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In this PhD thesis, we investigate instances of how quenched disorder, charge fluctuations, and reduced spatial symmetries can alter the nature of phases and/or phase transitions in many-body systems with active -- and often frustrated -- magnetic degrees of freedom. The thesis is divided into three parts, each of which focuses on the impact of one of the aforementioned perturbations on a different type of physical system. Part I is concerned with the influence of quenched disorder on spiral spin liquids, i.e., correlated states of matter in which a frustrated magnet evades order by fluctuating between a set of degenerate (or quasi-degenerate) coplanar spin spirals. Using the classical J1-J2 Heisenberg model on the honeycomb lattice as a prototype, we analyze effects driven both by (i) isolated impurities and (ii) nonvanishing concentrations of defects at zero temperature. We address (i) by employing perturbative analytical techniques to identify different order-by-quenched-disorder mechanisms and characterize the impurity-induced spin textures. Besides demonstrating that systems hosting spiral spin liquids are exceptionally susceptible to long-range deformations, we prove that the textures generally acquire an oscillatory out-of-plane component which carries direct information about the ground-state manifold and, remarkably, constitutes a bosonic analogue of Friedel oscillations in metals. We investigate (ii) by performing extensive numerical simulations as a means to characterize the zero-temperature phases realized for different types of disorder. Our results show that the competition between incompatible order-by-quenched-disorder mechanisms can, already at a low to moderate concentrations of defects, destabilize long-range order and induce 'spiral spin-glass' states, in which spins are frozen despite displaying spatial correlations akin to those of a spiral spin liquid. We interpret this finding in the light of effective low-energy theories, which allow us to make concrete statements about the fate of the system at nonzero temperatures. Finally, we discuss extensions of our conclusions to three-dimensional models, as well as their applications to experiments. In Part II, we examine how strong valence fluctuations affect quantum phase transitions marked by the breakdown of coherent Kondo screening in metals with lattices of local magnetic moments. To this end, we study a generalized Anderson lattice model via a parton mean-field theory that describes various Kondo-screened phases in addition to a fractionalized Fermi liquid -- an unscreened phase where electron-like quasiparticles from the conduction sea coexist with fractionalized excitations of a quantum spin liquid. Our results indicate that, at fixed chemical potential, abrupt valence changes can coincide with Kondo breakdown transition and render it discontinuous. We also show that, at a fixed total filling, this translates into phase-separation tendencies which, upon the inclusion of long-range Coulomb interactions, give rise to inhomogeneous states where Kondo-screened and unscreened regions coexist on a mesoscopic scale. We conclude by connecting our findings to experiments in heavy-fermion compounds and other condensed-matter systems. Finally, Part III discusses how the reduction of the spatial symmetries of an SU(2) or generalized SU(N) antiferromagnet can induce different types of magnetic order. This thread includes two separate projects. First, we study the precise means through which a collinear SU(2) Heisenberg antiferromagnet becomes a ferrimagnet once all symmetries between its magnetic sublattices are eliminated. Using general symmetry principles, we elucidate that this process is associated with the splitting of degenerate magnon bands and that the unequal thermal population of the latter is ultimately what generates a uniform magnetization. We thus establish that, in the systems under consideration, ferrimagnetism is induced by thermal fluctuations and that the resulting magnetization curve has a nonmonotonic temperature dependence. This prediction is then supported by an explicit linear spin-wave calculation for a layered variant of a J1-J2 Heisenberg model on a square lattice, which allows us to analytically derive the low-temperature behavior of the uniform magnetization. We show that the same T^4 power law and general qualitative behavior is obtained for a layered and distorted version of the Shastry-Sutherland model. Finally, we complement our results by positively identifying fluctuation-induced ferrimagnetism in Mn2Mo3O8 and discussing the prospects for its emergence in the high-pressure phases of SrCu2(BO3)2. Second, we demonstrate that the concept of altermagnetism, a type of magnetic order that has attracted great interest lately as a potential gateway to unconventional phases of matter and novel technological applications, can be extended to SU(N) magnets with N>2. To begin, we indicate how simple models for such generalized altermagnets can be constructed by reducing the spatial symmetries of SU(N) Heisenberg Hamiltonians with N-color antiferromagnetic ground states. We then apply this procedure to concrete examples with N<5 and conduct comparative analyses of our N=2 and N=3 models via linear spin-wave and flavor-wave calculations. As a result, we find that both systems share the same characteristic behavior of insulating altermagnets, namely a symmetric splitting between magnon bands with different sets of quantum numbers and definite chiralities. Finally, we show that the analogy between the models persists when they are supplemented with charge carriers to become metallic. Much like its SU(2) counterpart, the SU(3) altermagnet exhibits flavor-split electronic bands with crossings determined by a combination of spin and lattice symmetries.:1 Introduction I Magnetic frustration, quenched disorder, and their interplay 2 Elements of frustrated magnetism 2.1 Heisenberg magnetism in Mott insulators 2.2 Frustration and classical ground-state degeneracies 2.3 Order by disorder 2.3.1 Thermal order by disorder 2.3.2 Quantum order by disorder 2.3.3 Conceptual differences and possible competition between thermal and quantum order by disorder 2.3.4 Pseudo-Goldstone modes and experimental realizations 2.4 Classical spin liquids 2.5 Quantum spin liquids 3 Quenched disorder in magnetic systems 3.1 Types of disorder and their consequences 3.2 Random-field disorder: Stability of long-range order and the Imry-Ma argument 3.3 Random-mass disorder: Rounding of first-order phase transitions and the Harris criterion 3.4 Spin glasses: Basic concepts and experimental signatures 3.4.1 The spin glass transition 4 Spiral spin liquids 4.1 J1-J2 Heisenberg model on the honeycomb lattice 4.1.1 Teaser: Enhanced ground-state degeneracy at J2/J1 = 1/2 4.2 Classical ground states and the spiral contour 4.3 Influence of thermal fluctuations 4.3.1 Low temperatures: Thermal order by disorder and nematic order 4.3.2 Intermediate temperatures: Spin liquid regimes 4.4 Linear spin-wave theory and quantum order by disorder 5 Disorder effects in spiral spin liquids 5.1 Disordered J1-J2 Heisenberg model on the honeycomb lattice 5.2 Isolated impurities 5.2.1 Linear response theory: Spin textures and energy corrections 5.2.2 Order by quenched disorder 5.2.3 Friedel-like oscillations 5.2.4 In-plane textures and their impact on long-range order 5.3 Numerics on clean finite systems 5.4 Nonvanishing concentration of impurities 5.4.1 Bond defects of the same type and orientation 5.4.2 Bond defects of random orientation: Emergent glassiness from competing order-by-quenched-disorder mechanisms 5.4.3 Randomly distributed vacancies 5.5 Effects or thermal fluctuations and additional couplings 5.5.1 J1-J2 honeycomb model 5.5.2 Spiral spin liquids in three dimensions 5.6 Connections to experiments 5.7 Summary and outlook II Quantum phase transitions in local-moment metals 6 Single-impurity and lattice Kondo effects: Experimemtal signatures, theory basics, and Kondo breakdown 6.1 Dilute magnetic impurities in metals and the Kondo effect 6.1.1 The Anderson impurity model and local moment formation 6.1.2 From the Anderson to the Kondo model 6.1.3 Kondo temperature and the onset of screening 6.1.4 Breakdown of the Kondo effect in quantum impurity models 6.2 Heavy-fermion metals and Kondo lattices 6.2.1 Anderson and Kondo lattice models 6.2.2 Heavy Fermi liquid 6.2.3 RKKY interaction 6.2.4 Doniach picture and quantum criticality in heavy-fermion systems 6.2.5 Breakdown of screening in Kondo lattices 7 Kondo-breakdown transitions and phase-separation tendencies in valence-fluctuating heavy-fermion metals 7.1 Anderson-Heisenberg lattice model and relevant limits 7.2 Parton mean-field theory 7.2.1 Mean-field equations 7.2.2 Implementation: Lattice, numerics, and temperature regime 7.3 Mean-field phases and phase diagrams 7.3.1 Mean-field phases 7.3.2 Phase diagram and Kondo breakdown at a fixed chemical potential 7.3.3 Phase diagram and Kondo breakdown at a fixed filling 7.4 Discussion: Fluctuation effects, frustrated phase separation, and inhomogeneous states 7.5 Summary and outlook III Modifying magnetic order by symmetry reduction 8 Classification of nonrelativistic collinear magnetic states 8.1 Spin-space and spin point groups 8.1.1 Four classes of collinear magnetism 8.2 Application to collinear magnets in metallic systems 8.3 Application to collinear magnets in insulating systems 8.3.1 Linear spin-wave theory for a collinear Heisenberg magnet 8.3.2 Symmetries of the linear spin-wave spectrum 9 Fluctuation-induced ferrimagnetism 9.1 Ferrimagnetism from thermal fluctuations in a toy model 9.1.1 The toy model, its symmetries, and its ground state 9.1.2 Ground-state antiferromagnetism 9.1.3 Spin-wave spectrum 9.1.4 Uniform magnetization at T>0 9.1.5 Discussion: Nonmonotonic magnetization curves and the influence of magnetic anisotropies 9.2 Ferrimagnetism in a layered and distorted Shastry-Sutherland model 9.2.1 Background: The Shastry-Sutherland model 9.2.2 The layered and distorted Shastry-Sutherland model 9.2.3 Spin-wave theory: Magnon spectrum and ferrimagnetism 9.2.4 Application to SrCu(BO3)2 under pressure 9.3 Summary and outlook 10 From SU(2) to SU(N) altermagnetism 10.1 Prelude: Elements of the theory of su(N) algebras, their representations, and applications to magnetism 10.1.1 The {M,0} irreps of SU(N): Schwinger bosons and the Holstein-Primakoff transformation 10.2 Heisenberg models for SU(N) altermagnetism 10.2.1 Symmetries and good quantum numbers 10.2.2 SU(2) Heisenberg model on the checkerboard lattice 10.2.3 SU(3) Heisenberg model on the hexatriangular lattice 10.2.4 SU(4) Heisenberg model on the cross-cubic lattice 10.3 Spin-wave and flavor-wave theories: Magnon spectra and their symmetries 10.3.1 SU(2) altermagnet 10.3.2 SU(3) altermagnet 10.4 Magnon chirality 10.4.1 SU(2) altermagnet 10.4.2 SU(3) altermagnet 10.5 Electronic band structure: Spin- and flavor-split bands 10.5.1 SU(2) checkerboard Kondo lattice model 10.5.2 SU(3) hexatriangular Kondo lattice model 10.6 Summary and outlook Appendices A Ground-state degeneracy at J2/J1 = 1/2 (Chapter 4) B Iterative minimization algorithm (Chapter 5) C Details on derivations from Chapter 5 C.1 Derivation of Eq. (5.10) C.2 Derivation of Eq. (5.16) C.3 Asymptotic behavior of Friedel-like oscillations for a circular spiral contour D In-plane component of impurity-induced textures: A case study (Chapter 5) D.1 Implementation D.2 Nearest-neighbor bond defect D.3 Next-nearest-neighbor bond defect D.4 Vacancy D.4.1 Changes in textures with increasing J2/J1 E Different parton mean-field theories (Chapter 7) E.1 Fermionic SU(N) and Sp(2N) theories E.1.1 SU(N) large-N limit E.1.2 Sp(2N) large-N limit E.2 Compressibility of different homogeneous spin-liquid states Bibliography

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This PhD thesis focuses on the mutual interplay of frustration and quenched disorder in magnetic insulators. Frustrated quantum magnets are known to host a plethora of interesting many-body phenomena ranging from noncollinear N\'el ordering to spin liquid phases. In this thesis, the consequences of the breakdown of translation symmetry, a widely occurring phenomenon in real materials, are studied in several examples of frustrated spin systems. The thesis is split into two parts dedicated to different kinds of frustrated magnets and the effects of quenched random perturbations in them. In the first part, bond randomness in frustrated noncollinear ordering is considered. Noncollinear magnetic orders originating from the spontaneous breakdown of continuous spin rotation symmetries at zero temperature are found to be unstable in the presence of exchange randomness. It is shown that in this case, the frustrated N\'{e}el ordering is destroyed for any magnitude of random exchange disorder. The resulting disordered ground states, however, possess interesting distinctions depending on the precise nature of the broken spin rotation symmetry. For SU(2) Heisenberg spins, it is demonstrated that the weak disordered ground describes a classical spin glass at zero temperature with a finite correlation length. At higher disorder, enhanced quantum fluctuations are predicted to modify that ground state into a random-singlet-like form. On the other hand, for noncollinear XY spin systems with U(1) or SO(2) symmetry which have stable integer-valued vortex topological defects, it is instead found that the weak disorder and the strong disorder ground states are distinct even at the classical level. The former has a quasi-long range order spin arrangement, while the latter exhibits a truly short-range ordered state. These two phases are shown to be separated by a Kosterlitz-Thouless-like phase transition point where vortex unbinding takes place. The spontaneously broken chiral degeneracy of noncollinear N\'el ordering is witnessed to be robust up to the point of the vortex-driven phase transition. In the second part of the thesis, the focus is switched to the effects of quenched disorder on quantum spin liquids. These are quantum disordered phases of matter with long-range entanglement, topological order, and fractionalised excitations that often arise in frustrated spin systems. The U(1) Dirac spin liquid with its magnetic monopole excitations has been identified as a parent state for N\'{e}el, valence-bond solid, and algebraic spin liquid phases. In this thesis, the fate of this state is studied in the presence of quenched random perturbations. It is demonstrated that a wide class of random perturbations induce monopole-driven confinement of the fractionalised quasi-particles of the spin liquid, leading to the onset of a spin glass-like order. Finally, dilution effects in the $\rm Z_2$ spin liquid phase of the Kitaev model are discussed in the presence of generic symmetry allowed interactions. The spin-liquid state remains stable when the non-Kitaev perturbations and dilution are small. However, the low-energy properties of the ground state are altered. It is shown that the degeneracies from the Majorana zero modes, which are known to localise at defect sites of the Kitaev spin liquid, are generically lifted by the non-Kitaev perturbations. Consequently, a dilution-tuned impurity band with a finite density of states is found to emerge.

Dans cette thèse, on étudie des processus stochastiques issus de la physique statistique. Le mouvement Brownien fractionnaire, objet central des premiers chapitres, généralise le mouvement Brownien aux cas où la mémoire est importante pour la dynamique. Ces effets de mémoire apparaissent par exemple dans les systèmes complexes et la diffusion anormale. L’absence de la propriété de Markov rend difficile l’étude probabiliste du processus. On développe une approche perturbative autour du mouvement Brownien pour obtenir de nouveaux résultats, sur des observables liées aux statistiques des extrêmes. En plus de leurs applications physiques, on explore les liens de ces résultats avec des objets mathématiques, comme les lois de Lévy et la constante de Pickands. / In this thesis, we study stochastic processes appearing in different areas of statistical physics: Firstly, fractional Brownian motion is a generalization of the well-known Brownian motion to include memory. Memory effects appear for example in complex systems and anomalous diffusion, and are difficult to treat analytically, due to the absence of the Markov property. We develop a perturbative expansion around standard Brownian motion to obtain new results for this case. We focus on observables related to extreme-value statistics, with links to mathematical objects: Levy’s arcsine laws and Pickands’ constant. Secondly, the model of elastic interfaces in disordered media is investigated. We consider the case of a Brownian random disorder force. We study avalanches, i.e. the response of the system to a kick, for which several distributions of observables are calculated analytically. To do so, the initial stochastic equation is solved using a deterministic non-linear instanton equation. Avalanche observables are characterized by power-law distributions at small-scale with universal exponents, for which we give new results.

De nombreux systèmes complexes soumis à un ajout continu d'énergie réagissent à cet ajout par une accumulation de tension au cours du temps, interrompue par de soudaines libérations d'énergie appelées avalanches. Récemment, il a été remarqué que plusieurs propriétés élémentaires de la dynamique d'avalanche sont issues de processus de relaxation ayant lieu à une échelle microscopique, processus qui sont négligés dans la plupart des modèles. Lors de ma thèse, j'ai étudié deux modèles classiques d'avalanches, modifiés par l'ajout d'une forme de relaxation la plus simple possible. Le premier système est une interface viscoélastique tirée à travers un milieu désordonné. En champ moyen, nous prouvons que l'interface a un comportement périodique caractérisé par une nouvelle échelle temporelle (émergente), avec des avalanches qui touchent l'ensemble du système. Le calcul semi-analytique de la force de friction agissant sur la surface donne des résultats compatibles avec les expériences de friction classique. En dimension finie (2D), les événements touchant l'ensemble du système (trouvés en champ moyen) deviennent localisés, et les simulations numériques donnent des résultats en bon accord avec plusieurs caractéristiques importantes des tremblements de terre, tant qualitativement que quantitativement. Le second système incluant également une forme très simple de relaxation est un modèle jouet d'avalanche : c'est la percolation dirigée. Dans notre étude d'une variante non-markovienne de la percolation dirigée, nous avons observé que la classe d'universalité était modifiée mais seulement partiellement. En particulier, un exposant change de valeur tandis que plusieurs relations d'échelle sont préservées. Cette idée d'une classe d'universalité étendue, obtenue par l'ajout d'une perturbation non-markovienne offre des perspectives prometteuses pour notre premier système. / Many complex systems respond to a continuous input of energy by an accumulation of stress over time, interrupted by sudden energy releases called avalanches. Recently, it has been pointed out that several basic features of avalanche dynamics are induced at the microscopic level by relaxation processes, which are neglected by most models. During my thesis, I studied two well-known models of avalanche dynamics, modified minimally by the inclusion of some forms of relaxation. The first system is that of a viscoelastic interface driven in a disordered medium. In mean-field, we prove that the interface has a periodic behaviour (with a new, emerging time scale), with avalanche events that span the whole system. We compute semi-analytically the friction force acting on this surface, and find that it is compatible with classical friction experiments. In finite dimensions (2D), the mean-field system-sized events become local, and numerical simulations give qualitative and quantitative results in good agreement with several important features of real earthquakes. The second system including a minimal form of relaxation consists in a toy model of avalanches: the Directed Percolation process. In our study of a non-Markovian variant of Directed Percolation, we observed that the universality class was modified but not completely. In particular, in the non-Markov case an exponent changes of value while several scaling relations still hold. This picture of an extended universality class obtained by the addition of a non-Markovian perturbation to the dynamics provides promising prospects for our first system.