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.

In dieser Dissertation untersuchen wir unterschiedliche Aspekte im Zusammenhang mit Korrelationsfunktionen in der Fischnetz-Theorie. Zunächst betrachten wir einen der einfachsten Korrelatoren der Fischnetz Theorie, das konforme Box-Integral, in Minkowski Signatur. Während dieses Integral in Euklidischer Signatur eine konforme Symmetrie aufweist, wird diese Symmetrie in Minkowski-Raumzeit subtil gebrochen. Wir beschreiben die Brechung der konformen Symmetrie quantitativ, indem wir die funktionale Form des Box-Integrals in allen kinematischen Regionen untersuchen. Ausserdem untersuchen wir das Ausmass zu dem das Box integral durch seine Yangian-Symmetrie festgelegt ist. Als nächstes widmen wir uns den Basso-Dixon-Graphen, die ebenfalls konforme Vier-Punkt-Integrale sind und Verallgemeinerungen des Box-Integrals zu höheren Schleifenordnungen darstellen. Wir leiten die Yangian-Ward-Identitäten ab, die diese Klasse von Integralen erfüllen. Die Ward-Identitäten sind einhomogene Erweiterungen der partiellen Differentialgleichungen, die im homogenen Fall durch Appell-Hypergeometrische Funktionen gelöst werden. Die Ward-Identitäten können natürlicherweise auf eine Ein-Parameter-Familie von D-dimensionalen Integralen erweitert werden, die Korrelatoren in der verallgemeinerten Fischnetz-Theorie von Kazakov und Olivucci darstellen. Schliesslich untersuchen wir den Dilatationsoperator in einem Drei-Skalar-Sektor der Fischnetztheorie, der auch als Eklektisches Modell bezeichnet wird. In diesem Sektor der Dilatationsoperator nimmt nicht--diagonalisierbare Form an. Das führt dazu, dass die Zwei-Punkt-Korrelationsfunktionen eine logarithmische Abhängigkeit von der Raumzeitseparierung der Operatoren annimmt. Unter Zuhilfenahme von kombinatorischen Argumenten führen wir eine generierende Funktion ein, die das Jordan-Block-Spektrum eines verwandten Modells, der hypereklektischen Spinkette, vollständig charakterisiert. / We study various aspects of correlation functions in fishnet theory. We begin with the study of the simplest correlator in theory theory, represented by the conformal box integral, in Minkowski space. While this integral is conformally invariant in Euclidean space, this symmetry is subtly broken in Minkowski space. We quantify the extent to which conformal symmetry is broken by analysing the functional form of the box in each kinematic region. We propose a new method to calculate the box integral directly in Minkowski space, by introducing a family of configurations with two points at infinity. Furthermore, we investigate the extent to which the box integral is constrained by Yangian symmetry. We constrain the functional form of the box integral in all kinematic regions up to twelve undetermined constants, which we fix by three separate analytic continuations from the Euclidean region. Next, we study the Basso-Dixon graphs, which represent higher-loop versions of the box integral. We derive and study Yangian Ward identities for this class of integrals. These take the form of inhomogeneous extensions of the partial differential equations defining the Appell hypergeometric functions. The Ward identities naturally generalise to a one-parameter family of D dimensional integrals representing correlators in a generalised fishnet theory. Finally, we study the dilatation operator in a particular three scalar sector of the fishnet theory, which has been dubbed the eclectic model. This dilatation operator is non-diagonalisable in this sector. This leads to logarithmic spacetime dependence in the corresponding two-point functions. Using combinatorial arguments, we introduce a generating function which fully characterises the Jordan block spectrum of a related model: the hypereclectic spin chain. This function is found by purely combinatorial means and can be expressed in terms of the q-binomial coefficient.

No description available.

In 1981, Unruh suggested the possibility of simulating the dynamics of quantum fields in curved spacetimes using sound-waves propagating in moving fluids: a supersonic flow would indeed influence the dynamics of sound similarly to what happens to light when it’s dragged by the spacetime geometry in strong gravity environments. This simple yet groundbreaking observation has lead to the beginning of a whole new field of research, nowadays known as Analog Gravity. Due to their superfluid character, intrinsic quantum nature and impressive experimental tunability, Bose-Einstein condensates represent one of the most promising platforms to realize analog spacetimes, including black-hole geometries with horizons and ergoregions, as well as of time-dependent configurations relevant to cosmology. In this Thesis we go beyond the standard single-component BEC and focus on two-component mixtures of atomic condensates, possibly in the presence of a coherent coupling between the two-components: the availability of various branches of elementary excitations with different sound speed and effective mass may in fact lead to advantages in the implementation of interesting geometries and, eventually, to the exploration of a broader spectrum of physical processes. We first consider black-hole related phenomena (Hawking radiation and rotational superradiance) that have already been analysed with single-component systems, generalising the results to mixtures; we then proceed to tackle a problem (the decay from the false vacuum) which instead requires the additional degrees of freedom that only a mixture displays.

This thesis considers the problem of calculating and observing the mixing of modes of positive and negative frequency in inhomogeneous, dispersive media. Scattering of vacuum modes of the electromagnetic field at a moving interface in the refractive index of a dielectric medium is discussed. Kinematics arguments are used to demonstrate that this interface may, in a regime of linear dispersion, act as the analogue of the event horizon of a black hole to modes of the field. Furthermore, a study of the dispersion of the dielectric shows that five distinct configurations of modes of the inhomogeneous medium at the interface exist as a function of frequency. Thus it is shown that the interface is simultaneously a black- and white-hole horizon-like and horizonless emitter. The role, and importance, of negative-frequency modes of the field in mode conversion at the horizon is established and yields a calculation of the spontaneous photonic flux at the interface. An algorithm to calculate the scattering of vacuum modes at the interface is introduced. Spectra of the photonic flux in the moving and laboratory frame, for all modes and all realisable increase in the refractive index at the interface are computed. As a result of the various mode configurations, the spectra are highly structured in intervals with black-hole, white-hole and no horizon. The spectra are dominated by a negative-frequency mode, which is the partner in any Hawking-type emission. An experiment in which an incoming positive-frequency wave is populated with photons is assembled to observe the transfer of energy to outgoing waves of positive and negative frequency at the horizon. The effect of mode conversion at the interface is clearly shown to be a feature of horizon physics. This is a classical version of the quantum experiment that aims at validating the mechanism of Hawking radiation.

In this thesis we review the gauge/gravity duality and how it can be used to compute the thermodynamic properties and low-energy excitations of holographic quantum liquids - strongly-interacting field theories with a non-zero density of matter. We then study in detail the charge density excitations of two such liquids, the D3/D7 theory and the RN-AdS₄ theory, by computing the poles of their charge density Green's functions, and their charge density spectral functions. Although it is not a Landau Fermi liquid, the charge density excitations of the D3/D7 theory display many of the same properties as one, including a collisionless/hydrodynamic crossover as the temperature is increased. In contrast to this, the charge density (and energy density) excitations of the RN-AdS₄ theory do not share these properties but behave in a way that cannot be explained by Landau's theory of interacting fermionic quasiparticles. This is consistent with other results which indicate that this is not a Landau Fermi liquid.

Les moments magnétiques anomaux des leptons ont joué un rôle important dans le développement du modèle standard de la physique des particules. Aujourd’hui, celui du muon est mesuré très précisément et le sera avec une precision encore plus grande par une expérience qui débutera en 2017. Dans la mesure où la prédiction théorique pourra être faite avec des incertitudes comparables, un test rigoureux du modèle standard sera possible. Nous étudions ici le facteur limitant de cette prédiction, la contribution de la polarisation hadronique du vide à l’ordre dominant (HVP-LO). Nous calculons cette contribution numériquement à l’aide d’une version discrétisée de la théorie de l’interaction forte, la chromodynamique quantique sur réseau. Le calcul haute-performance permet de résoudre la théorie dans son régime hautement non-linéaire qui est le plus pertinent ici. Les algorithmes de simulation et les méthodes utilisées pour obtenir la polarisation hadronique, ainsi que les incertitudes associées, sont décrits. Ces méthodes sont ensuite appliquées à des simulations réalisées avec la collaboration Budapest-Marseille-Wuppertal. Dans un premier temps, elles sont implémentées dans une étude dédiée des effets de volume fini. Les méthodes les plus robustes sont ensuite utilisées pour calculer la polarisation hadronique avec des simulations qui comprennent N_f=2+1+1 saveurs de quarks. Celles-ci sont réalisées directement à la valeur physique des masses de quarks u, d, s et c, avec six tailles de maille et dans de gros volumes de 6 fm^3. Elles nous permettent de calculer la contribution HVP-LO au moment magnétique anomal du muon avec des erreurs contrôlées d’environ 3%. / The anomalous magnetic moments of leptons have played an important role in the development of the Standard Model of particle physics. Today, that of the muon is measured very precisely and will be so with even higher precision in an experiment that will begin in 2017. To the extent that the theoretical prediction can be made with comparable uncertainties, a rigorous test of the Standard Model will be possible. Here we study the limiting factor in this prediction, the leading-order hadronic vacuum polarization contribution (HVP-LO). We compute this contribution numerically with a discretized version of the theory of the strong interaction: lattice Quantum Chromodynamics. High-performance computing allows to solve the theory in its highly nonlinear regime, which is the one most relevant here. The simulation algorithms and the methods used to obtain the HVP, as well as the associated statistical and systematic uncertainties, are described. These methods are then applied to simulations performed with the Budapest-Marseille-Wuppertal collaboration. First they are implemented in a dedicated study of finite-volume effects. The most robust methods are then used to compute the HVP with simulations which include N_f=2+1+1 flavors of quarks. These are performed directly at the physical values of the u, d, s and c quark masses, with six lattice spacings and in large volumes of 6 fm^3. They allow us to compute the HVP-LO contribution to the anomalous magnetic moment of the muon with controlled errors of around 3%.

Esta tese trata a existência de Princpios de Grandes Desvios (PGD), no âmbito de sistemas fermiônicos em equilbrio. A motivação fsica detrás de nossos estudos são medidas experimentais de resistência elétrica de nanofios de silcio dopados com átomos de fósforo. Estas medidas mostram que efeitos quânticos no transporte de carga elétrica quase desaparecem para nanofios de comprimentos maiores que alguns nanômetros, mesmo para temperaturas muito baixas (4.2°K). A fim de provar matematicamente tal efeito, dividimos nosso trabalho em diversos passos: 1. No primeiro passo, para férmions não interagentes numa rede com desordem, mostramos que a incerteza quântica da densidade da corrente elétrica microscópica, em torno de seus valores macroscópicos(clássicos), é suprimida exponencialmente rápido em relação ao volume da região da rede onde um campo elétrico externo é aplicado. A desordem é modelada como um potencial elétrico aleatório juntamente com amplitudes aleatórias de saltos com valores complexos. O célebre modelo de Anderson de tight-binding é um exemplo particular do caso geral considerado aqui. Nossa análise matemática é baseada em estimativas de Combes-Thomas, o Teorema Ergódico de Akcoglu-Krengel e no formalismo de Grandes Desvios, em particular o Teorema de Gärtner-Ellis. 2. Em segundo lugar, provamos que, para férmions interagindo fracamente na rede, as funções geradoras J(s), s R de cumulantes de distribuições de probabilidades associadas com estados KMS pode ser escrito como o limite de logartmos de integrais gaussianas de Berezin. Mostramos que os determinantes das covariáncias associadas às integrais gaussianas são majorados uniformemente (via desigualdades de Hölder para normas Schatten). Tais covariâncias são também somáveis, em casos gerais de interesse, incluindo assim, sistemas que não são invariantes por translação. 3. No terceiro passo, analisamos expansões de logartmos de integrais gaussianas de Berezin, e assim combinando com métodos construtivos de teoria quântica de campos, mostramos a analiticidade de J(s) para s nas vizinhanças de 0. Finalmente, discutimos como combinar os passos 2-3, a fim de provar (matematicamente falando) os resultados experimentais mencionados acima para férmions interagindo em equilbrio. De fato, os resultados encontrados nesta tese, generalizam trabalhos prévios no âmbito do PGD usado para o estudo de sistemas quânticos. / This Thesis deals with the existence of Large Deviation Principles (LDP) in the scope of fermionic systems at equilibrium. The physical motivation beyond our studies are experimental measures of electric resistance of nanowires in silicon doped with phosphorus atoms. The latter demonstrate that quantum effects on charge transport almost disappear for nanowires of lengths larger than a few nanometers, even at very low temperature (4.2°K). In order to mathematically prove the latter, we divide our work in several steps: 1. In the first step, for noninteracting lattice fermions with disorder, we show that quantum uncertainty of microscopic electric current density around their (classical) macroscopic values is suppressed, exponentially fast with respect to the volume of the region of the lattice where an external electric field is applied. Disorder is modeled by a random external potential along with random, complex-valued, hopping amplitudes. The celebrated tight-binding Anderson model is one particular example of the general case considered here. Our mathematical analysis is based on Combes-Thomas estimates, the Akcoglu-Krengel ergodic theorem, and the large deviation formalism, in particular the Gärtner-Ellis theorem. 2. Secondly, we prove that for weakly interacting fermions on the lattice, the logarithm moment generating function J(s), s R of probability distributions associated with KMS states can be written as the limit of logarithms of Gaussian Berezin integrals. The covariances of the Gaussian integrals are shown to have a uniform determinant bound (via Hölder inequalities for Schatten norms) and to be summable in general cases of interest, including systems that are not translation invariant. 3. In the third step we analyze expansions of logarithms of Gaussian Berezin integrals, which combined with constructive methods of quantum field theory is useful to show the analyticity of J(s) for s in a neighborhood of 0. We finally discuss how to combine steps 2-3 in order to prove (mathematically speaking) for interacting fermions in equilibrium the experimental results above mentioned. In fact, the found results in this Thesis generalize previous works in the scope of LDP used to study quantum systems.