Since its introduction, the Renormalization Group has been the theoretical tool of choice for understanding and describing collective behavior characterized by an emergent scale invariance as in the vicinity of a continuous phase transition. It provides a powerful conceptual framework but, exact results being scarce, the search for generic and efficient approximation schemes has been very active from the very beginning. One recent line of research starts from an exact formulation of the Renormalization Group, in the form of a functional Renormalization Group (FRG) for scale-dependent generating functionals of correlation functions, and introduces potentially {it nonperturbative} approximations through ansatzes for the scale-dependent generating functional under study. The question that we address is to what extent such generic approximation schemes are able to describe specific problems in which the long-distance behavior involves strongly nonuniform configurations with, {it e.g.}, localized excitations.An example of such an approximation scheme within the FRG is the so-called derivative expansion of the effective average action (coarse-grained Gibbs free energy in the language of magnetic systems) which amounts to truncating the functional form of the latter in powers of the external momenta or equivalently in gradients of the fields. The derivative-expansion approximation focuses on the long wavelength properties and, in terms of coarse-grained configurations of the system, works as an expansion about uniform configurations. One may therefore wonder if such a scheme is able to capture the physics associated with nonuniform configurations containing, {it e.g.}, domain walls, spin waves, or localized defects such as the kinks and anti-kinks found in the instanton analysis of the 1-dimensional Ising model.In this work we assess the ability of the derivative expansion of the FRG to quantitatively describe the long-distance physics of a model across the whole range of space dimensions {it without a priori knowledge} of the relevant real-space coarse-grained configurations. To do so we first investigate how low orders of the derivative expansion describe the approach to the lower critical dimension d→dlc of the Ising-like scalar φ4 theory which is expected to be controlled by the proliferation of localized excitations (instantons). We show that the convergence of the fixed-point effective potential describing the critical point of the model is nonuniform in the field when the dimension d approaches dlcdlc, with the emergence of a boundary layer around the minimum of the potential. At the lowest nontrivial approximation level (known as LPA’) this allows us to make analytical predictions for the value of the lower critical dimension dlcdlc and for the behavior of the critical temperature as d→dlc, which are both found in fair agreement with the known exact results. We also consider for the same Ising-like scalar φ4 model the ordered phase in the limit where d→dlc. We focus on the way the effective potential in the FRG becomes convex as the proper spatial fluctuations, which now involve nonuniform configurations with a domain wall separating two phases, are included in the calculation.
Since its introduction, the Renormalization Group has been the theoretical tool of choice for understanding and describing collective behavior characterized by an emergent scale invariance as in the vicinity of a continuous phase transition. It provides a powerful conceptual framework but, exact results being scarce, the search for generic and efficient approximation schemes has been very active from the very beginning. One recent line of research starts from an exact formulation of the Renormalization Group, in the form of a functional Renormalization Group (FRG) for scale-dependent generating functionals of correlation functions, and introduces potentially {it nonperturbative} approximations through ansatzes for the scale-dependent generating functional under study. The question that we address is to what extent such generic approximation schemes are able to describe specific problems in which the long-distance behavior involves strongly nonuniform configurations with, {it e.g.}, localized excitations.An example of such an approximation scheme within the FRG is the so-called derivative expansion of the effective average action (coarse-grained Gibbs free energy in the language of magnetic systems) which amounts to truncating the functional form of the latter in powers of the external momenta or equivalently in gradients of the fields. The derivative-expansion approximation focuses on the long wavelength properties and, in terms of coarse-grained configurations of the system, works as an expansion about uniform configurations. One may therefore wonder if such a scheme is able to capture the physics associated with nonuniform configurations containing, {it e.g.}, domain walls, spin waves, or localized defects such as the kinks and anti-kinks found in the instanton analysis of the 1-dimensional Ising model.In this work we assess the ability of the derivative expansion of the FRG to quantitatively describe the long-distance physics of a model across the whole range of space dimensions {it without a priori knowledge} of the relevant real-space coarse-grained configurations. To do so we first investigate how low orders of the derivative expansion describe the approach to the lower critical dimension d→dlc of the Ising-like scalar φ4 theory which is expected to be controlled by the proliferation of localized excitations (instantons). We show that the convergence of the fixed-point effective potential describing the critical point of the model is nonuniform in the field when the dimension d approaches dlcdlc, with the emergence of a boundary layer around the minimum of the potential. At the lowest nontrivial approximation level (known as LPA’) this allows us to make analytical predictions for the value of the lower critical dimension dlcdlc and for the behavior of the critical temperature as d→dlc, which are both found in fair agreement with the known exact results. We also consider for the same Ising-like scalar φ4 model the ordered phase in the limit where d→dlc. We focus on the way the effective potential in the FRG becomes convex as the proper spatial fluctuations, which now involve nonuniform configurations with a domain wall separating two phases, are included in the calculation.
We revisit the approach to the lower critical dimension d_{lc} in the Ising-like φ^{4} theory within the functional renormalization group by studying the lowest approximation levels in the derivative expansion of the effective average action. Our goal is to assess how the latter, which provides a generic approximation scheme valid across dimensions and found to be accurate in d≥2, is able to capture the long-distance physics associated with the expected proliferation of localized excitations near d_{lc}. We show that the convergence of the fixed-point effective potential is nonuniform in the field when d→d_{lc} with the emergence of a boundary layer around the minimum of the potential. This allows us to make analytical predictions for the value of the lower critical dimension d_{lc} and for the behavior of the critical temperature as d→d_{lc}, which are both found in fair agreement with the known results. This confirms the versatility of the theoretical approach.
Fluctuations fortement non uniformes au sein du Groupe de Renormalisation Fonctionnelle Depuis son introduction, le Groupe de Renormalisation est devenu l'outil théorique de prédilection pour comprendre et décrire le comportement collectif caractérisé par une invariance d'échelle émergente à proximité d'une transition de phase continue. Il offre un cadre conceptuel puissant, mais les résultats exacts étant rares, la recherche de schémas d'approximation génériques et efficaces a été très active dès le départ. Une ligne de recherche récente part d'une formulation exacte du Groupe de Renormalisation, sous la forme d'un Groupe de Renormalisation Fonctionnelle (GRF) pour les fonctionnelles génératrices dépendantes d'échelle, et introduit des approximations potentiellement non perturbatives via des ansatzes pour la fonctionnelle génératrice dépendante d'échelle étudiée. La question que nous abordons est de savoir dans quelle mesure de tels schémas d'approximation génériques sont capables de décrire des problèmes spécifiques où le comportement à longue distance implique des configurations fortement non uniformes avec, par exemple, des excitations localisées. Un exemple d'un tel schéma d'approximation au sein du GRF est l'expansion dérivée de l'action moyenne effective (énergie libre de Gibbs moyenne à gros grain ("coarse-grained") dans le langage des systèmes magnétiques) qui consiste à tronquer la forme fonctionnelle de cette dernière en puissances des moments extérieurs ou, de manière équivalente, en gradients des champs. L'approximation par expansion dérivée se concentre sur les propriétés à grande longueur d'onde et, en termes de configurations "coarse-grained" du système, fonctionne comme une expansion autour de configurations uniformes. On peut donc se demander si un tel schéma est capable de saisir la physique associée à des configurations non uniformes contenant, par exemple, des parois de domaine, des ondes de spin ou des défauts localisés comme les créneaux et les anti-créneaux trouvés dans l'analyse des instantons du modèle d'Ising en 1 dimension. Dans ce travail, nous évaluons la capacité de l'expansion dérivée du GRF à décrire quantitativement la physique à longue distance d'un modèle sur toute la gamme des dimensions spatiales sans connaissance a priori des configurations "coarse-grained" pertinentes dans l'espace réel. Pour ce faire, nous examinons d'abord comment les ordres inférieurs de l'expansion dérivée décrivent l'approche de la dimension critique inférieure dlcdlc de la théorie scalaire de type Ising φ4 qui est censée être contrôlée par la prolifération d'excitations localisées (instantons). Nous montrons que la convergence du potentiel effectif du point fixe décrivant le point critique du modèle est non uniforme dans le champ lorsque la dimension dd s'approche de dlcdlc, avec l'émergence d'une couche limite autour du minimum du potentiel. Au niveau d'approximation non trivial le plus bas (connu sous le nom de LPA’), cela nous permet de faire des prédictions analytiques pour la valeur de la dimension critique inférieure dlcdlc et pour le comportement de la température critique à mesure que d→dlc, qui sont tous deux en assez bon accord avec les résultats exacts connus. Nous considérons également pour le même modèle scalaire de type Ising φ4 la phase ordonnée dans la limite où d→dlc. Nous nous concentrons sur la manière dont le potentiel effectif dans le GRF devient convexe à mesure que les fluctuations spatiales appropriées, qui impliquent maintenant des configurations non uniformes avec une paroi de domaine séparant deux phases, sont incluses dans le calcul.
Approach to the lower critical dimension of the φ4 theory in the derivative expansion of the Functional Renormalization Group
We revisit the approach to the lower critical dimension dlc in the Ising-like φ4 theory within the functional renormalization group by studying the lowest approximation levels in the derivative expansion of the effective average action. Our goal is to assess how the latter, which provides a generic approximation scheme valid across dimensions and found to be accurate in d≥2, is able to capture the long-distance physics associated with the expected proliferation of localized excitations near dlc. We show that the convergence of the fixed-point effective potential is nonuniform when d→dlc with the emergence of a boundary layer around the minimum of the potential. This allows us to make analytical predictions for the value of the lower critical dimension dlc and for the behavior of the critical temperature as d→dlc, which are both found in fair agreement with the known results. This confirms the versatility of the theoretical approach.
Continuous phase transitions provide a natural setting for understanding universality. Near criticality, the correlation length diverges, microscopic details become increasingly irrelevant, and physical observables obey scaling laws characterized by universal critical exponents. Within the Functional Renormalization Group (FRG), this scale dependence is described through the Effective Average Action, which progressively incorporates fluctuations as the momentum scale is lowered. A first part of this thesis is developed within the Local Potential Approximation (LPA), where we study multicritical fixed points of scalar field theories. Their universal properties are compared with results obtained at higher orders in the derivative expansion. Particular attention is devoted to the relation between global fixed-point potentials and their local representations through finite polynomial truncations. We show that the convergence of such expansions is constrained by the analytic structure of the global solution, and that expanding around a suitable nonzero stationary point significantly improves the reconstruction of multicritical branches. Beyond the structure of individual fixed points, a central part of this thesis concerns RG irreversibility. In two dimensions, Zamolodchikov’s c-theorem describes the flow from the ultraviolet to the infrared through a monotonically decreasing c-function. Within the FRG framework, we develop a variational procedure to reconstruct RG trajectories between fixed points and integrate the corresponding c-function along them. The resulting quantity displays monotonic behaviour and becomes increasingly stable as the truncation order is raised, providing a manifestation of RG irreversibility within the approximation considered. Finally, close to the upper critical dimensions, the linearized FRG flow around the Gaussian fixed point is naturally organized in terms of Hermite polynomials. Using the epsilon-expansion, we derive a curvature-based candidate c-function and obtain an analytic explanation for the alternating curvature of successive multicritical fixed points and for their progressive accumulation around the Gaussian fixed point.
The functional renormalization group (FRG) is a powerful tool that has facilitated the investigation of various strongly interacting theories, encompassing systems characterized by both bosonic and fermionic variables. Specifically, this project focuses on the analysis of theories with O(N) symmetry with bosonic scalar fields. These theories are of particular interest due to their broad applicability in real physical systems. For example, Z2 symmetry (the O(1) model) describes the well-known Ising universality class, which in turn models the liquid-gas transition. On the other hand, the O(2) model belongs to the universality class of the XY model, used to describe the transition from fluid to superfluid in 4 He, while the O(3) model, known as the Heisenberg model, describes the ferromagnetic transition in isotropic materials. Finally the O(0) model is related to the self-avoiding walk (SAW). To characterize these theories, various methods have been implemented within the framework of the renormalization group. Perturbation theory has been applied since the early days of the subject, specifically in the form of the ε-expansion, which has reached high loop order in recent years. Alternatively, a non-perturbative approach can be chosen, which is known as the non-perturbative renormalization group (NPRG). Within the NPRG, there are several approximation schemes, with the derivative expansion being particularly noteworthy. The ultimate goal of these efforts is to calculate the critical exponents, which define the given universality class. The standard non-perturbative method involves solving an equation that depends on a regulator or cut-off, which represents one of the main challenges of this approach. Although the regulators are designed so that the theory does not depend on them at the scales of interest, the results for the critical exponents can vary depending on the regulator used once approximations are made. In this project, we propose an approach that eliminates the need for an explicit mass cut-off significantly simplifying the calculation of critical exponents while mitigating the regulator dependence of the results. In particular, we show that it is possible to use dimensional regularization (DR) beyond the ε-expansion in the context of RG calculations of critical properties. Based on this we propose a new functional RG scheme called Functional Dimensional Regularization (FDR) and apply it to the O(N) model in three dimension, finding excellent agreement with state-of-the-art computations
We explore the application of the nonperturbative functional renormalization group (NPFRG), within its most common approximation scheme based on truncations of the derivative expansion, to the Z_{2}-symmetric scalar φ^{4} theory as the lower critical dimension d_{lc} is approached. We aim to assess whether the NPFRG-a broad, nonspecialized method which is accurate in d≥2-can capture the effect of the localized (droplet) excitations that drive the disappearance of the phase transition in d_{lc} and control the critical behavior as d→d_{lc}. We extend a prior analysis to the next (second) order of the derivative expansion to check the convergence of the results and the robustness of the conclusions. The study turns out to be much more involved. Through extensive numerical and analytical work we provide evidence that the convergence to d_{lc} is nonuniform in the field dependence and is characterized by the emergence of a boundary layer near the minima of the fixed-point effective potential. This is the mathematical mechanism through which the NPFRG within the truncated derivative expansion reproduces nontrivial features predicted by the droplet theory of Bruce and Wallace [Phys. Rev. Lett. 47, 1743 (1981)0031-900710.1103/PhysRevLett.47.1743; J. Phys. A: Math. Gen. 16, 1721 (1983)0305-447010.1088/0305-4470/16/8/018], namely, the existence of two distinct small parameters as d→d_{lc} that control different aspects of the critical behavior and that are nonperturbatively related. The second order of the derivative expansion fixes several issues that were encountered at the lower level and improves the compatibility with the droplet-theory predictions.