Thermalization, the evolution of an interacting many-body system towards a thermal Gibbs ensemble after initialization in an arbitrary non-equilibrium state, is currently a phenomenon of great interest, both in theory and experiment. As the time evolution of a quantum system is unitary, the proposed mechanism of thermalization in quantum many-body systems corresponds to the so-called eigenstate thermalization hypothesis (ETH) and the typicality of eigenstates. Although this formally solves the contradiction of thermalizing but unitary dynamics in a closed quantum many-body system, it does neither make any statement on the dynamical process of thermalization itself nor in which way the coupling of the system to an environment can hinder or modify the relaxation dynamics. In this thesis, we address both the question whether or not a quantum system driven away from equilibrium is able to relax to a thermal state, which fulfills detailed balance, and if one can identify universal behavior in the non-equilibrium relaxation dynamics. As a first realization of driven quantum systems out of equilibrium, we investigate a system of Ising spins, interacting with the quantized radiation field in an optical cavity. For multiple cavity modes, this system forms a highly entangled and frustrated state with infinite correlation times, known as a quantum spin glass. In the presence of drive and dissipation, introduced by coupling the intra-cavity radiation field to the photon vacuum outside the cavity via lossy mirrors, the quantum glass state is modified in a universal manner. For frequencies below the photon loss rate, the dissipation takes over and the system shows the universal behavior of a dissipative spin glass, with a characteristic spectral density $\\mathcal{A}(\\omega)\\sim\\sqrt{\\omega}$. On the other hand, for frequencies above the loss rate, the system retains the universal behavior of a zero temperature, quantum spin glass. Remarkably, at the glass transition, the two subsystems of spins and photons thermalize to a joint effective temperature, even in the presence of photon loss. This thermalization is a consequence of the strong spin-photon interactions, which favor detailed balance in the system and detain photons from escaping the cavity. In the thermalized system, the features of the spin glass are mirrored onto the photon degrees of freedom, leading to an emergent photon glass phase. Exploiting the inherent photon loss of the cavity, we make predictions of possible measurements on the escaping photons, which contain detailed information of the state inside the cavity and allow for a precise, non-destructive measurement of the glass state. As a further set of non-equilibrium systems, we consider one-dimensional quantum fluids driven out of equilibrium, whose universal low energy theory is formed by the so-called Luttinger Liquid description, which, due to its large degree of universality, is of intense theoretical and experimental interest. A set of recent experiments in research groups in Vienna, Innsbruck and Munich have probed the non-equilibrium time-evolution of one-dimensional quantum fluids for different experimental realizations and are pushing into a time regime, where thermalization is expected. From a theoretical point of view, one-dimensional quantum fluids are particular interesting, as Luttinger Liquids are integrable and therefore, due to an infinite number of constants of motion, do not thermalize. The leading order correction to the quadratic theory is irrelevant in the sense of the renormalization group and does therefore not modify static correlation functions, however, it breaks integrability and will therefore, even if irrelevant, induce a completely different non-equilibrium dynamics as the quadratic Luttinger theory alone. In this thesis, we derive for the first time a kinetic equation for interacting Luttinger Liquids, which describes the time evolution of the excitation densities for arbitrary initial states. The resonant character of the interaction makes a straightforward derivation of the kinetic equation, using Fermi\'s golden rule, impossible and we have to develop non-perturbative techniques in the Keldysh framework. We derive a closed expression for the time evolution of the excitation densities in terms of self-energies and vertex corrections. Close to equilibrium, the kinetic equation describes the exponential decay of excitations, with a decay rate $\\sigma^R=\\mbox\\Sigma^R$, determined by the self-energy at equilibrium. However, for long times $\\tau$, it also reveals the presence of dynamical slow modes, which are the consequence of exactly energy conserving dynamics and lead to an algebraic decay $\\sim\\tau^$ with $\\eta_D=0.58$. The presence of these dynamical slow modes is not contained in the equilibrium Matsubara formalism, while they emerge naturally in the non-equilibrium formalism developed in this thesis. In order to initialize a one-dimensional quantum fluid out of equilibrium, we consider an interaction quench in a model of interacting, dispersive fermions in Chap.~\\ref. In this scenario, the fermionic interaction is suddenly changed at time $t=0$, such that for $t>0$ the system is not in an eigenstate and therefore undergoes a non-trivial time evolution. For the quadratic theory, the stationary state in the limit $t\\rightarrow\\infty$ is a non-thermal, or prethermal, state, described by a generalized Gibbs ensemble (GGE). The GGE takes into account for the conservation of all integrals of motion, formed by the eigenmodes of the Hamiltonian. On the other hand, in the presence of non-linearities, the final state for $t\\rightarrow\\infty$ is a thermal state with a finite temperature $T>0$. . The spatio-temporal, dynamical thermalization process can be decomposed into three regimes: A prequench regime on the largest distances, which is determined by the initial state, a prethermal plateau for intermediate distances, which is determined by the metastable fixed point of the quadratic theory and a thermal region on the shortest distances. The latter spreads sub-ballistically $\\sim t^$ in space with $0<\\alpha<1$ depending on the quench. Until complete thermalization (i.e. for times $t<\\infty$), the thermal region contains more energy than the prethermal and prequench region, which is expressed in a larger temperature $T_{t}>T_$, decreasing towards its final value $T_$. As the system has achieved local detailed balance in the thermalized region, energy transport to the non-thermal region can only be performed by the macroscopic dynamical slow modes and the decay of the temperature $T_{t}-T_\\sim t^$ again witnesses the presence of these slow modes. The very slow spreading of thermalization is consistent with recent experiments performed in Vienna, which observe a metastable, prethermal state after a quench and only observe the onset of thermalization on much larger time scales. As an immediate indication of thermalization, we determine the time evolution of the fermionic momentum distribution after a quench from non-interacting to interacting fermions. For this quench scenario, the step in the Fermi distribution at the Fermi momentum $k\\sub$ decays to zero algebraically in the absence of a non-linearity but as a stretched exponential (the exponent being proportional to the non-linearity) in the presence of a finite non-linearity. This can serve as a proof for the presence or absence of the non-linearity even on time-scales for which thermalization can not yet be observed. Finally, we consider a bosonic quantum fluid, which is driven away from equilibrium by permanent heating. The origin of the heating is atomic spontaneous emission of laser photons, which are used to create a coherent lattice potential in optical lattice experiments. This process preserves the system\'s $U(1)$-invariance, i.e. conserves the global particle number, and the corresponding long-wavelength description is a heated, interacting Luttinger Liquid, for which phonon modes are continuously populated with a momentum dependent rate $\\partial_tn_q\\sim\\gamma |q|$. In the dynamics, we identify a quasi-thermal regime for large momenta, featuring an increasing time-dependent effective temperature. In this regime, due to fast phonon-phonon scattering, detailed balance has been achieved and is expressed by a time-local, increasing temperature. The thermal region emerges locally and spreads in space sub-ballistically according to $x_t\\sim t^{4/5}$. For larger distances, the system is described by an non-equilibrium phonon distribution $n_q\\sim |q|$, which leads to a new, non-equilibrium behavior of large distance observables. For instance, the phonon decay rate scales universally as $\\gamma_q\\sim |q|^{5/3}$, with a new non-equilibrium exponent $\\eta=5/3$, which differs from equilibrium. This new, universal behavior is guaranteed by the $U(1)$ invariant dynamics of the system and is insensitive to further subleading perturbations. The non-equilibrium long-distance behavior can be determined experimentally by measuring the static and dynamic structure factor, both of which clearly indicate the exponents for phonon decay, $\\eta=5/3$ and for the spreading of thermalization $\\eta_T=4/5$. Remarkably, even in the presence of this strong external drive, the interactions and their aim to achieve detailed balance are strong enough to establish a locally emerging and spatially spreading thermal region. The physical setups in this thesis do not only reveal interesting and new dynamical features in the out-of-equilibrium time evolution of interacting systems, but they also strongly underline the high degree of universality of thermalization for the classes of models studied here. May it be a system of coupled spins and photons, where the photons are pulled away from a thermal state by Markovian photon decay caused by a leaky cavity, a one-dimensional fermionic quantum fluid, which has been initialized in an out-of-equilibrium state by a quantum quench or a one-dimensional bosonic quantum fluid, which is driven away from equilibrium by continuous, external heating, all of these systems at the end establish a local thermal equilibrium, which spreads in space and leads to global thermalization for $t\\rightarrow\\infty$. This underpins the importance of thermalizing collisions and endorses the standard approach of equilibrium statistical mechanics, describing a physical system in its steady state by a thermal Gibbs ensemble.

We study different aspects of perturbatively renormalized quantum gauge theories in the presence of non-trivial background Lorentzian metrics and background connections. First, we show that the proof of nilpotency of the renormalized interacting BRST charge can be reduced to the cohomological analysis of the classical BRST differential. This result guarantees the self-consistency of a class of local, renormalizable field theories with vanishing 'gauge anomaly'' at the quantum level, such as the pure Yang-Mills theory in four dimensions. Self-consistency here means that the algebra of gauge invariant observables can be constructed as the cohomology of this charge. Second, we give a proof of background independence of the Yang-Mills theory. We define background independent observables in a geometrical formulation as flat sections of a cohomology algebra bundle over the manifold of background configurations, with respect to a flat connection which implements background variations. We observe that background independence at the quantum level is potentially violated. We, however, show that the potential obstructions can be removed by a finite renormalization. Third, we construct the advanced/retarded Green's functions and Hadamard parametrices for linearized Yang-Mills and Einstein equations in general linear covariant gauges. They play an essential role in formulating gauge theories in curved spacetimes. Finally, we study a superconformal gauge theory in three dimensions (the ABJM theory) which is conformally coupled to a curved background. The superconformal symmetry of this theory is described by a conformal symmetry superalgebra on manifolds which admit twistor spinors. By analyzing the relevant cohomology class of an appropriate BV-BRST differential, we show that the full superalgebra is realized at the quantum level.

The goal of this Thesis is to give a presentation of some key issues regarding the non-perturbative renormalization of the periodic scalar field theories. As an example of the non-perturbative methods, we use the differential renormalization group approach, particularly the Wegner-Houghton and the Polchinski renormalization group equations, in order to investigate the renormalization of a one-component periodic scalar field theory. The Wegner-Houghton equation provides a resummation of the loop-expansion, and the Polchinski equation is based on the resummation of the perturbation series. Therefore, these equations are exact in the sense that they contain all quantum corrections. In the framework of these renormalization group equations, field theories with periodic self interaction can be considered without violating the essential symmetry of the model: the periodicity. Both methods - the Wegner-Houghton and the Polchinski approaches - are inspired by Wilson's blocking construction in momentum space: the Wegner-Houghton method uses a sharp momentum cut-off and thus cannot be applied directly to non-constant fields (contradicts with the &amp;quot;derivative expansion&amp;quot;); the Polchinski method is based on a smooth cut-off and thus gives rise naturally to a &amp;quot;derivative expansion&amp;quot; for varying fields. However, the shape of the cut-off function (the &amp;quot;scheme&amp;quot;) is not fixed a priori within Polchinski's ansatz. In this thesis, we compare the Wegner--Houghton and the Polchinski equation; we demonstrate the consistency of both methods for near-constant fields in the linearized level and obtain constraints on the regulator function that enters into Polchinski's equation. Analytic and numerical results are presented which illustrate the renormalization group flow for both methods. We also briefly discuss the relation of the momentum-space methods to real-space renormalization group approaches. For the two-dimensional Coulomb gas (which is investigated by a real-space renormalization group method using the dilute-gas approximation), we provide a systematic method for obtaining higher-order corrections to the dilute gas result.

The aim of this project is to understand the structure of the Standard Model of the particle physics. Therefore quantum field theories (QFT) are studied in the both cases of abelian and non-abelian gauge theories i.e. quantum electrodynamics (QED), quantum chromodynamics (QCD) and electroweak interaction are reviewed. The solution to the mass problem arising in these theories i.e. spontaneous symmetry breaking is also studied. / Syftet med detta projekt är att förstå strukturen för partikelfysikens standardmodell. Därför studeras kvantfältsteorier (QFT) i båda fallen av abelska och icke-abelska gaugeteorier, dvs kvantelektrodynamik (QED), kvantkromodynamik (QCD) och elektrosvag växelverkan granskas. Lösningen på massproblemet som uppstår i dessa teorier, dvs. spontant symmetribrott studeras också.

The goal of the original research presented in this thesis is to study the strong coupling regime of Quantum Field Theories (QFTs) with different methods, making concrete predictions about the phase structure and the dynamics of these theories, and on their observables. The focus is on (gauge) field theories in three spacetime dimensions, which are an interesting laboratory to understand the properties of strong coupling in setups that are usually simpler than in the more familiar case of gauge theories in four dimensions. Importantly, topological effects play a relevant role in three dimensions, thanks to the presence of the so-called Chern-Simons term.The thesis contains a short introduction to QFTs in 3d, principles and applications of infrared dualities, large N techniques, and holography. Indeed, the web of infrared dualities, the large N expansion, and the holographic correspondence between QFT and gravity are the main tools which we use to investigate the strongly coupled regimes of 3d QFTs.Then, the original material is presented. In a first line of research, we focus on the study of the phase diagram of a 3d gauge theory making use of conjectured infrared dualities, extending such dualities to the case where more than one mass parameter can be dialed. In a second line of research, we study a class of 3d gauge theories by engineering their gravity dual in a string theory setup. We prove the existence of multiple phase transitions between phases characterized by both massless particles and topological sectors. In a third line of research, we use holography as a tool to explore the interplay between the physics of 4d QCD and 3d gauge theories. In particular, we analyze the properties of 3d domain walls, which appear as soliton-like solutions of 4d QCD in specific parametric regimes. Finally, we propose a boundary construction of 3d large N vector models, which appear as critical points of theories obtained by coupling degrees of freedom localized on a 3d boundary to a 4d bulk theory. This construction allows to prove new dualities and uncovers a new computational tool for 3d vector models. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

In Chapter 1, we give an introduction to the topic of open string field theory. The concepts presented include gauge invariance, tachyon condensation, as well as the star product.<p>In Chapter 2, we give a brief review of vacuum string field theory (VSFT), an approach to open string field theory around the stable vacuum of the tachyon. We discuss the sliver state explaining its role as projector in the space of half-string basis. We review the construction of D-brane solutions in vacuum string field theory. We show that in the sliver basis the star product correspond to a matrix product. <p>Using the material introduced in the previous chapters, in Chapter 3 we establish a translation dictionary between open and closed strings, starting from open string field theory. Under this correspondence, we show that (off--shell) level--matched closed string states are represented by star algebra projectors in open string field theory. As an outcome of our identification, we show that boundary states, which in closed string theory represent D-branes, correspond to the identity string field in the open string side. <p>We then turn to noncommutative field theories. In Chapter 4, we introduce the framework in which we will work. The tools introduced are solitons, projectors, and partial isometries.<p>The ideas of Chapter 4 are applied to specific examples in Chapter 5, where we present new solutions of noncommutative gauge theories in which coincident vortices expand into circular shells. As the theories are noncommutative, the naive definition of the locations of the vortices and shells is gauge-dependent, and so we define and calculate the profiles of these solutions using the gauge-invariant noncommutative Wilson lines introduced by Gross and Nekrasov. We find that charge 2 vortex solutions are characterized by two positions and a single nonnegative real number, which we demonstrate is the radius of the shell. We find that the radius is identically zero in all 2-dimensional solutions. If one considers solutions that depend on an additional commutative direction, then there are time-dependent solutions in which the radius oscillates, resembling a braneworld description of a cyclic universe. There are also smooth BIon-like space-dependent solutions in which the shell expands to infinity, describing a vortex ending on a domain wall.<p>In Chapter 6, we review the Fronsdal models for free high-spin fields that exhibit peculiar properties. We discuss the triplet structure of totally symmetric tensors of the free String Field Theory and their generalization to AdS background.<p>In Chapter 7, in the context of massless higher spin gauge fields in constant curvature spaces discussed in chapter 6, we compute the surface charges which generalize the electric charge for spin one, the color charges in Yang-Mills theories and the energy-momentum and the angular momentum for asymptotically flat gravitational fields. We show that there is a one-to-one map from surface charges onto divergence free Killing tensors. These Killing tensors are computed by relating them to a cohomology group of the first quantized BRST model underlying the Fronsdal action.<p><p> / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Cette thèse s'inscrit dans le cadre de la physique mathématique des interactions fondamentales. Elle porte sur l'étude des théories de champs qui décrivent les particules élémentaires. En particulier, les théories de champs de spin élevé (plus grand ou égal à 2) sont analysées. Mis à part pour le graviton, vecteur supposé des interactions gravitationnelles, il n'y a aucun indice que ces champs soient présents dans la nature. Cependant leur existence n'est pas impossible théoriquemement et ils interviennent dans la théorie des cordes, candidate pour une théorie quantique d'unification de toutes les forces fondamentales y compris la gravitation. En effet, les modes de vibration de la corde élémentaire sont décrits par des champs de spin élevé.  <p>Dans ce travail, la dimension de l'espace-temps est laissée arbitraire, ce qui entraine la possibilité d'avoir plusieurs sortes (= représentations) de champs différentes ayant le même spin.<p>Le premier aspect traité dans cette thèse concerne les dualités, symétries qui relient entre elles plusieurs théories. Il est montré que différentes représentations de champs de spin élevé sont duales au niveau de l'action. En particulier, en dimension quatre, la dualité échange la composante électrique et la composante magnétique d'un même champs. Cette propriété est ensuite utilisée pour introduire des sources magnétiques pour les champs de spin élevé. La construction généralise les travaux de Dirac sur le couplage au champ électromagnétique de monopoles magnétiques. Une condition de quantification est également dérivée pour des quantités conservées, qui généralise la condition de quantification de Dirac pour la charge électrique en présence de monopoles magnétiques.<p>La deuxième partie de la thèse est consacrée aux interactions de champs de spin élevé. L'analyse est effectuée dans le formalisme de champs et d'antichamps dévelopé par Batalin et Vilkovsky. Elle repose sur la procédure de déformation de l'équation maîtresse mise au point par Henneaux et Barnich. Les champs étudiés sont les champs de spin deux exotiques (c-à-d différents du graviton) ainsi que les champs de spin trois complètement symétriques. Pour les premiers, il est prouvé que toutes les interactions doivent être abélienne. Il n'y a donc pas d'équivalent de la théorie d'Einstein pour ces champs. Dans le cas des champs de spin trois, plusieurs vertex cohérents au premier ordre sont obtenus.<p><p><br><p><p>In this thesis, we consider two aspects of higher-spin gauge field theories: dualities and interactions.<p>The first aspect is related to the presence of dualities, i.e. 'hidden' symmetries among gauge field theories. Do two higher-spin theories corresponding to different irreducible representations of the Poincaré group have the same physical content. Duality relations were already known at the level of the equations of motion and Bianchi identities, here we prove (in some cases) that these dualities hold also at the level of the action. As a consequence, the dual theories are formally equivalent. For example, in five space-time dimensions the spin-two theory of Pauli and Fierz is dual to the theory of a mixed-symmetry spin-two field written by Curtright. <p><p>In four space-time dimensions the duality exchanges the electric and magnetic degrees of freedom of the field. This property leads us to introduce external magnetic sources for higher-spin fields, thereby generalizing to arbitrary spin the work of Dirac on the coupling of magnetic monopoles to the electromagnetic field. Similarly to the quantization condition on the product of the electric and magnetic charges for electromagnetism, there is a quantization condition on the product of conserved ``electric' and ``magnetic' charges for higher spins.<p><p>The second aspect of higher-spin gauge field theories that is analysed in this thesis is the problem of interactions. Self-interactions of exotic spin-two gauge fields are studied, as well as self-interactions of completely symmetric spin-three fields. This is done in the BRST field-antifield formalism developped by Batalin and Vilkovisky, using the technique of consistent deformations of the master equation proposed by Barnich and Henneaux. <p><p> / Doctorat en sciences, Spécialisation physique / info:eu-repo/semantics/nonPublished

Scale symmetry is notoriously fickle: even when (approximately) present at the classical level, quantum fluctuations often break it, sometimes rather dramatically. Indeed, contemporary physics encompasses the study of very different phenomena at very different scales, e.g., from the (nominally) meV scale of spin systems, via the eV of electronic band structures, to the GeV of elementary particles, and possibly even the 10¹⁹ GeV of quantum gravity. However, there are often – possibly surprising – analogies between systems across these seemingly disparate settings. Studying the possible emergence of quantum scale symmetry and its breakdown is one way to systematically exploit these similarities, and in fact allows one to make testable predictions within a unified technical framework (viz., the renormalization group). The aim of this thesis is to do so for a few explicit scenarios. In the first four of these, quantum scale symmetry emerges in the long-wavelength limit near a quantum phase transition, over length scales of the order of the correlation length. In the fifth example, quantum scale symmetry is restored at very high energies (i.e., at and above the Planck scale), but severely constrains the phenomenology at 'low' energies (e.g., at accelerator scales), despite scale invariance being badly broken there. We begin with the Gross–Neveu (= chiral) SO(3) transition in D = 2+1 spacetime dimensions, which notably has been proposed to describe the transition of certain spin-orbital liquids to antiferromagnets. The chiral fermions that suffer a spontaneous breakdown of their isospin symmetry in this setting are fractionalized excitations (called spinons), and are as such difficult to observe directly in experiment. However, as gapless degrees of freedom, they leave their imprint on critical exponents, which may hence serve as a diagnostic tool for such unconventional excitations. These may be computed using (comparatively) conventional field-theoretic techniques. Here, we employ three complementary methods: a three-loop expansion in D = 4 - ε spacetime dimensions, a second-next-leading order expansion in large flavour number N , and a non-perturbative calculation using the functional renormalization group in the improved local potential approximation. The results are in fair agreement with each other, and yield combined best-guess estimates that may serve as benchmarks for numerical simulations, and possibly experiments on candidate spin liquids. We next turn our attention to spontaneous symmetry breaking at zero temperature in quasi-planar (electronic) semimetals. We begin with Luttinger semimetals, i.e., semimetals where two bands touch quadratically at isolated points of the Brioullin zone; Bernal-stacked bilayer graphene (BBLG) within certain approximations is one example. Luttinger semimetals are unstable at infinitesimal 4-Fermi interaction towards an ordered state (i.e., the field theory is asymptotically free rather than safe). Nevertheless, since the interactions are marginal, there are several pathologies in the critical behaviour. We show how these pathologies may be understood as a collision between the IR-stable Gaußian fixed point and a critical fixed point distinct from the Gaußian one in d = 2 + ε spatial dimensions. Observables like the order-parameter expectation value develop essential rather than power-law singularities; their exponent, as shown herein by explicit computation for the minimal model of two-component ‘spinors’, is distinct from the mean-field one. More tellingly, although finite critical exponents often default to canonical power-counting values, the susceptibility exponent turns out to be one-loop exact, and, in said minimal model takes the value γ = 2γᵐᵉᵃⁿ⁻ᶠᶦᵉˡᵈ = 2. Such an exact yet non-mean-field prediction can serve as a useful benchmark for numerical methods. We then proceed to scenarios in D = 2 + 1 spacetime dimensions where Dirac fermions can arise from Luttinger fermions due to low rotational symmetry. In BBLG, the 'Dirac from Luttinger' mechanism can occur both due to explicit and spontaneous breaking of rotational symmetry. The explicit symmetry breaking is due to the underlying honeycomb lattice, which only has C₃ symmetry around the location of the band crossings (so-called K points). As a consequence, the quadratic band crossing points each split into four Dirac cones, which is shown explicitly by computing the two-loop self-energy in the 4-Fermi theory. Within our approximations, we can estimate the critical coupling up to which a semimetallic state survives; it is finite (unlike a quadratic band touching point with high rotational symmetry), but significantly smaller than a 'vanilla' Dirac semimetal. Based on the ordering temperature of BBLG, our rough estimate further shows that the (effective) coupling strength in BBLG may be close to the critical value, in sharp contrast to other quasi-planar Dirac semimetals (such as monolayer graphene). Rotational symmetry in BBLG may also be broken spontaneously, i.e., due to the presence of nematic order, whereby a quadratic band crossing splits into two Dirac cones. Such a scenario is also very appealing for BBLG, since the precise nature of the ordered ground state of BBLG has not been established unambiguously: whilst some experiments show an insulating ground state with a full bulk gap, others show a partial gap opening with four isolated linear band crossings. Here, we show within a simplified phenomenological model using mean-field theory that there exists an extended region of parameter space with coexisting nematic and layer-polarized antiferromagnetic order, with a gapless nematic phase on one side and a gapped antiferromagnetic phase on the other. We then show that the nematic-to-coexistence quantum phase transition has emergent Lorentz invariance to one-loop in D = 2 + ε as well as D = 4 - ϵ dimensions, and thus falls into the celebrated Gross-Neveu-Heisenberg universality class. Combining previous higher-order field-theoretic results, we derive best-guess estimates for the critical exponents of this transition, with the theoretical uncertainty coming out somewhat smaller than in the monolayer counterpart due to the enlarged number of fermion components. Overall, BBLG may hence be a promising candidate for experimentally accessible Gross–Neveu quantum criticality in D = 2 + 1 spacetime dimensions. Finally, we turn our attention to the 'low-energy' consequences of transplanckian quantum scale symmetry. Extensions to the Standard Model that tend to lower the Higgs mass have many phenomenologically attractive properties (e.g., it would allow one to accommodate a more stable electroweak vacuum). Dark matter is one well-motivated candidate for such an extension. However, even in the most conservative settings, one usually has to contend with a significantly enlarged number of free parameters, and a concomitant reduction of predictivity. Here, we investigate how asymptotic safety (i.e., imposing quantum scale symmetry at the Planck scale and above) may constrain the Higgs mass in Standard Model (plus quantum gravity) when coupled to Yukawa dark matter via a Higgs portal. Working in a toy version of the Standard Model consisting of the top quark and the radial mode of the Higgs, we show within certain approximations that the Higgs mass may be lowered by the necessary amount if the dark scalar undergoes spontaneous symmetry breaking, as a function of the dark scalar mass, which is the only free parameter left in the theory.:1 Introduction 1.1 Scale invariance – why and where 1.1.1 Fundamental quantum field theories 1.1.2 Universality 1.1.3 Novel phases of matter 1.2 Outline of this thesis 2 Renormalization Group: A Brief Review 2.1 Quantum fluctuations and generating functionals 2.2 Renormalization group flow 2.3 Basic notions 2.4 Scale transformations, scale symmetry and RG fixed points 2.5 Characterization and interpretation of RG fixed points 2.5.1 Formal aspects 2.5.2 Scaling at (quantum) phase transitions 2.5.3 Predictivity in fundamental physics 2.5.4 Effective asymptotic safety in particle physics and condensed matter 3 Gross–Neveu SO(3) Quantum Criticality in 2 + 1 Dimensions 3.1 Effective field theory 3.2 Renormalization and critical exponents 3.2.1 4 - ϵ expansion 3.2.1.1 Method 3.2.1.2 Flow equations 3.2.1.3 Critical exponents 3.2.2 Large-N expansion 3.2.2.1 Method 3.2.2.2 Critical exponents 3.2.3 Non-perturbative FRG 3.2.3.1 Flow equations 3.2.3.2 Representation of the effective potential 3.2.3.3 Choice of regulator 3.2.3.4 Limiting behaviour 3.3 Discussion 3.3.1 General behaviour and qualitative aspects 3.3.2 Quantitative estimates for D = 3 3.4 Summary and outlook 4 Luttinger Fermions in Two Spatial Dimensions 4.1 Introduction 4.2 Action from top-down construction 4.3 Renormalization 4.3.1 4-Fermi formulation 4.3.2 Yukawa formulation 4.4 Fixed-point analysis 4.5 Non-mean-field behaviour 4.5.1 Order-parameter expectation value 4.5.2 Susceptibility exponent 4.6 Bottom-up construction: Spinless fermions on kagome lattice 4.6.1 Tight-binding dispersion 4.6.2 From Hubbard to Fermi 4.6.3 Fate of particle-hole asymmetry 4.7 Discussion 5 Dirac from Luttinger I: Explicit Symmetry Breaking 5.1 From lattice to continuum 5.1.1 Fermions on Bernal-stacked honeycomb bilayer 5.1.2 Continuum limit 5.1.3 Interactions 5.2 Mean-field theory 5.3 Renormalization-group analysis 5.3.1 Flow equations 5.3.2 Basic flow properties 5.3.3 Phase diagrams 5.4 Discussion 5.5 Summary and outlook 6 Dirac from Luttinger II: Spontaneous Symmetry Breaking 6.1 Model 6.2 Phase diagram and transitions 6.3 Emergent Lorentz symmetry 6.3.1 Loop expansion near lower critical dimension 6.3.1.1 Minimal 4-Fermi model 6.3.1.2 Gross–Neveu–Heisenberg fixed point 6.3.1.3 Fate of rotational symmetry breaking 6.3.2 Loop expansion near upper critical dimension 6.3.2.1 Gross–Neveu–Yukawa–Heisenberg model 6.3.2.2 Gross–Neveu–Yukawa–Heisenberg fixed point 6.3.2.3 Fate of rotational symmetry breaking 6.4 Critical exponents 6.5 Discussion 7 Higgs Mass in Asymptotically Safe Gravity with a Dark Portal 7.1 Review: The asymptotic safety scenario for quantum gravity and matter 7.2 Review: Higgs mass, and RG flow in the SM and beyond 7.2.1 Higgs mass in the SM 7.2.2 Higgs mass bounds in bosonic portal models 7.2.3 Higgs mass in asymptotic safety 7.2.4 Higgs Portal and Asymptotic Safety 7.3 Higgs mass in an asymptotically safe dark portal model 7.3.1 The UV regime 7.3.2 Flow towards the IR 7.3.3 Infrared masses 7.3.4 From the UV to the IR – Contrasting effective field theory and asymptotic safety 7.4 Discussion 8 Conclusions Appendices A Position-space propagator for C₃-symmetric QBT B Two-sided Padé approximants for C₃-symmetric QBTs C Corrections to the mean-field nematic order-parameter effective potential due to explicit symmetry breaking D Self-energy in anisotropic Yukawa theory E Master integrals for anisotropic Yukawa theory Bibliography

The Gross-Neveu model is a quantum field theory of interacting N-flavor fermions in 1+1dimensions, with interaction term $(\bar{\psi}_f\psi_f )^2$. This model is studied using the property offactorized scattering. The spectrum of bound states including the kinks are discussed andthe thermodynamic state equations are derived using the thermodynamic Bethe ansatz.The full particle-particle integral kernel and corresponding S-matrix is derived startingfrom the Gross-Neveu version of the Y -system introduced by Zamolodchikov. / Gross-Neveu-modellen är en kvantfältteori som beskriver N identiska versioner av fundamentala fermioner i 1 + 1 dimensioner, växelverkande med potentialen $(\bar{\psi}_f\psi_f )^2$. Modellen studeras med utgångspunkt i partiklarnas så kallade faktoriserade spridning. Samtligafysikaliska bundna tillstånd inklusive solitonerna diskuteras och de termodynamiska tillståndsekvationerna härleds med hjälp av Bethe-ansatsen. Alla integralkärnor och motsvarande S-matriselement beräknas på sluten form utifrån Y-systemet som först beskrevs av Zamolodchikov.

In dieser Dissertation erweitern wir die Ideen des Yangian-Bootstrap-Algorithmus auf Feynman-Diagramme mit massiven Teilchen. Ausgehend von der massiven dual-konformen Symmetrie der N = 4 Super-Yang-Mills Theorie auf dem Coulomb-Zweig konstruieren wir einen Satz von bilokalen Yangian Level-Eins Generatoren und zeigen, dass sie eine unendliche Anzahl von planaren ein- und zwei-Schleifen-Diagrammen vernichten. Wir beschreiben außerdem wie der dual-konforme Level-Eins Impuls-Operator auf eine massive Verallgemeinerung des gewöhnlichen spezial-konformen Generators im Impulsraum abgebildet wird. Als nächstes wenden wir den Yangian-Bootstrap-Algorithmus mit großem Erfolg auf eine Reihe von massiven Ein-Schleifen-Diagrammen mit verallgemeinerten Propagatorexponenten und in beliebiger Anzahl von Raumdimensionen an. Im Spezialfall der dual-konformen Integrale, deren Propagatorexponenten sich zur Raumdimension addieren, finden wir neue sehr einfache Darstellungen durch hypergeometrische Funktionen, die eine natürliche Verallgemeinerung für Diagramme mit beliebig vielen äußeren Punkten erlauben. Außerdem diskutieren wir Aspekte des Yangian-Bootstrap-Algorithmus in Minkowski-Raumzeit am Beispiel des masselosen Box-Integrals. Wir zeigen, dass dessen Yangian-Symmetrie gemeinsam mit seinen diskreten Permutationssymmetrien das Box-Integrals bis auf 12 unbestimmte Konstanten komplett festlegt. Schließlich schlagen wir vor, dass das Auftreten von Yangian-Symmetrie in massiven Fischnetz-Diagrammen mit deren Rolle als Ein-Spur-Streuamplituden in einer massiven Fischnetz-Theorie zusammenhängen könnte. In Analogie mit der masselosen Fischnetz-Theorie zeigen wir, wie diese Theorie als Deformation der N = 4 Super-Yang-Mills Theorie auf dem Coulomb-Zweig definiert werden kann. Wir diskutieren eine bestimmte Klasse von planaren Grenzfällen, in der die off-shell Streuamplituden der Theorie eine massive dual-konforme Symmetrie sowie Yangian-Symmetrie aufweisen. / In this dissertation, we extend the ideas of the Yangian bootstrap algorithm to massive Feynman diagrams. Based on the massive dual-conformal symmetry of Coulomb branch N = 4 super-Yang-Mills theory, we construct a set of bi-local Yangian level-one generators and show that they annihilate infinite classes of massive planar Feynman integrals at one and two loops. We also describe how the dual-conformal level-one momentum generator maps to a massive deformation of the ordinary momentum space special conformal generator. We then apply the Yangian bootstrap to a set of massive one-loop integrals with generalised propagator powers and in an arbitrary number of space dimensions to great success. In the special case of dual-conformal integrals, whose propagator powers sum to the space dimension, we find very simple novel hypergeometric structures, suggesting a natural generalisation to diagrams with an arbitrary number of external points. In the particular case of the massless box integral we also discuss elements of the Yangian bootstrap in Minkowski space. We show that its Yangian and discrete permutation symmetries constrain it up to 12 undetermined constants. We then derive the values of these constants via analytic continuation from the box integral in the Euclidean region. Finally, we provide evidence that the appearance of Yangian symmetry for massive fishnet diagrams is related to their role as colour-ordered scattering amplitudes in a massive fishnet theory. We show how to construct this theory from Coulomb branch N = 4 super-Yang-Mills theory, paralleling the original construction of the massless fishnet theory. We discuss how a particular class of planar limits leads to the emergence of massive dual-conformal symmetry as well as massive Yangian symmetry for the theory’s off-shell scattering amplitudes.