Abstract We study the higher-derivative extension of the functional renormalization group (FRG). We consider FRG equations for a scalar field that consist of terms with higher functional derivatives of the effective action and arbitrary cutoff functions. We show that the ϵ expansion around the Wilson–Fisher fixed point is indeed reproduced by the local potential approximation of the FRG equations.
Abstract We study the higher-derivative extension of the functional renormalization group (FRG). We consider FRG equations for a scalar field that consist of terms with higher functional derivatives of the effective action and arbitrary cutoff functions. We show that the ϵ expansion around the Wilson–Fisher fixed point is indeed reproduced by the local potential approximation of the FRG equations.
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.
Scalar field theories with Z2 symmetry are the traditional playground of critical phenomena. In this work, these models are studied using functional renormalization group (FRG) equations at order ${\ensuremath{\partial}}^{2}$ of the derivative expansion and, differently from previous approaches, the spike plot technique is employed to find the relative scaling solutions in two and three dimensions. The anomalous dimension of the first few universality classes in d=2 is given, and the phase structure predicted by conformal field theory is recovered (without the imposition of conformal invariance), while in d=3 a refined view of the standard Wilson-Fisher fixed point is found. Our study enlightens the strength of shooting techniques in studying FRG equations, suggesting them as candidates to investigate strongly nonperturbative theories even in more complex cases.
Equations related to the Polchinski version of the exact renormalization group (RG) equations for scalar fields which extend the local potential approximation to first order in a derivative expansion, and which maintain reparameterization invariance, are postulated. Reparameterization invariance ensures that the equations determine the anomalous dimension η unambiguously and the equations are such that the result is exact to O (ε 2 ) in an ε-expansion for any multi-critical fixed point. It is also straightforward to determine η numerically. When the dimension d = 3 numerical results for a wide range of critical exponents are obtained in theories with O ( N ) symmetry, for various N and for ranges of η, are obtained within the local potential approximation. The associated η, which follows from the derivative approximation described here, is found for various N . The large N limit of the equations is also analysed. A corresponding discussion is also given in a perturbative RG framework and scaling dimensions for derivative operators are calculated to first order in ε.
We employ perturbative renormalization group and $\ensuremath{\epsilon}$-expansion to study multicritical single-scalar field theories with higher derivative kinetic terms of the form $\ensuremath{\phi}(\ensuremath{-}\ensuremath{\square}{)}^{k}\ensuremath{\phi}$. We focus on those with a Z2-symmetric critical point which are characterized by an upper critical dimension ${d}_{c}=2nk/(n\ensuremath{-}1)$ accumulating at even integers. We distinguish two types of theories depending on whether or not the numbers k and $n\ensuremath{-}1$ are relatively prime. When they are, the critical theory involves a marginal powerlike interaction ${\ensuremath{\phi}}^{2n}$ and the deformations admit a derivative expansion that at leading order involves only the potential. In this case we present the beta functional of the potential and use this to calculate some anomalous dimensions and operator product expansion coefficients. These confirm some conformal field theory data obtained using conformal-block techniques, while giving new results. In the second case where k and $n\ensuremath{-}1$ have a common divisor, the theories show a much richer structure induced by the presence of marginal derivative operators at criticality. We study the case k=2 with odd values of n, which fall in the second class, and calculate the functional flows and spectrum. These theories have a phase diagram characterized at leading order in $\ensuremath{\epsilon}$ by four fixed points which apart from the Gaussian UV fixed point include an IR fixed point with a purely derivative interaction.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].
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.