

Results

In this chapter, we present the numerical results that we have obtained by solving the self-consistent equations (9.18 -9.21). In our calculations, we consider two different orbital-dependent electronic bandwidths: 𝑊 3/2 and 𝑊 1/2 = 𝑊 5/2 , along with 𝑓 -electron count and Coulomb interactions as parameters. Depending upon case of study, we will be varying the electronic bandwidths, Coulomb interactions, and 𝑓 -electron count 𝑛 𝑓 . Table 11 After checking the limits of our model, we will perform a full calculation by varying orbital-dependent electronic bandwidths to analyze the effect of its anisotropies. For that, we also consider non-integer 𝑓 -electron count 𝑛 𝑓 = 2.5, which is justified through the experimental observations [287,[298][299][300]332] where 𝑓 -electron valencies situate between 2 and 3 for uranium-based heavy-fermions. Here, we consider the Coulomb interactions 𝑈 𝐽=4 = 17.21𝑒𝑉, 𝑈 𝐽=2 = 18.28𝑒𝑉 and 𝑈 𝐽=0 = 21.00𝑒𝑉 which was obtained from the local density approximation 5 𝑓 wave functions calculation [START_REF] Zwicknagl | Microscopic description of origin of heavy quasiparticles in UPt 3[END_REF] for UPt 3 . From these Coulomb parameters, one could notice the anisotropic as Δ𝑈 4 = 𝑈 𝐽=4 -𝑈 𝐽=0 = -3.79 eV and Δ𝑈 2 = 𝑈 𝐽=2 -𝑈 𝐽=0 = -2.72 eV. This anisotropy depends weakly upon the chemical environment are usually not screened whereas 𝑈 0 get screened. The choice of UPt 3 is motivated by the previous calculations [START_REF] Zwicknagl | Microscopic description of origin of heavy quasiparticles in UPt 3[END_REF]327] done within the dual model. This will permit us to make a direct comparison of our results with the previous calculations whenever possible.

Non-localized paramagnetic phase in isotropic hopping case

In this section, we present our results for non-interacting (𝑈 = 0) and atomic (𝑊 𝑗 𝑧 = 0) limits on the isotropic line (𝑊 = 𝑊 1/2 = 𝑊 1/2 = 𝑊 5/2 ). The solutions on the isotropic line are highly degenerate. Nevertheless, the analytical solutions for both limits can be determined. Thus, these limits set a benchmark for our numerical method to produce consistent results. Furthermore, the solutions at these limits provide us a solid starting point to perform global minimization for other sets of parameters 𝑊 3/2 and 𝑊 3/2 = 𝑊 1/2 .

Non-interacting limit

At first, we start from electronic occupation 𝑛 𝑓 = 3.00 in the non-interacting limit.

Thereafter, we will gradually decrease 𝑛 𝑓 to obtain the intermediate occupation 𝑛 𝑓 = 2.5. For 𝑛 𝑓 = 3.0 with 𝑈 = 0, all the slave-bosons 𝜑 𝑛𝑛 ′ on the Fock-Fock basis set are equivalent and the slave-boson matrix is diagonal. From this argument, we can easily determine the amplitudes of slave-bosons as

𝜑 𝑛𝑛 ′ = 𝛿 𝑛𝑛 ′ √ 64 . (11.1)
Instead of Fock-Fock basis set, one can freely choose multiplet-multiplet basis set to represent both physical and auxiliary states. The linear transformation (9.15) can be applied to the equation (11.1) to obtain 𝜑 ΓΓ . However, we perform our numerical work using multiplet-Fock basis, and we apply again a linear transformation to obtain 𝜑 Γ𝑛 as

𝜑 ΓΓ = 𝑛 ⟨𝑛|Γ⟩𝜑 Γ𝑛 . (11.2)
With the relations 11.1, and 11.2, we solved the RISBMF equations from 𝑛 𝑓 = 3.00 to 2.5. 64) 2 = 0.015625 and the orbitals are fully delocalized 𝑍 𝑗 𝑧 = 1. As soon as 𝑛 𝑓 is moved away from the half-filling 𝑛 𝑓 = 3.0, the symmetry of |𝜑 ΓΓ | 2 over all charge sectors breaks. However, the symmetry over the same particle sector is still preserved.

Atomic limit

At the atomic limit 𝑊 = 0, the ground-state verifies the Hund's rule, and is made up of the local atomic multiplets | 𝑓 2 ; 𝐽 = 4, 𝐽 𝑧 ⟩ and | 𝑓 3 ; 𝐽 = 9/2, 𝐽 𝑧 ⟩ with the conditions below 4

𝐽 𝑧 =-4 |𝜑 𝑓 2 ;𝐽=4,𝐽 𝑧 ⟩,| 𝑓 2 ;𝐽=4,𝐽 𝑧 ⟩ | 2 = 1 2 , (11.3)
9/2 .4) Expanding the expressions 11.3 and 11.4, we can deduce that the values of each slave-boson contributing to the ground-state solution at the atomic limit is (11.7) In order to solve the mean-field equations numerically at the atomic limit, we start from the solution obtained for 𝑛 𝑓 = 2.5 at 𝑈 = 0. Thereafter, we gradually increase the Coulomb integrals 𝑈 𝐽 with 𝑊 ≫ 𝑈 until 𝑈 0 = 21.00 eV, 𝑈 2 = 18.28 eV and 𝑈 4 = 17.21 eV. Thereafter, we slowly decrease the isotropic electronic bandwidth until 𝑊 = 0. We found that the ground state is highly degenerate i.e. any set of slave-boson matrix Φ fulfilling the conditions 11.3 and 11.4 with the energy 𝐸 0 can be a solution. This made the numerical resolution challenging. However, we were able to recover the analytical solutions 11.5 and 11.5 with ground-state energy 𝐸 0 . The numerical result is presented in the figure 11.2. This also shows the correctness and robustness of our numerical method to solve RISBMF equations.

𝐽 𝑧 =-9/2 |𝜑 | 𝑓 3 ;𝐽=9/2,𝐽 𝑧 ⟩,| 𝑓 3 ;𝐽=9/2,𝐽 𝑧 ⟩ | 2 = 1 2 . ( 11
|𝜑 |𝑀=2;𝐽=4,𝐽 𝑧 ⟩,|𝑀=2;𝐽=4,𝐽 𝑧 ⟩ | 2 = 1 18 , ( 11

From atomic limit to non-interacting limit: the appearance of three distinct regions

In this section, we will present our results on the isotropic line while moving from atomic limit to non-interacting limit by gradually increasing isotropic electronic bandwidth. Figure 11.2 presents the slave-boson probabilities on a multipletmultiplet basis. By analyzing the non-zeros slave-bosons probabilities, we can distinguish two thresholds on electronic bandwidth, delimiting three regions: the first region lies between 𝑊/𝑈 0 = 0 to ≈ 1.32, the second region lies between 𝑊/𝑈 0 ≈ 1.32 to ≈ 3.27, and the third region lies for 𝑊/𝑈 0 ≳ 3.27. Furthermore, the signatures of the thresholds can also be observed in the quasiparticle weight 𝑍 𝑗 𝑧 and mean-field energies 𝐸.

Let's start with the first region. As per Hund's rule, the ground state of the system at atomic-limit is made up of from 𝑓 2 and 𝑓 3 -electron configurations with total angular momentum 𝐽 = 4 and 𝐽 = 9/2 respectively. As soon as the isotropic electronic hopping gets a finite value 𝑊 > 0, the slave-bosons associated with 𝑓 4 configuration become non-zeros, suggesting valency configuration transition. Here, the ground-state is made up of local configurations: | 𝑓 The second region starts at 𝑊/𝑈 0 ≈ 1.32 and is marked by slave-boson associated with configurations | 𝑓 1 ; 𝐽 = 5/2⟩, | 𝑓 5 ; 𝐽 = 5/2⟩, | 𝑓 2 ; 𝐽 = 2⟩ and | 𝑓 2 ; 𝐽 = 4⟩ becoming non-zero. Thus, these additional local multiplets contribute to the formation of ground-state. The quasiparticle weight presents a large 𝑍 𝑗 𝑧 around 𝑊/𝑈 0 ≈ 1.32. Similar behavior is also observed in the energy and the mean-field parameters and is presented in the figurefigure 11.4. Similar jumps in energies were seen in the two sites model [START_REF] Efremov | Dual nature of 5 𝑓 electrons: Effect of intra-atomic correlations on hopping anisotropies[END_REF] resulting from the competition between Coulomb interaction and hopping. The third region starts approximately at 𝑊/𝑈 0 ≈ 3.27 where all the slave-bosons on all the charge sectors become non-zeros. Thus, the ground-state is composed of all the configurations: 𝑓 0 , 𝑓 1 , 𝑓 2 , 𝑓 3 , 𝑓 4 , 𝑓 5 , and 𝑓 6 as for non-interacting limit. From the figure 11.4, we can observe that the mean-field parameter 𝜆 cancels out, and 𝜇 changes its sign from negative to positive as in the non-interacting case. The quasiparticle weight is isotropic and recovers the value 𝑍 𝑗 𝑧 ≃ 1 for large 𝑊.

In this isotropic case, we were able to recover the non-interacting limit from the atomic limit. The results on both limits are similar to those obtained by Duc-Anh Le [327]. However, multiple differences were obtained. For low hopping, we obtained anisotropic 𝑍 𝑗 𝑧 , which was not observed in the previous study [327]. Furthermore, successive transitions in valency configurations were observed due to interplay between Hund's rule, Coulomb interaction, and increasing bandwidth: ( 𝑓 2 , 𝑓 3 ) → ( 𝑓 2 , 𝑓 3 , 𝑓 4 ) → ( 𝑓 1 , 𝑓 2 , 𝑓 3 , 𝑓 4 , 𝑓 5 ) → ( 𝑓 0 , 𝑓 1 , 𝑓 2 , 𝑓 3 , 𝑓 4 , 𝑓 5 , 𝑓 6 ). However, due to the isotropic nature of electronic bandwidth, no orbital-selective Mott transition was observed in this case which is not the case when the anisotropic in electronic bandwidths are present.

Ground-state phase diagram

After the isotropic case, we solved the RISBMF equations considering paramagnetic (PM) and ferromagnetic (FM) phases in the range of 0 ≤ 𝑊 1/2 ≤ 15 and 0 ≤ 𝑊 5/2 (= 𝑊 1/2 ) ≤ 15. In order to simply, we note 𝑊 = 𝑊 3/2 and 𝑊 ′ = 𝑊 5/2 = 𝑊 1/2 . Her, a paramagnetic phase is characterized by having symmetric 𝑗 𝑧 and -𝑗 𝑧 orbitals:

𝑛 𝑗 𝑧 = 𝑛 -𝑗 𝑧 and 𝑍 𝑗 𝑧 = 𝑍 -𝑗 𝑧 . Similarly, a ferromagnetic phase is characterized by having at least one orbital 𝑗 𝑧 being fully occupied and its counterpart -𝑗 𝑧 being empty: 𝑛 𝑗 𝑧 = 1, 𝑍 𝑗 𝑧 = 0 and 𝑛 𝑗 𝑧 = 0, 𝑍 𝑗 𝑧 = 0. Also, a saturated ferromagnetic phase (SFM) is characterized by having all the 𝑗 𝑧 orbitals occupied and all -𝑗 𝑧 empty.

From our numerical calculations, we have found altogether twelve phases with partially localized orbitals: five PM, five FM, and two SFM phases. Furthermore, each phase can be either one-electron localized or two-electron localized. From here, if a phase is paramagnetic with 𝑗 𝑧 = 5/2 and 1/2 localized, we will note it as PM In order to construct a phase diagram, we compare the energies. The energy for each phase is obtained from the equation below The first term of the equation (11.8) gives the kinetic energy, and the second term is related to the energy due to local interactions. The total energy of the system for a given phase depends only on the contribution from its non-localized orbitals.

𝐸[Φ] = - 𝑗 𝑧 𝑊 𝑗 𝑧 2 𝑍 𝑗 𝑧 [Φ]𝑛 𝑗 𝑧 [Φ](1 -𝑛 𝑗 𝑧 [Φ]) + Γ𝑛 𝐸 Γ 𝜑 2 Γ𝑛 . ( 11
In order the effect of anisotropies, we present the energies of each partially localized phase along the line 𝑊 + 𝑊 ′ = 7 and 𝑊 + 𝑊 ′ = 14 in the figure 11.5. The representation of the energies along these lines permits us to include all the phases in the ground state phase diagram (see figures 11.6 and 11.7). First, we can clearly remark that the energy of the partially localized phase decreases linearly with increasing 𝑊 𝑗 𝑧 due to a decrease of kinetic energy. However, it is partially compensated by energy due to local interaction. Furthermore, we observe that the FM and PM phases with the same localized orbitals are degenerate. The crossing of energies curves translating phase transitions happens smoothly except near isotropic line where a jump in energy for non-localized paramagnetic phase is seen (see section 11.1.3).

For smaller electronic bandwidths with anisotropy, the two-electrons localized phases are found to be more stable than a one-electron localized phase. However, phases with one-electron localized can be stabilized for larger hopping when 𝑊 > 𝑊 ′ and 𝑊 ′ > 𝑊. Depending upon the anisotropy, different sets of orbitals can be localized. For 𝑊 > 𝑊 ′ , either 𝑗 𝑧 = 5/2, 1/2 or 𝑗 𝑧 = 5/2 or 𝑗 𝑧 = 1/2 orbitals can be localized. For 𝑊 ′ > 𝑊, either 𝑗 𝑧 = 5/2, 3/2 or 𝑗 𝑧 = 3/2 can be localized. 11.8: Schematic view of an electron hopping from a site-𝑖 with 𝑓 𝑀 configuration to a site-𝑗 with 𝑓 𝑀 ′ configuration. For 𝑀 ′ < 𝑀, this hopping becomes favorable. For instance, the hopping from 𝑀 = 3 to 𝑀 = 2 is more favorable than other way around since the hopping from 𝑀 = 2 to 𝑀 = 3 will create more energetic states with 𝑀 = 1 and 𝑀 = 5 particles.

Let's us consider 𝑊 ≲ 5.20 eV with 𝑊 ′ = 0 eV, in this case SFM5 2 , 1 2 was found to be stable. The formation of this phase follows the Hund's rule with the multiplets | 𝑓 2 ; 𝐽 = 4, 𝐽 𝑧 = 3⟩ = |000101⟩ and | 𝑓 3 ; 𝐽 = 9/2, 𝐽 𝑧 = 9/2⟩ = |000111⟩ forming the macroscopic phase. From these multiplets, we can easily deduce that only the electron in 𝑗 𝑧 = 3/2 orbital from 𝑓 3 configuration can hop to 𝑓 2 , since hopping from 𝑓 2 to 𝑓 3 will give less favorable high energetic 𝑓 4 electron state (see figure 11.8). Similar results were also obtained in two-sites model [START_REF] Efremov | Dual nature of 5 𝑓 electrons: Effect of intra-atomic correlations on hopping anisotropies[END_REF] where a phase with total magnetization 𝒥 𝑧 = 15/2 was obtained. The 𝒥 𝑧 = 15/2 (see figure 8

Physical signatures and manifestations of duality

In this section, we will treat the various aspects of partially localized phases through quasiparticle weight 𝑍 𝑗 𝑧 , electronic occupancies 𝑛 𝑗 𝑧 , magnetization 𝑚 𝑧 = 𝑗 𝑧 𝑛 𝑗 𝑧 𝑗 𝑧 and 𝑓 -electron valency configurations along 𝑊 + 𝑊 ′ = 7 and 𝑊 + 𝑊 ′ = 14 (see figure 11.7).

Quasiparticle weight

An orbital is localized when 𝑍 𝑗 𝑧 = 0 and delocalized when 𝑍 𝑗 𝑧 ≠ 0. With these definitions on hand, we can now analyze the quasiparticle weight of our groundstate phase diagram up on anisotropy of non-interacting electronic bandwidth. 11.9(a) and (c) presents 𝑍 𝑗 𝑧 for non-localized orbitals on the line 𝑊 +𝑊 ′ = 7 for FM and PM phases respectively. Since there is no equivalent counterpart phase of SFM in PM phase, 11.9(c) misses 𝑍 𝑗 𝑧 partially. The left side of each graph in figure 11.9 corresponds to the anisotropic region with 𝑊 > 𝑊 ′ where only orbital

𝑗 𝑧 = 3/2 is delocalized. For FM5 2 , 1 2
quasiparticle weights of delocalized orbitals are orbital dependent: 𝑍 -3/2 ≠ 𝑍 3/2 . This orbital dependency can be explained through the hopping of an electron from one local mulitplet to another. In this phase, the electrons in 𝑗 𝑧 = 3/2 can hop from 𝑓 3 -configuration to 𝑓 2 -configuration which produces more energetic favorable 𝑓 3 -configuration than the hopping of electrons in 𝑗 𝑧 = -3/2 from 𝑓 3 → 𝑓 3 which produces 𝑓 4 -configuration. Thus, the orbital 𝑗 𝑧 = 3/2 is more delocalized than 𝑗 𝑧 = -3/2. Furthermore, we observe that 𝑍 3/2 in PM5 ) with only one electron localized is present. In this phase, similar as above, the quasiparticle weights are orbital-dependent with 𝑗 𝑧 = 1/2 being more delocalized than 𝑗 𝑧 = 5/2.

Occupations and magnetization

In this section, we will analyze the occupancies and the magnetization of the groundstate phase diagram. Here, we define the magnetization as

𝑚 𝑧 = 𝑗 𝑧 𝑛 𝑗 𝑧 𝑗 𝑧 .
(11.9)

We found that the orbitals 𝑗 𝑧 and -𝑗 𝑧 will be localized when sum of their occupancies equals to 1: 𝑛 𝑗 𝑧 + 𝑛 -𝑗 𝑧 = 1. From this condition, we can deduce that in PM phase, a localized 𝑗 𝑧 -orbital will have occupancy of 𝑛 𝑗 𝑧 = 0.5 whereas in FM phase it will be either 0 or 1. Figure 11.10 presents occupations of delocalized orbitals and the figure 11.11 presents the net magnetization in FM phase of our ground-state phase diagram for 𝑊 + 𝑊 ′ = 7 and 𝑊 + 𝑊 ′ = 14.

Starting from FM does not present a discontinuity, it suggests that the transition is second-order. Meanwhile, all other phase transitions seem to be of first-order since the discontinuities in magnetization are present.

On the other hand for 𝑊 ′ > 𝑊 and in FM5 up on orbitals with 𝑛 1/2 < 𝑛 5/2 . This is again consistent with our 𝑍 𝑗 𝑧 where electrons in 𝑗 𝑧 = 1/2 were more delocalized than 𝑗 𝑧 = 5/2.

Fermi surfaces

The quasiparticle weight 𝑍 𝑗 𝑧 normalizes the electronic bandwidth as 𝑊 𝑗 𝑧 → 𝑊 𝑗 𝑧 𝑍 𝑗 𝑧 . From the equation (9.30), we can evaluate the chemical potential 𝜆 𝑗 𝑧 fixing electronic occupation 𝑛 𝑗 𝑧 per orbital. With this in hand, we present this renormalized effective bandwidths in the figures 11.12 and 11.13 for one-electron localized phases with 𝑊 = 0, 𝑊 ′ = 14.0 and two-electron localized phases with 𝑊 = 10.0, 𝑊 ′ = 0.0 respectively. Similarly, the Fermi surfaces also become orbital dependent. Furthermore, we also present a schematic view of Fermi surfaces corresponding to each delocalized orbitals on 2D square lattice with dispersion 𝜖 k,𝑗 𝑧 = First, we can observe the symmetry between 𝑗 𝑧 and -𝑗 𝑧 orbitals in PM phases, since bandwidths for both orbitals are renormalized similarly. On the other hand, for its counterpart FM phase, all the 𝑊 𝑗 𝑧 are renormalized differently. This difference in bandwidth normalization will reduce the number of Fermi surface sheets by twice in PM phases compared to their degenerate FM phases. For example, we can refer to the present four sheets of Fermi surfaces. Moreover, this difference between Fermi surfaces in two degenerate phases is a direct result of orbital-dependent occupancies. Similar situation can be also observed in twoelectron localized phases (see figure 11.13).

Here, the quasiparticle weight 𝑍 𝑗 𝑧 is the inverse of the effective mass 𝑚 * [324]. We can also relate the quasiparticle weight as 𝑍 𝑗 𝑧 ≡ [1 -𝜕𝑅𝑒(Σ(𝜔))/𝜕𝜔]| 𝜔=0 ] -1 with comparison to the previous definition of our effective mass in the chapter 5. Furthermore, as the effective mass gets larger, the electrons move slowly. This can be easily understood from free electron model where 𝑣 𝐹 = ℏ -1 𝜕𝜖 k /𝜕k| k=k 𝐹 = ℏk 𝐹 /𝑚 with 𝜖 k = ℏk 2 /2𝑚. In the figures 11.12 and 11.13, the darker to lighter blue color shades around Fermi surface represent schematically the lower to higher Fermi velocity. And, the width of the shades represents schematically the density of states at the Fermi level, which is determined by 1/𝑊 𝑗 𝑧 𝑍 𝑗 𝑧 . Here, we observe that the delocalized orbital with larger Fermi surfaces will have smaller Fermi velocity with larger effective mass. This can be understood through Coulomb repulsion that increases when occupancy increases, decreasing the degree of delocalization.

𝑓 -electron valency configurations

In this section, we will analyze the slave-boson probabilities described as (11.10) where 𝛼, 𝛽 can be a set of basis set representing physical Hilbert space and auxiliary Hilbert space in 𝑀 particle charge sector. Here, 𝑝(𝑀) may be interpreted as the weight for finding a state characterized by 𝛼, 𝛽 in 𝑓 𝑀 -electron configuration. Thus analyzing 𝑝(𝑀) per 𝑓 -electron configuration permits us to have more insight over the multi-configurational nature of 5 𝑓 orbitals [333, 334] and the effect of anisotropy on it.

𝑝(𝑀) = 𝛼𝛽∈𝑀 |𝜑 𝛼𝛽 | 2 ,
Here, we rewrite the completeness Eq. ( 9.28) and the conservation of number of fermions from the Eq. ( 9 (11.12) From the Eqs. (11.11) and (11.12), we can show that the variations on 𝑝(2) and 𝑝(3) depend linearly upon 𝑝(4) as 𝑝(4) = 𝑝(2) -0.5 and 𝑝(4) = 1 2 (0.5 -2𝑝( 3)). In the figure 11.14, we plot Δ𝑝(2) = 𝑝(2) -0.5, Δ𝑝(3) = 𝑝(3) -0.5 and Δ𝑝 = 𝑝(4) along the line 𝑊 +𝑊 ′ = 7 and 𝑊 +𝑊 ′ = 14. First, we can observe that our results are consistent with the theoretical prediction since the small variations in 𝑝( 4) coincide with Δ𝑝(2). This small 𝑝(4) values creates the small deviations of 𝑝(2) and 𝑝(3) from half-half weight. However, the differences between the two-particle sectors and three-particle sector enlarges with increasing anisotropy.

Chapter 12

Conclusions and perspectives

In this thesis, we looked at the link between duality in 5 𝑓 electrons with orbitalselective partially localized phases and the origin of heavy-fermionic behavior in uranium-based compounds. For that, we have carried out theoretical modal calculations using rotationally-invariant slave-bosons (RISB), considering the typical Coulomb parameters for uranium-based heavy-fermion compounds. This part emphasizes two major aspects: it contains a comprehensive discussion on algorithmic and numerical aspects and on the microscopic mechanism leading to putative orbital-selective localization.

The numerical treatment of RISB presents a great challenge due to the large number of parameters to be optimized. The procedure requires 116 non-linear selfconsistent equations to be solved. In this thesis, we have developed an efficient algorithm based both on the local search of minimal and on the global search of minimal to solve RISB equations. For this, we combine gradient-based methods for the local optimization and non-gradient methods like basin hopping, simulated annealing, and differential evolution for global optimization. A complete schematic view is present in the figure 10.3. Before performing the full optimization calculations, one needs a good initial input parameter. For that, we have developed a strategy (see figure 10.1) based on grid search combined with the reductions of Hilbert space defined by the charge sectors of 𝑓 -electrons. This permits us to scan all the partially localized phases spanning through all the charge sectors.

Our study started with the verification of the relevance of our numerical method. This was confirmed by the faithful reproduction of the analytical results at the atomic limit, where the multiplets | 𝑓 2 ; 𝐽 = 4⟩, | 𝑓 3 ; 𝐽 = 9/2⟩ formed the ground state, and on the non-interacting limit all the orbitals were fully delocalized. From atomic limit, increasing the electronic bandwidth in isotropic case (𝑊 3/2 = 𝑊 5/2 = 𝑊) induces transitions in 𝑓 -electron valency configurations to the more energetic states. More importantly, the transition in valency configurations happens stepwise: ( 𝑓 2 , 𝑓 3 ) at 𝑊 = 0, ( 𝑓 2 , 𝑓 3 ) → ( 𝑓 2 , 𝑓 3 , 𝑓 4 ) for 0 < 𝑊/𝑈 0 ≲ 1.32, ( 𝑓 2 , 𝑓 3 , 𝑓 4 ) → ( 𝑓 1 , 𝑓 2 , 𝑓 3 , 𝑓 4 , 𝑓 5 ) at 𝑊/𝑈 4 ≃ 1.62, and ( 𝑓 1 , 𝑓 2 , 𝑓 3 , 𝑓 4 , 𝑓 5 ) → ( 𝑓 0 , 𝑓 1 , 𝑓 2 , 𝑓 3 , 𝑓 4 , 𝑓 5 , 𝑓 6 ) at 𝑊/𝑈 0 ≃ 3.27. final goal is certainly to combine the present ansatz with material-specific ab-initio calculations. Hence, this thesis puts a new brick on the understanding of the dualism of 5 𝑓 electrons and paves a new way for more realistic band structure calculations.

Appendix B Matrix DMFT/CPA for Néel ordered antiferromagnetic phase

In this chapter of the appendix, we present a complete detail of our calculations and techniques that we have used to obtain DMFT equations. The first step would be the diagrammatic expansion. From now, every 2 × 2 matrices are represented by bold letters, and every 4 × 4 matrices are represented by bold letters and two bars upon it.

B.1 Diagrammatic development of Green function

In this diagrammatic expansion, we consider only direct part connecting from site 𝑖 to site 𝑗. Thus, Green's function 𝐺 𝜎 𝑖𝑗 for Kondo alloy model can be expressed as

𝐺 𝜎 𝑖𝑗 = 𝑎𝑙𝑙𝑝𝑎𝑡 ℎ𝑠 Π 𝜎 𝑖𝑖 𝛾 𝑖𝑖 1 P † 𝑖 WP 𝑖 1 Π 𝜎 𝑖 1 𝑖 1 𝛾 𝑖 1 𝑖 2 P † 𝑖 1 WP 𝑖 2 . . . Π 𝜎 𝑖 𝑝 𝑖 𝑝 𝛾 𝑖 𝑝 𝑗 P † 𝑖 𝑝 WP 𝑗 Π 𝜎 𝑗 𝑗 , (B.1)
multiplying above equation by the projectors P 𝑖 on left side and P † 𝑗 on the right side, and taking average over disorder, we get

⟨P 𝑖 𝐺 𝜎 𝑖𝑗 P † 𝑗 ⟩ = 𝑎𝑙𝑙𝑝𝑎𝑡 ℎ𝑠 ⟨P 𝑖 Π 𝜎 𝑖𝑖 P † 𝑖 ⟩𝛾 𝑖𝑖 1 W⟨P 𝑖 1 Π 𝜎 𝑖 1 𝑖 1 P † 𝑖 1 ⟩𝛾 𝑖 1 𝑖 2 W . . . ⟨P 𝑖 𝑝 Π 𝜎 𝑖 𝑝 𝑗 𝑝 P † 𝑗 𝑝 ⟩𝛾 𝑖 𝑝 𝑗 W⟨P 𝑗 Π 𝜎 𝑗 𝑗 P † 𝑗 ⟩ . (B.2)
We simplify the notation as ⟨P 𝑖 Π 𝜎 𝑖𝑖 P † 𝑖 ⟩ = ⟨ Π𝜎 𝑖 ⟩, the equation B.2 is rewritten as

⟨G 𝜎 𝑖𝑗 ⟩ = ⟨ Π 𝜎 𝑖𝑖 ⟩𝛿 𝑖𝑗 + 𝑎𝑙𝑙𝑝𝑎𝑡 ℎ𝑠 ⟨ Π 𝜎 𝑖𝑖 ⟩𝛾 𝑖𝑖 1 W⟨ Π 𝜎 𝑖 1 𝑖 1 ⟩𝛾 𝑖 1 𝑖 2 W⟨ Π 𝜎 𝑖 2 𝑖 2 ⟩ . . . 𝛾 𝑖 𝑝 𝑗 W⟨ Π 𝜎 𝑗 𝑗 ⟩ (B.3) = ⟨ Π 𝜎 𝑖𝑖 ⟩𝛿 𝑖𝑗 + ⟨ Π 𝜎 𝑖𝑖 ⟩ 𝑎𝑙𝑙𝑝𝑎𝑡 ℎ𝑠 𝛾 𝑖𝑖 1 W⟨ Π 𝜎 𝑖 1 𝑖 1 ⟩𝛾 𝑖 1 𝑖 2 W⟨ Π 𝜎 𝑖 2 𝑖 2 ⟩ . . . 𝛾 𝑖 𝑝 𝑗 W⟨ Π 𝜎 𝑗 𝑗 ⟩ (B.4) = ⟨ Π 𝜎 𝑖𝑖 ⟩𝛿 𝑖𝑗 + ⟨ Π 𝜎 𝑖𝑖 ⟩ 𝑙 𝛾 𝑖𝑙 W 𝑎𝑙𝑙𝑝𝑎𝑡 ℎ𝑠 ⟨ Π 𝜎 𝑙𝑙 ⟩𝛾 𝑙 1 𝑙 2 W⟨ Π 𝜎 𝑙 2 𝑙 2 ⟩ . . . 𝛾 𝑙 𝑝 𝑗 W⟨ Π 𝜎 𝑗 𝑗 ⟩ (B.5) = ⟨ Π 𝜎 𝑖𝑖 ⟩𝛿 𝑖𝑗 + ⟨ Π 𝜎 𝑖𝑖 ⟩ 𝑙 𝛾 𝑖𝑙 W⟨G 𝜎 𝑖𝑗 ⟩ . (B.6)
Finally,

⟨G 𝜎 𝑖𝑗 ⟩ = Π 𝜎 𝑖 (𝛿 𝑖𝑗 + 𝑙 𝛾 𝑖𝑙 W⟨G 𝜎 𝑙 𝑗 ⟩) , (B.7)
where

Π 𝜎 𝑖 = ⟨ Π 𝜎 𝑖𝑖 ⟩ = x𝑖 x𝑖 Π 𝜎 𝑖𝑖 x𝑖 ŷ𝑖 Π 𝜎 𝑖𝑖 ŷ𝑖 x𝑖 Π 𝜎 𝑖𝑖 ŷ𝑖 ŷ𝑖 Π 𝜎

𝑖𝑖

. We would like take Fourier transform of above equation so multiplying both sides by 1 𝑁 𝑖𝑗 𝑒 𝑖k ′ R j -𝑖kR i , so we get

G 𝜎 kk ′ = 1 𝑁 𝑖𝑗 𝑒 𝑖(k ′ -k)R i Π 𝜎 𝑖 + 1 𝑁 𝑖𝑗 𝑒 𝑖k ′ R j -𝑖kR i Π 𝜎 𝑖 𝛾 𝑖𝑙 WG 𝜎 𝑙 𝑗 , (B.8)
with few lines of algebraic calculations, we obtain

G 𝜎 kk ′ = Π 𝜎 k ′ -k + k ′′ Π 𝜎 k ′′ -k 𝛾 k ′′ G 𝜎 k ′′ k ′ . (B.9)

B.2 Green's function matrix in reduced Brillouin zone

Here, we are treating Néel ordered antiferromagnetic (AF) phase with wave ordering vector Q = (𝜋, 𝜋) considering two lattice sites in a unit antiferromagnetic cell. To do so, we have to rewrite the equations on the reduced Brillouin zone (RBZ) considering the bipartite organization in AF lattice with two sub-lattices 𝐴 and 𝐵 and 𝛿 0 between the distance between the first neighboring sites. In this way, the local propagator ANTIFERROMAGNETIC PHASE terms are written in RBZ i.e. k ′ , k ∈ 𝑅𝐵𝑍 as:

Π 𝜎 (k ′ -k) = 1 𝑁 𝑖 Π 𝜎 𝑖 𝑒 𝑖(k ′ -k)R i = 1 𝑁 𝑖∈𝐿 𝐴 Π 𝜎 𝑖 𝑒 𝑖(k ′ -k)R i + 1 𝑁 𝑖∈𝐿 𝐵 Π 𝜎 𝑖 𝑒 𝑖(k ′ -k)R i = 1 𝑁 Π 𝜎 𝐴 𝑖∈𝐿 𝐴 𝑒 𝑖(k ′ -k)R i + Π 𝜎 𝐵 𝑖∈𝐿 𝐴 𝑒 𝑖(k ′ -k)(R i +𝛿 0 ) = 1 𝑁 Π 𝜎 𝐴 𝑁 2 𝛿 kk ′ + Π 𝜎 𝐵 𝑁 2 𝛿 kk ′ = Π 𝜎 𝐴 + Π 𝜎 𝐵 2 𝛿 kk ′ , (B.10)
where Π 𝜎 𝐴 and Π 𝜎 𝐵 are local propagators for a site in a sublattice 𝐴 or 𝐵 respectively. Similarly, the other three local propagators are expressed as

Π 𝜎 (k ′ +Q-k) = Π 𝜎 𝐴 -Π 𝜎 𝐵 2 𝛿 kk ′ , (B.11) Π 𝜎 (k ′ -k+Q) = Π 𝜎 𝐴 -Π 𝜎 𝐵 2 𝛿 kk ′ , (B.12) Π 𝜎 (k ′ +Q-k+Q) = Π 𝜎 𝐴 + Π 𝜎 𝐵 2 𝛿 kk ′ . (B.13)
To be more clear and concise, we define

Π + 𝜎 = Π 𝜎 𝐴 +Π 𝜎 𝐵 2 and Π - 𝜎 = Π 𝜎 𝐴 -Π 𝜎 𝐵 2 . G 𝜎 kk ′ is expanded in RBZ as G 𝜎 kk ′ = Π 𝜎 k ′ -k + k ′′ Π 𝜎 k ′′ -k 𝛾 k ′′ WG 𝜎 k ′′ k ′ = Π 𝜎 k ′ -k + 𝑘 ′′ 𝛾 k ′′ Π 𝜎 k ′′ -k WG 𝜎 k ′′ k ′ + 𝛾 (k ′′ +Q) Π 𝜎 (k ′′ +Q)-k WG 𝜎 (k ′′ +Q),k ′ = Π + 𝜎 𝛿 kk ′ + 𝑘 ′′ 𝛾 k ′′ Π + 𝜎 WG 𝜎 k ′′ k ′ 𝛿 kk ′′ + 𝛾 (k ′′ +Q) Π - 𝜎 WG 𝜎 (k ′′ +Q),k ′ 𝛿 kk ′′ = Π + 𝜎 𝛿 kk ′ + Π + 𝜎 E k G 𝜎 kk ′ + Π - 𝜎 E k+Q G 𝜎 k+Q,k ′ , (B.14) with E k = 𝛾 k W and E k+Q = 𝛾 k+Q W are dispersion matrices. ANTIFERROMAGNETIC PHASE similarly, G 𝜎 k,k ′ +Q = Π - 𝜎 𝛿 kk ′ + Π + 𝜎 E k G 𝜎 k,k ′ +Q + Π - 𝜎 E k+Q G 𝜎 k+Q,k ′ +Q , (B.15) G 𝜎 k+Q,k ′ = Π - 𝜎 𝛿 kk ′ + Π - 𝜎 E k G 𝜎 k,k ′ + Π + 𝜎 E k+Q G 𝜎 k+Q,k ′ , (B.16) G 𝜎 k+Q,k ′ +Q = Π + 𝜎 𝛿 kk ′ + Π - 𝜎 E k G 𝜎 k,k ′ +Q + Π + 𝜎 E k+Q G 𝜎 k+Q,k ′ +Q . (B.17)
We would like to express the matrix Ḡ𝜎 kk ′ which is a tensor whose components are

G 𝜎 kk ′ , G 𝜎 k+Q,k ′ , G 𝜎 k,k ′ +Q and G 𝜎 k+Q,k ′ +Q . We define a 4 × 4 matrix as Ḡ𝜎 kk ′ = G 𝜎 kk ′ G 𝜎 kk ′ +Q G 𝜎 k+Qk ′ G 𝜎 k+Qk ′ +Q
. Inserting relations B.14, B.15, B.16 and B.17 along with few calculations, we get • a 0 0 0

Ḡ𝜎 kk ′ =        Π + 𝜎 Π - 𝜎 Π - 𝜎 Π + 𝜎 -1 - E k 0 0 E k+Q        -1 𝛿 kk ′ (B.
= 1 2 (1 ⊗ 1 + 𝜎 𝑧 ⊗ 1)K 1 2 (1 ⊗ 1 + 𝜎 𝑧 ⊗ 1) • b 0 0 0 = 1 2 (1 ⊗ 1 + 𝜎 𝑧 ⊗ 1)K 1 2 (𝜎 𝑥 ⊗ 1 -𝑖𝜎 𝑦 ⊗ 1) • c 0 0 0 = 1 2 (𝜎 𝑥 ⊗ 1 + 𝑖𝜎 𝑦 ⊗ 1)K 1 2 (1 ⊗ 1 + 𝜎 𝑧 ⊗ 1) • d 0 0 0 = 1 2 (𝜎 𝑥 ⊗ 1 + 𝑖𝜎 𝑦 ⊗ 1)K 1 2 (𝜎 𝑥 ⊗ 1 -𝑖𝜎 𝑦 ⊗ 1) ANTIFERROMAGNETIC PHASE • 0 0 0 a = 1 2 (𝜎 𝑥 ⊗ 1 -𝑖𝜎 𝑦 ⊗ 1)K 1 2 (𝜎 𝑥 ⊗ 1 + 𝑖𝜎 𝑦 ⊗ 1) • 0 0 0 b = 1 2 (𝜎 𝑥 ⊗ 1 -𝑖𝜎 𝑦 ⊗ 1)K 1 2 (1 ⊗ 1 -𝜎 𝑧 ⊗ 1) • 0 0 0 c = 1 2 (1 ⊗ 1 -𝜎 𝑧 ⊗ 1)K 1 2 (𝜎 𝑥 ⊗ 1 + 𝑖𝜎 𝑦 ⊗ 1) • 0 0 0 d = 1 2 (1 ⊗ 1 -𝜎 𝑧 ⊗ 1)K 1 2 (1 ⊗ 1 -𝜎 𝑧 ⊗ 1)

B.4 Local Green's function matrix

We define a local Green's function matrix as (B.32) where 𝛼 = 𝐴, 𝐵 and ᾱ = 𝐵, 𝐴.

G

B.5 Dynamical local bath

In this section of this appendix, we will be deriving the equation of dynamical local bath. At first, we write the expression of the dynamical local bath as 𝐴𝑙𝑙𝑜 𝑦 (0) for 𝑛 𝑐 = 0.70 and 𝑇 𝐾 = 0.019. Top: for 𝑥 in the vicinity of the critical point 𝑥 ★ = 0.08 which is characterized by 𝑚 ★ = 0, we observe the emergence of a non-monotonicity at low energy. This leads to the gradual formation of a multiple-branches dispersion for 𝑥 > 𝑥 ★ . Center: for 𝑥 in the vicinity of 𝑛 𝑐 , the maximum is realized at 𝜔 < 0 for 𝑥 < 𝑛 𝑐 and at 𝜔 > 0 for 𝑥 > 𝑛 𝑐 resulting in a second change of sign of 𝑚 ★ . Bottom: for 𝑥 close to 1, we observe signatures of the singularity Σ 𝐾 (𝜔) = 𝑟 2 𝜔+𝜆 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. . . .

∆

Figure 1 . 2 :

 12 Figure 1.2: Radial extend of 4 𝑓 electrons in Sm3+ and Pu 3+ respectively with both relativistic and nonrelativistic effects. 𝑥-axis in radial distance from the nucleus whereas 𝑃(𝑅) radial probability to find an electron at a distance 𝑟 from the nucleus. From[START_REF] Clark | The chemical complexities of plutonium[END_REF]

Figure 2 . 1 :

 21 Figure 2.1: Schematic view of spin-flip process during electron scatter in Kondo model.

Figure 2 . 3 :

 23 Figure 2.3: Schematic view of Doniach's phase diagram. Blue line indicate Kondo temperature 𝑇 𝐾 and red line indicate magnetic order temperature scale 𝑇 𝑅𝐾𝐾𝑌 .

Figure 2 . 4 :

 24 Figure 2.4: Schematic representation of Kondo lattice with impurity concentration 𝑥 greater than electronic filling 𝑛 𝑐 in large Kondo coupling scenario. Impurity spins are represented by blue double arrows and conduction electrons are represented by green arrows. Red glow represent the formation of spin-singlet formation between impurity spins and conduction spins screening the local moments.

Figure 2 . 5 :

 25 Figure 2.5: Left: large Fermi surface of YbRh 2 Si 2 seen through (a) band renormalization calculation and (b) ARPES, where 𝑓 -electron enter the description of Fermi surface at the temperature 𝑇 < 𝑇 𝐾 . Right: small Fermi surface of YbCo 2 Si 2 (a) LDA calculation and (b) ARPES, at the temperature 𝑇 > 𝑇 𝐾 . From [135]

Figure 2 . 6 :

 26 Figure 2.6: Schematic representation of dilution of magnetic impurities. In both case, impurity spin (in blue arrow) forms a spin-singlet with conduction spin, represented by red glow. Right: the conduction electrons hop from singlet site to bachelor site, left: the conduction electrons hops from non-magnetic site to non-magnetic site.

Figure 3 . 1 :

 31 Figure 3.1: Schematic view of matrix-DMFT approach of mapping a lattice problem (a) into a two local site local problem (b).

Figure 3 . 2 :

 32 Figure 3.2: Figure illustrating the bipartite nature of the lattice for a Néel ordered antiferromagnetic phase. Dotted rectangle in the figure represent a unit cell composed two lattice sites belonging to each sublattices 𝐴 and 𝐵.

CHAPTER 3 .Figure 3 . 3 :

 333 Figure 3.3: Schematic view of the generalized matrix-DMFT approach of mapping a lattice problem (a) onto four effective sites (b) in a bipartite system interacting with dynamical effective local bath in each case, where • denotes sub-lattice A and • denotes sub-lattice B, red arrows are Kondo impurity spins, and the black arrows denotes the dynamical local baths.

  .39) CHAPTER 3. MODEL, METHOD AND APPROXIMATIONS Here, a cavity site can belongs to either sublattice 𝐴 or 𝐵 tagged by index 𝛼. As before, we define (4 × 4) dynamical local bath matrix as

Figure 3 . 4 :

 34 Figure 3.4: (3.4a) DMFT loop for paramagneticKondo phase and (3.4b) for magnetically ordered phases

Figure 4 . 1 :

 41 Figure 4.1: Magnetic phase diagram of CeRu 2 (Si 𝑥 Ge 1-𝑥 ). Substitution of Si by Ge increases the unit cell volume. A cascade of magnetic to non-magnetic phase transition is observed for increase Si concentration. From [119]

  Figure 4.2: phase diagrams (a) 1D 'chain', (b) 2D 'square', (c) 3D 'cubic' lattices at T = 0. K = paramagnetic Kondo phase, F = ferromagnetic phase, sat. F = saturated ferromagnetic phase and AF = antiferromagnetic phase

Figure 4 . 3 :

 43 Figure 4.3: from left to right: ground state phase diagrams of the Kondo alloys for 𝑛 𝑐 =0.30, 0.70 and 0.30. AFII = Néel ordered anti-ferromagnetic phase, and F = ferromagnetic phase

Figure 4 . 4 :

 44 Figure 4.4: Schematic description of the Kondo alloys phase diagrams depicted on figure 4.3, with Kondo (K) and magnetically ordered (MO) phases. In order to fix the energy scale, the K-MO transition is arbitrarily chosen to be realized here at 𝑇 𝐾 ≈ 10 Kelvin for the Kondo lattice (𝑥 = 1) and at around 1 Kelvin in the dilute limit 𝑥 ≪ 1. Dashed lines describe four examples of Kondo alloys: Ce 𝑥 La 1-𝑥 Pt 2 Si 2 (circle), Ce 𝑥 La 1-𝑥 Ni 2 Ge 2 (star), Ce 𝑥 La 1-𝑥 Ru 2 Si 2 (square) and Ce 𝑥 La 1-𝑥 Pd 2 Si 2 (triangle).

Figure 5 . 1 :

 51 Figure 5.1: Left figures: Electronic density curve showing Kondo resonance peak (below) at on-resonance ARPES spectra. Right figure: Comparison of ARPES derived hybridized band and PAM hybridized band. From [231].

Figure 5 . 2 :

 52 Figure 5.2: First Brilloin zone of the square lattice, with indications of the points Γ = (0, 0), 𝑋 = (0, 𝜋), and 𝑀 = (𝜋, 𝜋).

Figure 5 . 3 :

 53 Figure 5.3: Spectral function evaluated 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 k axis corresponds to the high symmetry lines Γ -𝑋 -𝑀 -Γ in the square lattice first Brillouin zone (see figure 5.2). The corresponding electronic density of states 𝜌(𝜔) is plotted on the right side. Each individual cases of this figure are indicated in the phase diagram depicted in the figure 5.9.

Figure 5 . 4 :

 54 Figure 5.4: Spectral function evaluated 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 ★ = 0.08. The wavevector k axis corresponds to the high symmetry lines Γ -𝑋 -𝑀 -Γ in the square lattice first Brillouin zone (see figure 5.2). The corresponding electronic density of states 𝜌(𝜔) is plotted on the right side. Each individual case of this f are indicated in the phase diagram depicted in the figure 5.9.

Figure 5 . 6 :

 56 Figure 5.6: Fermi surface structure for 𝑇 𝐾 /𝑊 = 0.175 for 𝑛 𝑐 = 0.70 at 𝑥 = 𝑛 𝑐 -0.1, 𝑥 = 𝑛 𝑐 and 𝑥 = 𝑛 𝑐 + 0.1 from left to right. A clear evidence of Lifshitz-like transition is marked by the discontinuity of Fermi surface at 𝑥 = 𝑛 𝑐 .

Figure 5 . 5 :

 55 Figure 5.5: 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 figure5.2). 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. The red lines for 𝑇 𝐾 /𝑊 = 0.082 and 0.0058 separates the regimes 𝑥 < 𝑥 ★ and 𝑥 > 𝑥 ★ . In both cases, 𝑥 ★ increases with increasing 𝑛 𝑐 . However, for 𝑇 𝐾 /𝑊 = 0.0058, this is not visible because 𝑥 ★ lies between 𝑥 = 0.01 and 𝑥 = 0.20 for all values of 𝑛 𝑐 considered.

Figure 5 . 7 :

 57 Figure 5.7: 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 Fermi-surface. (b) 𝑇 𝐾 /𝑊 = 0.0031 with ferromagnetic ground states: we observe Zeeman splitting effect only.

Figure 5 . 8 :

 58 Figure 5.8: 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 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.

  [247]. For small to intermediate values of 𝑇 𝐾 , a paramagnetic Kondo phase with three distinct zones are identified: a dilute Kondo at 𝑥 ≪ 1, a dense Kondo at 𝑥 ≈ 1 separated by a large zone of intermediate state 𝑥 ★ < 𝑥 < 𝑛 𝑐 . When increasing the Kondo coupling, 𝑥 ★ tends towards 𝑛 𝑐 and for strong coupling only dilute and dense Kondo phases are obtained, separated by a Lifshitz transition. Now, we analyze the different regions of paramagnetic Kondo phases by means of effective self-energy Σ 𝐴𝑙𝑙𝑜 𝑦 (𝜔) and its corresponding effective mass 𝑚 ★ with a focus on spectral function, which is related to experimental observation of ARPES signals.

62 CHAPTER 5 .Figure 5 . 9 :

 62559 Figure 5.9: 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. (▲) indicates the individual cases presented in the figure 5.3 and figure 5.4 for 𝑛 𝑐 = 0.70. 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 ★ is obtained at concentration 𝑥 ★ (dashed line), and we find 𝑚 ★ < 0 in the intermediate region 𝑥 ★ < 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. Results for Bethe lattice can be found in the appendix C.2. 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.

Figure 5 .

 5 Figure 5.10: (a) Real part of the self-energy Σ ′ 𝐴𝑙𝑙𝑜 𝑦 (0)/𝑊, (b) Effective mass 𝑚 ★ /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 ★ which is characterized by a vanishing of 𝑚 ★ when 𝑇 𝐾 relatively small. At intermediate concentrations 𝑥 ★ < 𝑥 < 𝑛 𝑐 we find 𝑚 ★ < 0.

Figure 5 . 11 :

 511 Figure 5.11: Imaginary part of self-energy Σ ′′ (0)/𝑇 𝐾 for 𝑇 𝐾 /𝑊 = 0.169, 0.019 and 0.006. Σ ′′ (0)/𝑇 𝐾 is negative and increases with increasing 𝑇 𝐾 with a maximum at 𝑥 = 𝑛 𝑐 .

Figure 5 . 12 :

 512 Figure 5.12: Evolution of the critical concentration 𝑥 * with respect to the Kondo coupling 𝑇 𝐾 /𝑊 presented here on 𝑙𝑜 𝑔 -𝑙𝑜 𝑔 scale. Solid line represent 𝑛 𝑐 = 0.90, dashed line represent 𝑛 𝑐 = 0.70 and dotted line represent 𝑛 𝑐 = 0.30 with a slope 𝛾 of 0.83 ± 0.013, 0.78 ± 0.006, and 0.71 ± 0.016 respectively.

Figure 5 . 13 :

 513 Figure 5.13: Frequency dependence of the real part of the self-energy, 𝜔 -Σ ′ 𝐴𝑙𝑙𝑜 𝑦 (𝜔) + Σ ′𝐴𝑙𝑙𝑜 𝑦 (0) for 𝑛 𝑐 = 0.70 and 𝑇 𝐾 = 0.019. Top: for 𝑥 in the vicinity of the critical point 𝑥 ★ = 0.08 which is characterized by 𝑚 ★ = 0, we observe the emergence of a non-monotonicity at low energy. This leads to the gradual formation of a multiplebranches dispersion for 𝑥 > 𝑥 ★ . Center: for 𝑥 in the vicinity of 𝑛 𝑐 , the maximum is realized at 𝜔 < 0 for 𝑥 < 𝑛 𝑐 and at 𝜔 > 0 for 𝑥 > 𝑛 𝑐 resulting in a second change of sign of 𝑚 ★ . Bottom: for 𝑥 close to 1, we observe signatures of the singularityΣ 𝐾 (𝜔) = 𝑟 2𝜔+𝜆 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.

Figure 6 . 1 :

 61 Figure 6.1: Evolution of local potential scattering 𝑆 𝑎 (0) for 𝑛 𝑐 = 0.70 relatively large Kondo coupling 𝑇 𝐾 /𝑊 = 0.169 represented on Argand diagram. Solid line represent LPS for 𝒦 -site whereas dashed line represent LPS for 𝒩-site. (•) indicates Kondo lattice (𝑥 = 1.00), short vertical solid line indicates the concentrations 𝑥 = 𝑛 𝑐 ± 0.01 and (✖) the most diluted case (𝑥 = 0.01). Light red background highlights the electronic bandwidth 𝑊.

Figure 6 . 2 :

 62 Figure 6.2: Evolution of local potential scattering 𝑆 𝑎 (0) for 𝑛 𝑐 = 0.70 relatively low coupling 𝑇 𝐾 /𝑊 = 0.019 represented on Argand diagram. Solid line represent LPS for 𝒦 -site whereas dashed line represent LPS for 𝒩-site. (•) indicates Kondo lattice (𝑥 = 1.00), short vertical solid line indicates the concentrations 𝑥 = 𝑛 𝑐 ± 0.01 and (✖) the most diluted case (𝑥 = 0.01). Light red background highlights the electronic bandwidth 𝑊.

Figure 8 . 1 :

 81 Figure 8.1: Wigner-Seitz radius (𝑅 𝑊 𝑆 ) of 5𝑑, 4 𝑓 and 5 𝑓 metals as a function of atomic number Z, where Wigner-Seitz radius is defined as (4𝜋/3)𝑅 3𝑊 𝑆 = 𝑉 while 𝑉 being the equilibrium volume of the primitive unit cell. From[272]

Figure 8 . 2 :

 82 Figure 8.2: Hill plot for various uranium based compounds showing the transition temperatures of itinerant or localized 𝑓 -electron phases with respect to inter-atomic uranium-uranium distance. From [279].

Figure 8 . 3 :

 83 Figure 8.3: Schematic view 𝑗 𝑗-coupling scheming for an local 𝑓 -electron. (a) spinorbit splitting and (b) crystal-field splitting. Since, 5f electrons are closer to the nuclei, thus the crystal-field splitting is smaller and less relevant.

Figure 8 .

 8 4 presents a phase diagram obtained through exact diagonalization for a two-sites cluster with five electrons, characterized by 𝑧-component of the total angular momentum 𝒥 𝑧 = 𝐽 1𝑧 + 𝐽 2𝑧 , where 𝐽 1𝑧 and 𝐽 2𝑧 are angular momentum projections on site 1 and 2 respectively. In this study, orbital-dependent partial localizations of 𝑓 -electrons occurs with five partially localized phases: two ferromagnetically correlated phases with 𝒥 𝑧 = 15/2, 11/2 and three antiferromagnetically correlated phases with 𝒥 𝑧 = 1/2, 3/2, 5/2 were found. For 𝑡 3/2 > 𝑡 1/2 , at weak hopping, phase with 𝒥 𝑧 = 15/2 determined by fully localized 𝑗 𝑧 = 5/2, 1/2 and delocalized 𝑗 𝑧 = 3/2 was obtained. It was seen that the increase in hopping 𝑡 3/2 partially breaks the Hund's rule with 𝑡 3/2 > 𝑡 1/2 , thus at intermediate hopping a phase with 𝒥 𝑧 = 5/2 and at high hopping a phase 𝒥 𝑧 = 11/2, 1/2 was observed. Furthermore, for 𝑡 3/2 < 𝑡 1/2 , a phase with 𝒥 𝑧 = 3/2 was observed with empty 𝑗 -3/2 .

Figure 8 . 4 :

 84 Figure 8.4: Phase diagram obtained through exact diagonalization for two-sites cluster, derived from total magnetization 𝒥 𝑧 = 𝐽 1𝑧 + 𝐽 2𝑧 , where 𝐽 1𝑧 and 𝐽 2𝑧 are angular momentum projections on site 1 and 2 respectively. Here, only 𝑓 2 and 𝑓 3 configurations are considered. 𝑡 3/2 and 𝑡 1/2 = 𝑡 5/2 are nearest-site hopping along orbitals 𝑗 𝑧 = 3/2 and 𝑗 𝑧 = 1/2, 5/2 respectively. Extracted from[START_REF] Efremov | Dual nature of 5 𝑓 electrons: Effect of intra-atomic correlations on hopping anisotropies[END_REF] 305]

Figure 10

 10 Figure 10.1: Schematic view of intial optimizing process to get guess parameters while considering various 𝑓 -configurations using the Eqs. 9.28-9.31. Whenever the physical constraints are satisfied, the scheme ejects an output with 𝜆, 𝜇 and slaveboson vector Φ.

  Figure 11.1 shows the numerical results with slave-boson probabilities. For 𝑛 𝑓 = 3.00, all the slave-boson amplitudes are equivalent with |𝜑 ΓΓ | 2 = (1/ √

Figure 11 . 1 :

 111 Figure 11.1: Evolution of slave boson probabilities |𝜑 ΓΓ | 2 on multiplet-multiplet basis along with 𝑓 -electron occupation from 𝑛 𝑓 = 3.0 to 𝑛 𝑓 = 2.5 for charge sectors 𝑓 𝑀 configurations for non-interacting case (see table 11.1).

. 5 )Figure 11 . 2 :

 5112 Figure 11.2: Diagonal slave bosons probabilities |𝜑 ΓΓ | 2 on the isotropic line (𝑊 = 𝑊 ′ ) for all six charge sectors | 𝑓 𝑀 ; 𝐽, 𝐽 𝑧 ⟩. The black arrows on the x-axis points the towards the threshold values of 𝑊/𝑈 0 where transition in valency configuration is observed.

Figure 11 . 3 :

 113 Figure 11.3: Quasiparticules weight 𝑍 𝑗 𝑧 as a function of electronic bandwidth 𝑊/𝑈 0 on the isotropic line (𝑊 5/2 = 𝑊 3/2 = 𝑊 1/2 ) for orbitals 𝑗 𝑧 = 5/2, 3/2 and 1/2 for paramagnetic phase. The inset shows a zoom of 𝑍 𝑗 𝑧 around small 𝑊/𝑈 0 .

2 ;

 2 𝐽 = 4, 𝐽 𝑧 ⟩, | 𝑓 3 ; 𝐽 = 9/2, 𝐽 𝑧 ⟩, | 𝑓 3 ; 𝐽 = 5/2, 𝐽 𝑧 ⟩, | 𝑓 3 ; 𝐽 = 3/2, 𝐽 𝑧 ⟩, | 𝑓 4 ; 𝐽 = 4, 𝐽 𝑧 ⟩, | 𝑓 4 ; 𝐽 = 2, 𝐽 𝑧 ⟩ and | 𝑓 4 ; 𝐽 = 0, 𝐽 𝑧 ⟩.Due to Hund's rule, the quasiparticles weight 𝑍 𝑗 𝑧 associated with each orbital 𝑗 𝑧 is anisotropic such that 𝑍 5/2 ≠ 𝑍 3/2 ≠ 𝑍 1/2 . The inset in the figure11.3 presents this anisotropic behavior.

4 Figure 11 . 4 :

 4114 Figure 11.4: 𝐸 -𝐸 0 on the isotropic line from 𝑊/𝑈 4 = 0 to large 𝑊/𝑈 0 large. The inset shows the mean-field parameters 𝜆 and 𝜇.

. 8 )Figure 11 . 5 :

 8115 Figure 11.5: Energies of partially localized phases along the line 𝑊 + 𝑊 ′ = constant: (a) 𝑊 + 𝑊 ′ = 7 and (b) 𝑊 + 𝑊 ′ = 14 while 𝑊 = 𝑊 3/2 and 𝑊 ′ = 𝑊 5/2 = 𝑊 1/2 . Colors of each line correspond to the color of each phase in figure 11.6.

  Figure 11.6 compiles our results and presents the ground-state phase diagram. Below, we present detailed information about the various one-electron and two-electron localized phases on the different parts of our phase diagram.

Figure 11 . 6 :

 116 Figure 11.6: Phase diagram with partially localized phases for the electronic bandwidths 0 ≤ 𝑊 ≤ 15 and 0 ≤ 𝑊 ′ ≤ 15. Solid lines are guides to the eyes separating two phases. (FM/PM) indicates that ferromagnetic and paramagnetic phases are degenerate.

Figure 11 . 7 :

 117 Figure 11.7: Figure showing the line 𝑊 + 𝑊 ′ = 7 and 𝑊 + 𝑊 ′ = 14 on the phase diagram.

Figure

  Figure 11.8: Schematic view of an electron hopping from a site-𝑖 with 𝑓 𝑀 configuration to a site-𝑗 with 𝑓 𝑀 ′ configuration. For 𝑀 ′ < 𝑀, this hopping becomes favorable. For instance, the hopping from 𝑀 = 3 to 𝑀 = 2 is more favorable than other way around since the hopping from 𝑀 = 2 to 𝑀 = 3 will create more energetic states with 𝑀 = 1 and 𝑀 = 5 particles.

|

  𝑓 3 ; 𝐽 = 4, 𝐽 𝑧 = 3/2⟩, | 𝑓 3 ; 𝐽 = 3/2, 𝐽 𝑧 = 3/2⟩ and | 𝑓 4 ; 𝐽 = 4, 𝐽 𝑧 = 4⟩ contributing to the formation of the ground-state for For 𝑊 ′ > 𝑊, only the orbitals 𝑗 𝑧 = 5/2 and 3/2 are localized simultaneously in either PM, FM or SFM. For low 𝑊 2 of local multiplets | 𝑓 2 ; 𝐽 = 4, 𝐽 𝑧 = 4⟩, and | 𝑓 3 ; 𝐽 = 9/2, 𝐽 𝑧 = 9/2⟩ whereas FM 5 multiplets | 𝑓 3 ; 𝐽 = 9/2, 𝐽 𝑧 = 7/2⟩ and | 𝑓 4 ; 𝐽 = 4, 𝐽 𝑧 = 4⟩ than SFM 5

Figure 11 . 9 :

 119 Figure 11.9: Quasiparticle weights of non-localized orbitals along the line: 𝑊+𝑊 ′ = 7 ((a) and (c)) and 𝑊 + 𝑊 ′ = 14 ((b) and (d)) in the phase diagram. Only delocalized orbitals are presented here for PM phases ((c) and (d)) and for FM phases ((a) and (b)). Localized orbitals per phase are indicated on the top the figure, and aside of each line for each orbital. Vertical dotted lines are guides for eyes marking frontier between phases whereas solid vertical gray line indicate isotropic point.

Figure

  Figure 11.9(a) and (c) presents 𝑍 𝑗 𝑧 for non-localized orbitals on the line 𝑊 +𝑊 ′ = 7 for FM and PM phases respectively. Since there is no equivalent counterpart phase of SFM in PM phase, 11.9(c) misses 𝑍 𝑗 𝑧 partially. The left side of each graph in figure11.9 corresponds to the anisotropic region with 𝑊 > 𝑊 ′ where only orbital

  even though both phases are degenerate. Thus, the internal magnetic configuration can lead to the different quasiparticle weights per orbitals, even for degenerate phases. 𝑗 𝑧 = 3/2 orbital (𝑍 3/2 = 1) and localized -3/2 orbital (Z -3/2 = 0). In this case, only hopping of 𝑗 𝑧 = 3/2 is possible from 𝑓 3 to 𝑓 2 -configuration. Upon further decrease of anisotropy 𝑊 ≃ 𝑊 ′ , a sharp discontinuity in Z 𝑗 𝑧 is observed which marks the onset of non-localized paramagnetic phase (PM 𝑛𝐿 ). With 𝑊 ′ > 𝑊, orbitals 𝑗 𝑧 = 5/2 and 3/2 get localized. The quasiparticle weight for delocalized 𝑗 𝑧 = 1/2 and -1/2 remains same for FM phase which was not the case when 𝑗 𝑧 = 5/2 and 1/2 were localized. This can be explained since either an electron in 𝑗 𝑧 = 1/2 or -1/2 can hop from the local multiplet | 𝑓 4 ; 𝐽 = 4, 𝐽 𝑧 ⟩ to another local multiplet | 𝑓 3 ; 𝐽 = 9/2, 𝐽 𝑧 ⟩, to generate again | 𝑓 4 ; 𝐽 = 4, 𝐽 𝑧 ⟩, forming the macroscopic FM figure 11.9(b) and (d), we present 𝑍 𝑗 𝑧 on the line 𝑊 + 𝑊 ′ = 14. Similar conclusions can be drawn for FM 5

2 ,Figure 11 . 10 :

 21110 Figure 11.10: Electronic occupations for delocalized orbitals along 𝑊 + 𝑊 ′ = 7: (a) and (c), and along 𝑊 + 𝑊 ′ = 14: (b) and (d). Upper digrams (a) and (b) are for FM phases whereas middle diagrams (c) and (d) are for PM phases. Localized orbitals are indicated on the top the figure, and aside of each line for each orbital. Vertical dotted lines are guides for eyes marking frontier between phases whereas solid vertical gray line indicate isotropic point.

Figure 11 . 11 :

 1111 Figure 11.11: Magnetization is presented for only FM phases along 𝑊 + 𝑊 ′ = 7 (left), along 𝑊 + 𝑊 ′ = 14 (right). Localized orbitals are indicated on the top the figure. Vertical dotted lines are guides for eyes marking frontier between phases whereas solid vertical gray line indicate isotropic point.

𝑊 𝑗𝑧 4 (

 4 Figure 11.12: Up: renormalized electronic bandwidth (𝑊 𝑗 𝑧 𝑍 𝑗 𝑧 ) for 𝑗 𝑧 = 3/2 localized orbital, calculated at the parameters 𝑊 = 0 and 𝑊 ′ = 14 for paramagnetic phase (a) and ferromagnetic phase (b). For each bar, black color indicates the electronic bandwidth below Fermi level with occupied states and gray color indicates the electronic bandwidth with unoccupied states. Down: schematic view of Fermi surfaces on 2D square lattice. Thin black line represent the Fermi surfaces. Darker blue color shades around Fermi surfaces indicate low Fermi velocity 𝑣 𝐹 ∝ 𝑍 𝑗 𝑧 𝑊 𝑗 𝑧 with high effective mass 𝑚 * , whereas lighter blue color shades indicate lighter effective with higher Fermi velocity. The width of the shades is determined by 1/𝑊 𝑗 𝑧 𝑍 𝑗 𝑧 representing schematically the density of states at Fermi level.

Figure 11 .

 11 Figure 11.13: Up: renormalized electronic bandwidth (𝑊 𝑗 𝑧 𝑍 𝑗 𝑧 ) with 𝑗 𝑧 = 5/2 and 𝑗 𝑧 = 1/2 localized orbitals calculated at the parameters 𝑊 = 10 and 𝑊 ′ = 0 for paramagnetic phase (a), ferromagnetic phase (b) and saturated ferromagnetic phase (c). For each bar, black color indicates the electronic bandwidth below Fermi level with occupied states and gray color indicates the electronic bandwidth with unoccupied states. Down: schematic view of Fermi surfaces on 2D square lattice. Thin black line represent the Fermi surfaces. Darker blue color shades around Fermi surfaces indicate low Fermi velocity 𝑣 𝐹 ∝ 𝑍 𝑗 𝑧 𝑊 𝑗 𝑧 with high effective mass 𝑚 * , whereas lighter blue color shades indicate lighter effective with higher Fermi velocity. The width of the shades is determined by 1/𝑊 𝑗 𝑧 𝑍 𝑗 𝑧 representing schematically the density of states at Fermi level.

Figure 11 . 14 :

 1114 Figure 11.14: Variations of slave bosons probabilities Δ𝑝(𝑀) per 𝑓 -electron configuration represented here for lines 𝑊 + 𝑊 ′ = 7 ((a) and (c)) and 𝑊 + 𝑊 ′ = 14 ((b) and (d)). Slave bosons probabilities for FM phases are presented in the upper panel ((a) and (b)) whereas PM phases are presented in the lower panel ((c) and (d)). Here, Δ𝑝(2) = 𝑝(2) -0.5, Δ𝑝(3) = 𝑝(3) -0.5 and Δ𝑝 = 𝑝(4).

  each a, b, c and d are the blocks of 2 × 2 matrices and K. With the help of three Pauli matrices 𝜎 𝑥 , 𝜎 𝑦 and 𝜎 𝑧 as 2 × 2, we can write the expressions below which will be useful to map 4 × 4 Green's function matrices to 2 × 2 Green's functions matrices.

  𝛼 denotes sublattice type: A or B. Now, we would like to express G 𝜎 𝑙𝑜𝑐,𝛼 in reciprocal k-space. To do so, we use the Fourier transform asG 𝜎 𝑙𝑜𝑐,𝛼 = 1 𝑁 kk ′ ∈𝐵𝑍 G 𝜎 kk ′ 𝑒 𝑖(k-k ′ )R 𝛼 .Here, k,k' are in normal first Brillouin zone. Since, here we are considering antiferromagnetic phase with ordering vector (Q = (𝜋, 𝜋)) with two lattice sites per unit cell. We have the Brillouin zone reduced by half. We have to rewrite above equation on reduced Brillouin zone (RBZ).kk ′ 𝑒 𝑖(k-k ′ )R 𝛼 𝛿 kk ′ +G 𝜎 k+Qk ′ 𝑒 𝑖(k-k ′ )R 𝛼 𝑒 𝑖QR 𝛼 + G 𝜎 kk ′ +Q 𝑒 𝑖(k-k ′ )R 𝛼 𝑒 -𝑖QR 𝛼 + G 𝜎 k+Qk ′ +Q 𝑒 𝑖(k-k ′ )R 𝛼 } , (B.21) the factor 𝑒 ±𝑖QR 𝛼 is whether +1 or -1 depending upon sublattice type 'A' or 'B'. Rewriting 𝑒 ±𝑖QR 𝛼 as 𝛼,finally our equation becomes : kk -G 𝜎 k+Qk -G 𝜎 kk+Q + G 𝜎 k+Qk+Q } . (B.23) We define a new 4 × 4 local Green's function matrix Ḡ𝜎 𝑙𝑜𝑐 composed of G 𝜎 𝑙𝑜𝑐,𝛼∈𝐴and G 𝜎 𝑙𝑜𝑐,𝛼∈𝐵 as below where M . with the help of usual matrix relations:{A -1 B -1 C -1 } = {CBA} -1 and A -1 + B -1 = A -1 (B + A)B -1, and with few lines of straight forward algebraic calculations, we arrive to the expression ⊗ 1)(𝜎 𝑥 ⊗ 1) M𝜎-1 𝐷 (𝜎 𝑥 ⊗ 1) -(1 ⊗ E k ) M𝜎 𝐷 (Π + 𝜎 + 𝜎 𝑧 ⊗ Π - 𝜎 ) -1 .Again, with few lines of few lines matrix calculations, we arrive our final expressionG 𝜎 𝑙𝑜𝑐,𝛼 = 2 𝑁 k∈𝑅𝐵𝑍 (Π -1 𝛼𝜎 -E k Π ᾱ𝜎 E k ) -1 .

5. 8

 8 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 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. . . . . . . . . . . . . . . . . . . . . 5.9 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. (▲) indicates the individual cases presented in the figure 5.3 and figure 5.4 for 𝑛 𝑐 = 0.70. In the Kondo phases the solid line indicates the discontinuity of the self-energy observed at 𝑥 = 𝑛 𝑐for 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 ★ is obtained at concentration 𝑥 ★ (dashed line), and we find 𝑚 ★ < 0 in the intermediate region 𝑥 ★ < 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. Results for Bethe lattice can be found in the appendix C.2. 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 (a) Real part of the self-energy Σ ′ 𝐴𝑙𝑙𝑜 𝑦 (0)/𝑊, (b) Effective mass 𝑚 ★ /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 ★ which is characterized by a vanishing of 𝑚 ★ when 𝑇 𝐾 relatively small. At intermediate concentrations 𝑥 ★ < 𝑥 < 𝑛 𝑐 we find 𝑚 ★ < 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Imaginary part of self-energy Σ ′′ (0)/𝑇 𝐾 for 𝑇 𝐾 /𝑊 = 0.169, 0.019 and 0.006. Σ ′′ (0)/𝑇 𝐾 is negative and increases with increasing 𝑇 𝐾 with a maximum at 𝑥 = 𝑛 𝑐 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Evolution of the critical concentration 𝑥 * with respect to the Kondo coupling 𝑇 𝐾 /𝑊 presented here on 𝑙𝑜 𝑔 -𝑙𝑜 𝑔 scale. Solid line represent 𝑛 𝑐 = 0.90, dashed line represent 𝑛 𝑐 = 0.70 and dotted line represent 𝑛 𝑐 = 0.30 with a slope 𝛾 of 0.83 ± 0.013, 0.78 ± 0.006, and 0.71 ± 0.016 respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Frequency dependence of the real part of the self-energy, 𝜔-Σ ′ 𝐴𝑙𝑙𝑜 𝑦 (𝜔)+ Σ ′

67 6. 1 72 6. 2 73 6. 3

 671722733 Evolution of local potential scattering 𝑆 𝑎 (0) for 𝑛 𝑐 = 0.70 relatively large Kondo coupling 𝑇 𝐾 /𝑊 = 0.169 represented on Argand diagram. Solid line represent LPS for 𝒦 -site whereas dashed line represent LPS for 𝒩-site. (•) indicates Kondo lattice (𝑥 = 1.00), short vertical solid line indicates the concentrations 𝑥 = 𝑛 𝑐 ± 0.01 and (✖) the most diluted case (𝑥 = 0.01). Light red background highlights the electronic bandwidth 𝑊. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of local potential scattering 𝑆 𝑎 (0) for 𝑛 𝑐 = 0.70 relatively low coupling 𝑇 𝐾 /𝑊 = 0.019 represented on Argand diagram. Solid line represent LPS for 𝒦 -site whereas dashed line represent LPS for 𝒩-site. (•) indicates Kondo lattice (𝑥 = 1.00), short vertical solid line indicates the concentrations 𝑥 = 𝑛 𝑐 ±0.01 and (✖) the most diluted case (𝑥 = 0.01). Light red background highlights the electronic bandwidth 𝑊. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charge occupation on 𝒦 and 𝒩-sites with the dilution of magnetic impurities 𝑥 for different Kondo temperature 𝑇 𝐾 . Left: occupation for Kondo site, Right: occupation for non-Kondo site. The solid lines represent theoretical values expected for 𝑇 𝐾 /𝑊 → ∞. 𝑛 𝒦 and 𝑛 𝒩 verifies the sum rule as: 𝑥𝑛 𝒦 + (1 -𝑥)𝑛 𝒩 = 𝑛 𝑐 . . . . . . . . . . . . . . 74 8.1 Wigner-Seitz radius (𝑅 𝑊 𝑆 ) of 5𝑑, 4 𝑓 and 5 𝑓 metals as a function of atomic number Z, where Wigner-Seitz radius is defined as (4𝜋/3)𝑅 3 𝑊 𝑆 = 𝑉 while 𝑉 being the equilibrium volume of the primitive unit cell. From [272] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 8.2 Hill plot for various uranium based compounds showing the transition temperatures of itinerant or localized 𝑓 -electron phases with respect to inter-atomic uranium-uranium distance. From [279]. . . . 85 8.3 Schematic view 𝑗 𝑗-coupling scheming for an local 𝑓 -electron. (a) spinorbit splitting and (b) crystal-field splitting. Since, 5f electrons are closer to the nuclei, thus the crystal-field splitting is smaller and less relevant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Phase diagram obtained through exact diagonalization for two-sites cluster, derived from total magnetization 𝒥 𝑧 = 𝐽 1𝑧 + 𝐽 2𝑧 , where 𝐽 1𝑧 and 𝐽 2𝑧 are angular momentum projections on site 1 and 2 respectively. Here, only 𝑓 2 and 𝑓 3 configurations are considered. 𝑡 3/2 and 𝑡 1/2 = 𝑡 5/2 are nearest-site hopping along orbitals 𝑗 𝑧 = 3/2 and 𝑗 𝑧 = 1/2, 5/2 respectively. Extracted from [60, 305] . . . . . . . . . . . . . . . . . . 10.1 Schematic view of intial optimizing process to get guess parameters while considering various 𝑓 -configurations using the Eqs. 9.28-9.31. Whenever the physical constraints are satisfied, the scheme ejects an output with 𝜆, 𝜇 and slave-boson vector Φ. . . . . . . . . . . . . . . . 10.2 Schematic view of local minimization for a given phase with a set of 𝜑 Γ𝑛 non-zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Schematic view of the complete numerical process. The light green background shows the local minimization process, and its output is injected into a global minimization routine. . . . . . . . . . . . . . . 11.1 Evolution of slave boson probabilities |𝜑 ΓΓ | 2 on multiplet-multiplet basis along with 𝑓 -electron occupation from 𝑛 𝑓 = 3.0 to 𝑛 𝑓 = 2.5 for charge sectors 𝑓 𝑀 configurations for non-interacting case (see table 11.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Diagonal slave bosons probabilities |𝜑 ΓΓ | 2 on the isotropic line (𝑊 = 𝑊 ′ ) for all six charge sectors | 𝑓 𝑀 ; 𝐽, 𝐽 𝑧 ⟩. The black arrows on the x-axis points the towards the threshold values of 𝑊/𝑈 0 where transition in valency configuration is observed. . . . . . . . . . . . . . . . . . . . . 11.3 Quasiparticules weight 𝑍 𝑗 𝑧 as a function of electronic bandwidth 𝑊/𝑈 0 on the isotropic line (𝑊 5/2 = 𝑊 3/2 = 𝑊 1/2 ) for orbitals 𝑗 𝑧 = 5/2, 3/2 and 1/2 for paramagnetic phase. The inset shows a zoom of 𝑍 𝑗 𝑧 around small 𝑊/𝑈 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 𝐸 -𝐸 0 on the isotropic line from 𝑊/𝑈 4 = 0 to large 𝑊/𝑈 0 large. The inset shows the mean-field parameters 𝜆 and 𝜇. . . . . . . . . . . . . 11.5 Energies of partially localized phases along the line 𝑊 +𝑊 ′ = constant: (a) 𝑊 + 𝑊 ′ = 7 and (b) 𝑊 + 𝑊 ′ = 14 while 𝑊 = 𝑊 3/2 and 𝑊 ′ = 𝑊 5/2 = 𝑊 1/2 . Colors of each line correspond to the color of each phase in figure 11.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Phase diagram with partially localized phases for the electronic bandwidths 0 ≤ 𝑊 ≤ 15 and 0 ≤ 𝑊 ′ ≤ 15. Solid lines are guides to the eyes separating two phases. (FM/PM) indicates that ferromagnetic and paramagnetic phases are degenerate. . . . . . . . . . . . . . . . . 119 11.7 Figure showing the line 𝑊 + 𝑊 ′ = 7 and 𝑊 + 𝑊 ′ = 14 on the phase diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 11.8 Schematic view of an electron hopping from a site-𝑖 with 𝑓 𝑀 configuration to a site-𝑗 with 𝑓 𝑀 ′ configuration. For 𝑀 ′ < 𝑀, this hopping becomes favorable. For instance, the hopping from 𝑀 = 3 to 𝑀 = 2 is more favorable than other way around since the hopping from 𝑀 = 2 to 𝑀 = 3 will create more energetic states with 𝑀 = 1 and 𝑀 = 5 particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 11.9 Quasiparticle weights of non-localized orbitals along the line: 𝑊 + 𝑊 ′ = 7 ((a) and (c)) and 𝑊 +𝑊 ′ = 14 ((b) and (d)) in the phase diagram. Only delocalized orbitals are presented here for PM phases ((c) and (d)) and for FM phases ((a) and (b)). Localized orbitals per phase are indicated on the top the figure, and aside of each line for each orbital. Vertical dotted lines are guides for eyes marking frontier between phases whereas solid vertical gray line indicate isotropic point. . . . 122 11.10Electronic occupations for delocalized orbitals along 𝑊 + 𝑊 ′ = 7: (a) and (c), and along 𝑊 + 𝑊 ′ = 14: (b) and (d). Upper digrams (a) and (b) are for FM phases whereas middle diagrams (c) and (d) are for PM phases. Localized orbitals are indicated on the top the figure, and aside of each line for each orbital. Vertical dotted lines are guides for eyes marking frontier between phases whereas solid vertical gray line indicate isotropic point. . . . . . . . . . . . . . . . . . . . . . . . . . . 124 11.11Magnetization is presented for only FM phases along 𝑊 + 𝑊 ′ = 7 (left), along 𝑊 + 𝑊 ′ = 14 (right). Localized orbitals are indicated on the top the figure. Vertical dotted lines are guides for eyes marking frontier between phases whereas solid vertical gray line indicate isotropic point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3 Model, method and approximations

  3.1 The Kondo alloy model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Treatment of disorder due to Kondo impurity substitution . . . . . . 3.2.1 An overview of methods . . . . . . . . . . . . . . . . . . . . . . 3.2.2 A brief introduction to dynamical mean-field theory (DMFT) 3.2.3 Matrix DMFT/CPA . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.1 Paramagnetic Kondo phase . . . . . . . . . . . . . . . 3.2.3.2 Dynamical local bath . . . . . . . . . . . . . . . . . . 3.2.3.3 Antiferromagnetic phase . . . . . . . . . . . . . . . . 3.2.3.4 Local Green's function . . . . . . . . . . . . . . . . . 3.3 Treatment of local Kondo interaction . . . . . . . . . . . . . . . . . . . 3.3.1 An overview of methods . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Magnetically ordered phases: Weiss mean-field approximation 3.3.3 Paramagnetic Kondo phases: "large N" slave-bosons meanfield approximation . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Numerical approach and algorithm scheme . . . . . . . . . . . . . . .

  ⟨G 𝑖𝑗 ⟩ 𝑑𝑖𝑠 = ⟨P 𝑖 𝐺 𝑖𝑗 P † 𝑗 ⟩ 𝑑𝑖𝑠 = 𝑎𝑙𝑙𝑝𝑎𝑡 ℎ𝑠 ⟨P 𝑖 Π 𝑖𝑖 𝛾 𝑖𝑖 1 P † 𝑖 WP 𝑖 1 Π 𝑖 1 𝑖 1 𝛾 𝑖 1 𝑖 2 P † 𝑖 1 WP 𝑖 2 . . . Π 𝑖 𝑝 𝑖 𝑝 𝛾 𝑖 𝑝 𝑗 P † 𝑖 𝑝 WP 𝑗 Π 𝑗 𝑗 P † 𝑗 ⟩ 𝑑𝑖𝑠 .

	𝑖 𝐺 𝑖𝑗 P † 𝑗 =	x𝑖 x𝑗 𝐺 𝑖𝑗 x𝑖 ŷ𝑗 𝐺 𝑖𝑗 ŷ𝑖 x𝑖 𝐺 𝑖𝑗 ŷ𝑖 ŷ𝑗 𝐺 𝑖𝑗	.	(3.11)
	Multiplying the Eq. (3.10) by P 𝑖 on left side and P † 𝑗 on the right side and by taking
	the average over disorder denoted by ⟨. . . ⟩ 𝑑𝑖𝑠 , we get

3.2.3.2 Dynamical local bath

	CHAPTER 3. MODEL, METHOD AND APPROXIMATIONS
	Like in the standard DMFT formalism, the relation between local Green's function
	with cavity and without cavity is expressed as
	.19)

3.2.3.2.2 Local actions Local

  𝒦 and 𝐺 𝒩 is obtained from local effective action for 𝒦 and 𝒩 site. Green's functions for 𝐺 𝒦 (𝑖𝜔) 𝒦 -sites and for 𝐺 𝒩 (𝑖𝜔) 𝒩-sites are obtained respectively from the local actions below

		CHAPTER 3. MODEL, METHOD AND APPROXIMATIONS
	with
	𝜅 𝒦 𝒩 (𝑖𝜔) = -	1 -2𝑥 + 𝑥(𝑖𝜔 + 𝜇 -Σ 𝒦 )𝐺 𝒦 -(1 -𝑥)(𝑖𝜔 + 𝜇)𝐺 𝒩 ) 𝑥𝐺 𝒦 -(1 -𝑥)𝐺 𝒩	,	(3.28)
	where 𝐺
			.27)

Table 4 .

 4

1: Table that resumes local Green's function invariance according to the transformations for K, F and AFII phases

  4 𝑓 electrons are directly observed by ARPES experiments on varieties of Kondo lattice systems. The ARPES experiments carried on varieties of Kondo lattice systems like CeRu 2 Si 2 [223-225], CeRu 2 Ge 2 [226], CeBi [227],CeNiSn [228] and YbRh 2 Si 2

) 6 .2 Results: Local Potential scattering

 6

Here, we present our LPS results at Fermi level (𝜔 = 0) for 2D square lattice, however, supplementary calculations for Bethe lattice were also performed, and we got similar results as obtained in

[138]

, and are presented in the appendix C.3. As before in the chapter 5, we consider two Kondo coupling: a relatively large Kondo coupling 𝑇 𝐾 /𝑊 = 0.169 and an intermediate Kondo coupling 𝑇 𝐾 /𝑊 = 0.019 in order to analyze LPS to find additional signatures of the critical concentrations 𝑥 = 𝑥 ★ where the cancellation of effective mass 𝑚 * occurs (see section 5.3) and 𝑥 = 𝑛 𝑐 where Lifshitz-like transition occurs (see chapter 5.4).

  In this case, we can derive INHOMOGENEITY IN KONDO ALLOYS two relations for charge occupations in two different regimes: 𝑥 < 𝑛 𝑐 and 𝑥 > 𝑛 𝑐 and is presented in the Eq 6.5. 𝑥)𝑛 𝒩 = 𝑛 𝑐 /2 ,gives𝑛 𝒩 = 𝑛 𝑐 -𝑥/2(1 -𝑥) . 𝑥)𝑛 𝒩 = 𝑛 𝑐 /2 , gives 𝑛 𝒦 = 𝑛 𝑐 /2𝑥.From the above figure6.3, we can observe that at the large Kondo coupling 𝑇 𝐾 /𝑊 = 0.350, our results follow the theoretical large Kondo coupling limit (see Eq. 6.5). For dense regime 𝑥 > 𝑛 𝑐 , the occupation 𝑛 𝒦 rises steadily upon depleting Kondo atoms up to 𝑥 = 𝑛 𝑐 , and 𝑛 𝒩 is zero. This is because all the conduction electrons are occupied, forming singlets with impurities electrons. In our case, 𝑛 𝒩 is not rigorously zero since the Kondo coupling is not infinite. The dense-dilute transition concentration at 𝑥 = 𝑛 𝑐 is marked by a kink in 𝑛 𝒦 and 𝑛 𝒩 for the large 𝑇

	𝑥 < 𝑛 𝑐 , 𝑛 𝒦 = 0.5 , 𝑥𝑛 𝒦 + (1 -𝑥 > 𝑛 𝑐 , 𝑛 𝒩 = 0 , 𝑥𝑛 𝒦 + (1 -(6.5)

𝐾 . For dilute regime 𝑥 < 𝑛 𝑐 , 𝑛 𝒩 increases gradually whereas 𝑛 𝒦 becomes a plateau with a value 0.50. Lowering 𝑇 𝐾 increases the probability of charge occupation in non-Kondo sites even at 𝑥 ≈ 1.0 and suppress the kink present at 𝑥 = 𝑛 𝑐 . Thus, charge inhomogeneity is present in Kondo alloys between Kondo and non-Kondo sites upon dilution, and the intensity of charge inhomogeneity decreases with decreasing Kondo coupling.

Table 8 .

 8 1: Table resuming Barnes slave-boson representation In the table 8.1, |𝑣𝑎𝑐⟩ denotes vacuum state and 𝑒 † 𝑖 , 𝑑 † 𝑖 are bosonic operators corresponding to an empty and doubly-occupied site whereas 𝑓 † 𝑖𝜎 is fermionic creation operator for an electron of spin 𝜎 in a site 𝑖. Similarly, 𝑒 † 𝑖 , 𝑑 † 𝑖 obey Bose commutation relations and 𝑓 †

Table 8 . 3 :

 83 Table resuming Li, Wölfe and Hirschfeld's slave bosons representation From the above mapping (8.3), we can observe that this method introduces altogether two auxiliary fermionic quasiparticule operators : 𝑓 𝑖↑ , 𝑓 𝑖↓ and six bosonic operators : 𝑒 𝑖 , p 𝑖 , 𝑑 𝑖 where,

	p 𝑖 =	𝑝 𝑖↑↑ 𝑝 𝑖↑↓
		𝑝 𝑖↓↑ 𝑝 𝑖↓↓ .

Table 9 .

 9 1: Table showing RISB mapping for zero and one particle sector (𝑀) and 𝑀 = 𝑗 𝑧 𝜂 𝑗 𝑧 .

Table 9 .

 9 [START_REF] Landau | The theory of a Fermi liquid[END_REF] shows the RISB mapping only for zero (𝑀 = 0) and (𝑀 = 1) particle sectors, where the slave bosons 𝜙 † 𝑗 𝑧 𝑗 ′ 𝑧 connects physical Fock states |𝜂 𝑗 𝑧 ⟩ to quasiparticle state |𝜂 ′ 𝑗 𝑧 ⟩. In this particular case, the atomic multiplets are also Fock states. However, for 𝑀 ≥ 2, the is not necessarily true. Thus, it would be natural to define a new basis set associating both atomic mulitplets and Fock states. Even though, the basis sets can be chosen arbitrarily, we choose multiplet-Fock basis set where a multiplet state |Γ⟩ is an eigenstate state of local interaction as Û |Γ⟩ = 𝐸 Γ |Γ⟩ and a Fock state |𝑛⟩ an eigenstate state of the occupation number operator 𝑗 𝑧 𝑓 † 𝑗 𝑧 𝑓 𝑗 𝑧 . More precisely, multiplet basis |Γ⟩ is chosen for initial physical Hilbert space whereas Fock state |𝑛⟩ is chosen for auxiliary Hilbert space. Here, a multiplet state with 𝑀-particles, total angular momentum 𝐽, and it's projection over 𝑧-direction 𝐽 𝑧 is labelled as |Γ⟩ = | 𝑓 𝑀 ; 𝐽, 𝐽 𝑧 ⟩. Similarly, a Fock state |𝑛⟩

  insures a proper normalization of a state. The underline in |Γ⟩ distinguishes |Γ⟩ from |Γ⟩ since |Γ⟩ lives in Hilbert space of quasiparticle and boson states whereas |Γ⟩ lives in Hilbert space of physical electron. From the equation (9.4), we identify the slave-bosons 𝜙 Γ𝑛 associated to each pair of atomic multiplet |Γ⟩ and quasiparticles Fock state |𝑛⟩.

  5 , 𝑓 6 𝜆, 𝜇, Φ

				𝜆 1 , 𝜇 1 , Φ 1
				𝜆 2 , 𝜇 2 , Φ 2
	𝑓 1 , 𝑓 2 , 𝑓 3 ,	input	Eqs.
	𝑓 4 , 𝑓 5 , 𝑓 6	Φ 𝑖𝑛𝑖𝑡 𝑎𝑙	9.28-9.31	. . .
				. . .
			𝜆 and 𝜇 grid	𝜆 𝑛 , 𝜇 𝑛 , Φ 𝑛
			. . .
	Output	Output	Output
	𝜆, 𝜇, Φ		𝜆, 𝜇, Φ	𝜆, 𝜇, Φ

  input of 𝜑 𝐺𝑢𝑒 𝑠𝑠 Schematic view of local minimization for a given phase with a set of 𝜑 Γ𝑛 non-zeros.

	Γ𝑛	for a given phase
	evaluation of
	𝜆, 𝜇
	local minimization of 𝐸[Φ]
	calculation of 𝜑 Γ𝑛 from Eq. 9.31
	with constraints 9.28 and 9.29
	evaluation of 𝐸[Φ] from Eq. 9.32
	Figure 10.2:

Table 11 . 1 :

 111 .1 resumes the parameters used on various cases. Table resuming the parameters for different case of study.

		𝑛 𝑓	𝑈 0	Δ𝑈 2	Δ𝑈 4	𝑊 3/2 𝑊 5/2 = 𝑊 1/2
	Non-interacting
	limit	from 3.0 to 2.5 0	0	0	𝑊	𝑊
	Atomic limit	(2+3)/2 = 2.5	21.00 eV -2.72 eV -3.79 eV 0	0
	General case	(2+3)/2 = 2.5	21.00 eV -2.72 eV -3.79 eV 𝑊	𝑊 ′

Table 11 .

 11 2: Orbitals occupations (𝑛 𝑗 𝑧 ) and quasiparticules weights (𝑍 𝑗 𝑧 ) for various partially localized phases. Here, PM signifies paramagnetic phase, FM signifies ferromagnetic phase and SFM signifies saturated ferromagnetic phase and the notation PM5

	5 2 , 1 2	.

  𝑖𝑗 ⟩ 𝑑𝑖𝑠 = ⟨G 𝑖𝑗 ⟩ 𝑑𝑖𝑠 -⟨G 𝑖𝛼 ⟩G-1 𝑙𝑜𝑐 ⟨G 𝛼𝑗 ⟩ 𝑑𝑖𝑠 in the equation B.33, we get 𝑖𝛼 W⟨G 𝜎 𝑖𝛼 ⟩ 𝑑𝑖𝑠 [⟨G 𝜎 𝛼𝛼 ⟩ 𝑑𝑖𝑠 ] -1 𝑗 𝛾 𝛼𝑗 ⟨G 𝜎 𝛼 𝑗 ⟩ 𝑑𝑖𝑠 W . (B.34) In order to express the Eq. B.34 in reciprocal k-space is reduced Brillouin zone, we perform the same type of method as in B.21. Thus, we can easily express each term of the Eq. B.34 in RBZ as 𝛾 k WG 𝜎 kk W + 𝛼𝛾 k 𝛾 k+Q WG 𝜎 kk+Q W + 𝛼𝛾 k+Q 𝛾 k WG 𝜎 k+Qk W + 𝛾 k+Q 𝛾 k+Q WG 𝜎 k+Qk+Q W} , (B.37) where 𝛼 = 1 if a site belongs to sublattice A or -1 if a site belongs to sublattice B. In order to find the expression of ∆ 𝜎 𝛼 , we describe a tensor ∆ 𝜎 composed of ∆ 𝜎 𝛼∈𝐴 and ∆ 𝜎 𝛼∈𝐵 as : The next step would be rewriting B.38 using 𝛿 0 between A and B sites, and we obtain 𝛾 𝑖+𝛿 0 𝛾 𝛼+𝛿 0 𝑗+𝛿 0 WG 𝜎 𝑖+𝛿 0 𝑗+𝛿 0 W We insert Eqs. B.35, B.36 and B.37 in each of the terms of B.39 and treating with Pauli In this section, we present the mapping of slave-bosons on multiplet-Fock basis set. Altogether there are 64 multiplets with 116 slave-bosons. The table below resumes the multiplets and slave bosons according to different changer sectors.

	√
	|Γ 49 ⟩ =	5|111001⟩ + 3|110110⟩	𝜙 49,|111001⟩ ,
	(𝛼) 𝑖𝑗 ⟩ 𝑑𝑖𝑠 W , kk + 𝛼𝛾 k WG 𝜎 {𝛾 k WG 𝜎 kk+Q + 𝛼𝛾 k+Q WG 𝜎 𝑖𝛼 ⟩ 𝑑𝑖𝑠 = 𝛾 𝑖𝛼 W⟨G 𝜎 1 𝑁 k∈𝑅𝐵𝑍 k+Qk + 𝛾 k+Q WG 𝜎 k+Qk+Q } , 𝛾 𝛼 𝑗 ⟨G 𝜎 𝛼𝑗 ⟩ 𝑑𝑖𝑠 W = 1 𝑁 k∈𝑅𝐵𝑍 {𝛾 k G 𝜎 kk W + 𝛼𝛾 k G 𝜎 kk+Q W + 𝛼𝛾 k+Q G 𝜎 k+Qk W + 𝛾 k+Q G 𝜎 k+Qk+Q W} , 𝛾 𝑖𝛼 𝛾 𝛼𝑗 W⟨G 𝜎 𝑖 𝑗 𝑖𝑗 ⟩ 𝑑𝑖𝑠 W = 1 𝑁 k∈𝑅𝐵𝑍 1 |Γ 18 ⟩ = √ 2|001010⟩ -√ 5|010001⟩ -3 |Γ 50 ⟩ = |111010⟩ -4 |Γ 51 ⟩ = |111100⟩ 1/2 |Γ 34 ⟩ = |010110⟩ -√ 5|011001⟩ + |100101⟩ 2 2 |Γ 52 ⟩ = 3|100111⟩ -√ 5|011011⟩ 𝑈 4,2 𝜙 52,|100111⟩ , (B.35) (B.36) 𝜙 18,|001010⟩ , 𝜙 49,|110110⟩ 𝜙 33,|001110⟩ , 𝜙 50,|111010⟩ 𝜙 33,|010101⟩ , 𝜙 51,|111100⟩ 𝜙 33,|100011⟩ 𝜙 34,|010110⟩ , Appendix E 𝑖𝑗 𝜙 18,|010001⟩ 0 |Γ 19 ⟩ = 4|001100⟩ -|010010⟩ -5|100001⟩ 𝜙 19,|001100⟩ , 𝜙 19,|010010⟩ , 𝜙 34,|011001⟩ , 𝜙 52,|011011⟩ √ √ 𝜙 34,|100101⟩ -1/2 |Γ 35 ⟩ = |011010⟩ -√ 5|100110⟩ + |101001⟩ 𝜙 35,|011010⟩ , 1 |Γ 53 ⟩ = 2|101011⟩ -5|011101⟩ 𝜙 53,|101011⟩ , 𝜙 53,|011101⟩ Slave-bosons mappings 𝜙 19,|100001⟩ -1 |Γ 20 ⟩ = √ 2|010100⟩ -√ 5|100010⟩ 𝜙 20,|010100⟩ , 𝜙 20,|100010⟩ -2 |Γ 21 ⟩ = 3|011000⟩ -√ 5|100100⟩ 𝜙 21,|011000⟩ , 𝜙 21,|100100⟩ 𝜙 35,|100110⟩ , 𝜙 35,|101001⟩ -3/2 |Γ 36 ⟩ = |011100⟩ -5 8 |101010⟩ + |110001⟩ 𝜙 36,|011100⟩ , 𝜙 36,|101010⟩ , 𝜙 36,|110001⟩ 0 |Γ 54 ⟩ = 4|110011⟩ -|101101⟩ -5|011110⟩ 𝜙 54,|110011⟩ , 𝜙 54,|101101⟩ , 𝜙 54,|011110⟩ -1 |Γ 55 ⟩ = √ 2|110101⟩ -√ 5|101110⟩ 𝜙 55,|110101⟩ , 𝜙 55,|101110⟩ √ {𝛾 k ∆ 𝜎 = ∆ 𝜎 𝐴 0 0 ∆ 𝜎 . 𝑓 𝑛 𝐽 𝐽 𝑧 |Γ 𝑛 ⟩ 𝐸 Γ 𝜙 Γ𝑛 0 0 |Γ 22 ⟩ = |001100⟩ -|010010⟩ + |100001⟩ 𝑈 0 𝜙 22,|001100⟩ , 𝜙 22,|010010⟩ , 5/2 5/2 |Γ 37 ⟩ = |010011⟩ -|001101⟩ 𝑈 3, 5 2 -2 |Γ 56 ⟩ = 3|111001⟩ -5|110110⟩ 𝜙 56,|111001⟩ , 𝜙 37,|010011⟩ , 𝜙 56,|110110⟩ 𝜙 37,|001101⟩ (B.38) 0 0 0 |Γ 1 ⟩ = |000000⟩ 0 𝜙 1,|000000⟩ 𝜙 22,|100001⟩ 3/2 |Γ 38 ⟩ = |001110⟩ -|100011⟩ 𝜙 38,|001110⟩ , 0 0 |Γ 57 ⟩ = |110011⟩ -|101101⟩ + |011110⟩ 𝑈 4,0 𝜙 57,|110011⟩ , 𝐵 ∆ 𝜎 = 0 -𝑖 𝛾 𝑖𝛼 WG 𝜎 𝑖𝛼 0 0 𝛾 𝑖+𝛿 0 𝛼+𝛿 0 WG 𝜎 𝑖+𝛿 0 𝛼+𝛿 0 G 𝜎 𝑙𝑜𝑐,𝐴 0 0 G 𝜎 𝑙𝑜𝑐,𝐵 -1 𝑗 𝛾 𝛼 𝑗 G 𝜎 𝛼 𝑗 W 0 0 𝛾 𝛼+𝛿 0 𝑗+𝛿 0 G 𝜎 𝛼+𝛿 0 𝑗+𝛿 0 W (B.39) 1 5/2 5/2 |Γ 2 ⟩ = |000001⟩ 0 𝜙 2,|000001⟩ 3/2 |Γ 3 ⟩ = |000010⟩ 𝜙 3,|000010⟩ 1/2 |Γ 4 ⟩ = |000100⟩ 𝜙 4,|000100⟩ -1/2 |Γ 5 ⟩ = |001000⟩ 3 9/2 9/2 |Γ 23 ⟩ = |000111⟩ 𝑈 3, 9 2 𝜙 38,|100011⟩ 𝜙 57,|101101⟩ , 𝜙 22,|000111⟩ 7/2 |Γ 24 ⟩ = |001011⟩ 1/2 |Γ 39 ⟩ = |010110⟩ -|100101⟩ 𝜙 39,|010110⟩ , 𝜙 57,|011110⟩ 𝜙 22,|001011⟩ 5/2 |Γ 25 ⟩ = |010011⟩ + |001101⟩ 𝜙 25,|010011⟩ , 𝜙 25,|001101⟩ 𝜙 39,|100101⟩ -1/2 |Γ 40 ⟩ = |011010⟩ -|101001⟩ 5 5/2 5/2 |Γ 58 ⟩ = |011111⟩ 𝑈 5, 5 2 𝜙 58,|011111⟩ 𝜙 40,|011010⟩ , 𝜙 40,|101001⟩ 3/2 |Γ 59 ⟩ = |101111⟩ 𝜙 59,|101111⟩ 𝜙 5,|001000⟩ -3/2 |Γ 6 ⟩ = |010000⟩ 𝜙 6,|010000⟩ -3/2 |Γ 7 ⟩ = |100000⟩ 𝜙 7,|100000⟩ 2 4 4 |Γ 8 ⟩ = |000011⟩ 𝑈 4 3/2 |Γ 26 ⟩ = |001110⟩ + 32 5 |010101⟩ + |100011⟩ 𝜙 26,|001110⟩ , -3/2 |Γ 41 ⟩ = |011100⟩ -|110001⟩ 1/2 |Γ 60 ⟩ = |110111⟩ 𝜙 60,|110111⟩ 𝜙 41,|011100⟩ , 𝜙 26,|010101⟩ , -1/2 |Γ 61 ⟩ = |111011⟩ 𝜙 61,|111011⟩ 𝜙 41,|110001⟩ 𝜙 26,|100011⟩ -5/2 |Γ 42 ⟩ = |110010⟩ -|101100⟩ -3/2 |Γ 62 ⟩ = |111101⟩ 𝜙 62,|111101⟩ 𝜙 42,|110010⟩ , 𝜙 8,|000011⟩ 1/2 |Γ 27 ⟩ = |010110⟩ + 4 5 |011001⟩ + |100101⟩ 𝜙 27,|010110⟩ , 𝜙 42,|101100⟩ -5/2 |Γ 63 ⟩ = |111110⟩ 𝜙 63,|111110⟩ (B.33) . with 𝛼 is a cavity site either belongs to sublattice A or B. Inserting ⟨G (𝛼) ∆ 𝜎 𝛼 = 3 |Γ 9 ⟩ = |000101⟩ 𝜙 9,|000101⟩ 2 |Γ 10 ⟩ = √ 5|000110⟩ + 3|001001⟩ 𝜙 27,|011001⟩ , 𝜙 27,|100101⟩ 4 4 4 |Γ 43 ⟩ = |001111⟩ 𝑈 4,4 𝜙 43,|010011⟩ 6 0 0 |Γ 64 ⟩ = |111111⟩ 𝑈 6,0 𝜙 64,|111111⟩ 𝜙 10,|001100⟩ , 𝜙 10,|001001⟩ 1 |Γ 11 ⟩ = √ 5|001010⟩ + √ 2|010001⟩ 𝜙 11,|001010⟩ , 𝜙 11,|010001⟩ 0 |Γ 12 ⟩ = 2|001100⟩ + 3|010010⟩ + |100001⟩ 𝜙 12,|001100⟩ , 𝜙 12,|010010⟩ , 𝜙 12,|100001⟩ -1 |Γ 13 ⟩ = √ 5|010100⟩ + √ 2|100010⟩ 𝜙 13,|010100⟩ , 𝜙 13,|100010⟩ -2 |Γ 14 ⟩ = √ 5|011000⟩ + 3|100100⟩ 𝜙 14,|011000⟩ , 𝜙 14,|100100⟩ -1/2 |Γ 28 ⟩ = |011010⟩ + 4 5 |100110⟩ + |101001⟩ 3 |Γ 44 ⟩ = |010111⟩ 𝜙 44,|010111⟩ √ 𝜙 28,|011010⟩ , 2 |Γ 45 ⟩ = 5|100111⟩ + 3|011011⟩ 𝜙 45,|100111⟩ , with 𝜙 28,|100110⟩ , 𝜙 28,|101001⟩ -3/2 |Γ 29 ⟩ = |011100⟩ + 32 5 |101010⟩ + |110001⟩ 𝜙 45,|011011⟩ 1 |Γ 46 ⟩ = √ 5|101011⟩ + √ 2|011101⟩ 𝜙 46,|101011⟩ , 𝑈 3, 9 2 = 9𝑈 2 + 33𝑈 4 14 , 𝑈 4,4 = 𝑈 0 + 5𝑈 2 + 12𝑈 4 3 , 𝑈 5, 5 2 = 2𝑈 0 + 10𝑈 2 + 18𝑈 4 3 𝜙 29,|011100⟩ , 𝜙 29,|101010⟩ , 𝜙 29,|110001⟩ -5/2 |Γ 30 ⟩ = |110010⟩ + |101100⟩ 𝜙 30,|110010⟩ , -7/2 |Γ 31 ⟩ = |110100⟩ 𝜙 48,|101110⟩ 𝜙 31,|110100⟩ -1 |Γ 48 ⟩ = 5|110101⟩ + 2|101110⟩ 𝜙 48,|110101⟩ , 𝜙 30,|101100⟩ 𝜙 46,|011101⟩ 0 |Γ 47 ⟩ = 2|110011⟩ + 3|101101⟩ + |011110⟩ 𝜙 47,|101101⟩ , √ √ 𝜙 47,|011110⟩ 𝑈 3, 5 2 = 4𝑈 0 + 5𝑈 2 + 9𝑈 4 6 , 𝑈 4,0 = 3 , 4𝑈 0 + 5𝑈 2 + 9𝑈 4 𝜙 47,|110011⟩ , 𝑈 3, 3 2 = 15𝑈 2 + 6𝑈 4 7 , 𝑈 4,2 = 𝑈 0 + 8𝑈 2 + 9𝑈 4 3 , 𝑈 6,0 = 𝑈 0 + 5𝑈 2 + 9𝑈 4 3 .	,

𝜎 = 𝑖𝑗 𝛾 𝑖𝛼 𝛾 𝛼𝑗 W⟨G 𝑖𝑗 𝛾 𝑖𝛼 𝛾 𝛼𝑗 W⟨G 𝜎 𝑖𝑗 ⟩ 𝑑𝑖𝑠 W -𝑖 𝛾 𝑖𝑗 𝛾 𝑖𝛼 𝛾 𝛼 𝑗 WG 𝜎 𝑖𝑗 W 0 0

  Table that resumes local Green's function invariance according to the transformations for K, F and AFII phases . . . . . . . . . . . . . . . . 4.2 commensurate AF phases with and their respective wave ordering vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Table resuming Barnes slave-boson representation . . . . . . . . . . . 8.2 Table resuming Kotliar and Ruckenstien's slave bosons representation 8.3 Table resuming Li, Wölfe and Hirschfeld's slave bosons representation 9.1 Table showing RISB mapping for zero and one particle sector (𝑀) and 𝑀 = 𝑗 𝑧 𝜂 𝑗 𝑧 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Table resuming the parameters for different case of study. . . . . . . . 11.2 Orbitals occupations (𝑛 𝑗 𝑧 ) and quasiparticules weights (𝑍 𝑗 𝑧 ) for various partially localized phases. Here, PM signifies paramagnetic phase, FM signifies ferromagnetic phase and SFM signifies saturated ferromagnetic phase and the notation PM 5 𝑗 𝑧 = 5/2 and 𝑗 𝑧 = 1/2 orbitals localized. . . . . . . . . . . . . . . . . .

	2 , 1 2	means paramagnetic phase with

1Unconventional because it can not be explained by BCS theory

1The spin-orbit interaction scales as 𝑍 4 with the atomic number.

Remerciements Acknowledgments

agréables. Pour cela, je souhaite remercier Jonathan, Julie, Jeremy, et Alexandre Bouzdine, Catherine Pépin, Sophie Tencé, Claudine Lacroix accepting to be part of the jury of this thesis. I would thank Daniel Malterre for agreeing to read my manuscript even if circumstances prevented him from participating in this thesis jury.

I would like to thank the National Research Agency (ANR) and Deutsche Forschungsgemeinschaft (DFG) for the grant to conduct this thesis. Thank you to the Basque Country and to the doctoral school for their financial support during this thesis.

calculation of 𝜑 Γ𝑛 from Eq. 9.31 with constraints 9.28 and 9.29 evaluation of 𝐸[Φ] from Eq. 9.32 calculation of 𝜑 Γ𝑛 from Eq. 9.31 with constraints 9.28 and 9.29 evaluation of 𝐸[Φ] from Eq. 9.32 calculation of 𝜑 Γ𝑛 from Eq. 9.31 with constraints 9.28 and 9.29

One-electron localized phases

Let's start from the situation 𝑊 > 𝑊 ′ where we found FM 5 . From our calculations, the localization of orbital 𝑗 𝑧 = 5/2 was more energetically favorable than the localization of 𝑗 𝑧 = 1/2. This can be understood since the Coulomb matrix element ⟨𝑗 𝑧 𝑗 𝑧 | Û | -𝑗 𝑧 -𝑗 𝑧 ⟩ in the Hamiltonian (9.1) is 0.23 eV larger for 𝑗 𝑧 = 5/2 than 𝑗 𝑧 = 1/2. Moreover, PM phases are degenerate with FM phases, and they can be obtained by the linear combination of two FM phases with its counterpart obtained by flipping the quantization axis. Either way, none of these one-electron localized phases were found to be in the ground-state phase diagram. This result is in contrast with the previous result obtained within RISB [327] and also in the two-site model [START_REF] Efremov | Dual nature of 5 𝑓 electrons: Effect of intra-atomic correlations on hopping anisotropies[END_REF].

For the anisotropy 𝑊 < 𝑊 ′ , we were able to localize the orbital 𝑗 𝑧 = 3/2 both in FM or PM phases. As before, PM and FM phases are degenerate and PM phase can be obtained by the combination of two FM phases. The local configurations | 𝑓 A possible explication of localization of one-electron can be done in an analogous manner as Mott localization. Following the same scenario, the Coulomb interaction ⟨𝑗 𝑧 𝑗 𝑧 | Û |-𝑗 𝑧 -𝑗 𝑧 ⟩ between the electrons on 𝑗 𝑧 and -𝑗 𝑧 orbitals compete with the orbitaldependent electronic bandwidth as

. Thus, under a certain threshold in

, the localization of an electron in an orbital 𝑗 𝑧 can happen.

Two-electrons localized phases

Here, we analyze the phases with two localized electrons. At first, we concentrate on the region 𝑊 > 𝑊 ′ , where we obtained localization of electrons in the orbitals 𝑗 𝑧 = 5/2 and 1/2.

CHAPTER 12. CONCLUSIONS AND PERSPECTIVES

For 𝑊 ≫ 𝑈 0 , we recover the results of non-interacting limit with 𝑍 𝑗 𝑧 ≃ 1.00.

With the results in isotropic line in hand, we varied the anisotropy (𝑊 3/2 ≠ 𝑊 5/2 (= 𝑊 1/2 ) in electronic bandwidth with 𝑓 -electron count 𝑛 𝑓 = 2.5. We found that the intra-atomic correlations enhances the anisotropies resulting in the partial localization of electrons on the sub-dominant hopping channel. Altogether we have found twelve partially localized phases: three one-electron localized PM phases, three one-electron localized FM phases, two two-electron localized PM phases, two two-electron localized FM phases, and two two-electron localized SFM. Thereafter, we constructed an orbital-selective partially localized phase diagram by comparing their energies. Figure 11.6 𝑊 𝑗𝑧 plays a vital role. Furthermore, our phase diagram is consistent with the previous results [START_REF] Efremov | Dual nature of 5 𝑓 electrons: Effect of intra-atomic correlations on hopping anisotropies[END_REF]327].

To gain further insight into the nature of the partially localized phases, we have analyzed Fermi surfaces, orbital occupations, magnetization, and 𝑓 -electron valency configurations. With FM and PM being degenerate but having different occupations per non-localized orbitals, FM phases have twice more unique Fermi surface sheets than their counterpart PM phase. As expected, we observed that the magnetization of two-electron localized phases is larger than that of one-electron localized phases. We also found that increasing electronic bandwidth changes the 𝑓 -electron valency configurations, where four electrons states can form the ground-state satisfying Hund's rule for two-electron localized phases. Furthermore, we also find that the bandwidth anisotropy increases the weight of 𝑓 2 configuration. This study has several perspectives. At first, the model calculations can be extended to the parameters 𝑊 3/2 > 15 eV and 𝑊 5/2 (𝑊 1/2 ) > 15. In this case, the more energetic configurations like 𝑓 5 , 𝑓 1 can participate in the formation of partially localized phases. This may stabilize one-electron localized phases like FM 5 2 (PM 5 2) as ground-state for 𝑊 3/2 > 𝑊 5/2 , which was observed in previous study [327]. One could also seek the possibility of having an antiferromagnetic phase in the further analysis of partially localized phases. Furthermore, this present study provides crucial information to perform a future model calculation with the inclusion of magnetic field and also more qualitative band-structure calculations for other uranium-based heavy-fermions. A second direction is to account for more general DOS profiles. A

Part III

Appendices

Appendix A

Evaluation of energies and ground state phase diagrams

Total energy in each phase is determined by taking average of the Hamiltonian per site as 𝐸/𝒩 = ⟨𝐻⟩/𝒩, where 𝒩 is the total number of sites in our system. Below, we present the equations to calculate energy for each considered phase.

A.1 Expressions of energies

Paramagnetic Kondo phase

Ferromagnetic phase

Antiferromagnetic phase

A.2 Evaluation of ground state phase diagram

Since we have noticed that ground energies (without interaction, 𝐽 𝐾 = 0) for each method are not exactly same. So the phase diagram were constructed by evaluating the energies differences

Putting the equations (B. 22, B.23) in (B.24) we get

where Ḡ𝜎 kk is defined in the section B.2. With some simple algebraic calculations we obtain a simplified equation as

(B.27) Inserting the expression B.18 in the above Eq. B.27 and knowing that,

We get

Putting all three equations obtained above into the equation B.39, and we get an equation

From the above equation, one can also derive the equation for dynamical local bath for sub-lattices 𝛼 =A or B:

Appendix C

Results on Bethe lattice

Parallelly, with 2D lattice, we have equally evaluated DMFT equations for Bethe lattice with infinite coordination. In this appendix, we present the phase diagram, some results on local potential scattering, and static magnetic susceptibility that we have obtained for Bethe lattice. However, for the calculations on Bethe lattice, we have considered only the paramagnetic Kondo case. The aim of these extra calculations is to verify the pertinence of our DMFT study of the 2D lattice.

C.1 Equations for local Green's functions

Here, we present the equations for dynamical local bath and the local Green's functions for Bethe lattice. Since we consider Bethe lattice with infinite coordination, the Green's functions can be replaced by local Green's functions in the equations for the dynamical local bath in the Eq 3.21. This lead us to a much simpler equation as

Furthermore, the equation C.1 can be inserted in the expressions of local Green's functions

With few lines of algebraic calculations, we can obtain the equations as

C.2 Phase diagram

Au contrary to the 2D lattice, here we don't consider magnetically ordered phase.

The aim is to compare the dilute-dense paramagnetic Kondo phase diagram with a 2D lattice.

C.3 Local potential scattering

The local potential is described in section 6.1.1. We use the same definition, and we analyze local potential scattering for Bethe lattice in paramagnetic phase as for 2D lattice. It will permit us to compare our result with previously published results on local potential scattering in [138]. In this figure, we can clearly observe that the discontinuity seen in 2D lattice at 𝑥 = 𝑛 𝑐 disappears, and it is translated by a jump near 𝑥 = 𝑛 𝑐 . 𝑆 𝒩 (0) remains inside the noninteracting electronic bandwidth and the 𝑆 𝒦 (0) slowly crosses the bandwidth at the upper edge of the electronic bandwidth. Again, we observe that the imaginary part can take negative values.

C.3.1 Large Kondo coupling

C.3.2 Low Kondo coupling

Appendix D Toy model for negative mass

In this part of the appendix, we present a simple toy model to explain the possible origin of the cancellation of the effective mass corresponding to the exceptional point. We consider an impurity site that is coupled with the two reservoirs (a) and (b) via the hybridizations 𝑉 𝑎 and 𝑉 𝑏 respectively. In this case, the local Green function at the impurity can be written as

D.1 Two reservoirs model

where Δ 𝑏 = 𝜋𝜌 𝑏 (0)𝑉 2 𝑏 , 𝜌 𝑏 (0) being density of states of reservoir (b) at Fermi level. We rewrite λ = -𝜆 + 𝑖Δ 𝑏 . Furthermore, the local self-energy describing the reservoir (a) can be written as

APPENDIX D. TOY MODEL FOR NEGATIVE MASS

Henceforth, the effective mass can be expressed as

If the imaginary part Δ 𝑏 is zero, the effective mass is always larger or equal to 1. Otherwise, 𝑚 * can be decreased or even negative if the arg( λ) > 𝜋/4.

D.2 Application to Σ 𝑎𝑙𝑙𝑜 𝑦 (𝜔)

We rewrite the expression of self-energy alloy (see Eq. 5.5 in the section 5.2.1) as

At 𝜔 = 0 and with 𝐺 𝒩 (0) = 𝐺 ′ 𝒩 (0) + 𝑖𝐺 ′′ 𝒩 (0), we can identify from the equation (D.4) and (D.1) where the reservoir (a) acts as Kondo sites and reservoir (b) acts as non-Kondo sites. Thus, the hybridization 𝑉 2 𝑎 will get dressed by a factor 𝑥, the probability of having a Kondo site and similarly, (1 -𝑥) will dress the quantities related to non-Kondo sites. Thus, we get

Thus, dependent upon the value of ℜ( λ) and ℑ( λ), 𝑚 * can be either negative or positive. Hence, we can obtain both nature: particle or hole.

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