It is expected that conformal symmetry is an emergent property of many systems at their critical point. This imposes strong constraints on the critical behavior of a given system. Taking them into account in theoretical approaches can lead to a better understanding of the critical physics or improve approximation schemes. However, within the framework of the nonperturbative or functional renormalization group and, in particular, of one of its most used approximation schemes, the derivative expansion (DE), nontrivial constraints apply only from third order [usually denoted O(∂^{4})], at least in the usual formulation of the DE that includes correlation functions involving only the order parameter. In this work we implement conformal constraints on a generalized DE including composite operators and show that new constraints already appear at second order of the DE [or O(∂^{2})]. We show how these constraints can be used to fix nonphysical regulator parameters.
It is expected that conformal symmetry is an emergent property of many systems at their critical point. This imposes strong constraints on the critical behavior of a given system. Taking them into account in theoretical approaches can lead to a better understanding of the critical physics or improve approximation schemes. However, within the framework of the nonperturbative or functional renormalization group and, in particular, of one of its most used approximation schemes, the derivative expansion (DE), nontrivial constraints apply only from third order [usually denoted O(∂^{4})], at least in the usual formulation of the DE that includes correlation functions involving only the order parameter. In this work we implement conformal constraints on a generalized DE including composite operators and show that new constraints already appear at second order of the DE [or O(∂^{2})]. We show how these constraints can be used to fix nonphysical regulator parameters.
The behavior of many critical phenomena at large distances is expected to be invariant under the full conformal group, rather than only isometries and scale transformations. When studying critical phenomena, approximations are often required, and the framework of the nonperturbative, or functional, renormalization group is no exception. The derivative expansion is one of the most popular approximation schemes within this framework, due to its great performance on multiple systems, as evidenced in the past few decades. Nevertheless, it has the downside of breaking conformal symmetry at a finite order. This breaking is not observed at the leading order of the expansion, denoted the local potential approximation, and it only appears once one considers, at least, the next-to-leading order of the derivative expansion [O(∂^{2})] when including composite operators. In this work, we study the constraints arising from conformal symmetry for the O(N) models using the derivative expansion at order O(∂^{2}). We explore various values of N and minimize the breaking of conformal symmetry to fix the nonphysical parameters of the approximation procedure. We compare our prediction for the critical exponents with those coming from a more usual procedure, known as the principle of minimal sensitivity.
Conformal invariance is expected to be an emergent property of many systems in their critical regime. However, approximation schemes generically spoil this property. This is in particular the case of the derivative expansion, a widely used approximation scheme in the framework of the functional renormalization group. In this article, we consider Ward identities associated with conformal invariance in the 3-d Ising universality class with truncations at order 4 (next-to-next-to-leading order) in the derivative expansion, with Z2 invariant composite operators. Our results confirm that the regulating functions which yield a small breaking of conformal invariance also present a small sensitivity of the universal critical exponents with the choice of this regulating function. We also show that, in the vicinity of regulator-parameter values for which the conformal constraints are best satisfied, the breaking of conformal invariance reduces as the order of the derivative expansion is increased, providing a new indication of the convergence of this approximation scheme.
Field-theoretical calculations performed in an approximation scheme often present a spurious dependence of physical quantities on some unphysical parameters associated with the details of the calculation setup (such as the renormalization scheme or, in perturbation theory, the resummation procedure). In the present article, we propose to reduce this dependence by invoking conformal invariance. Using as a benchmark the three-dimensional Ising model, we show that, within the derivative expansion at order 4, performed in the nonperturbative renormalization group formalism, the identity associated with this symmetry is not exactly satisfied. The calculations which best satisfy this identity are shown to yield critical exponents which coincide to a high accuracy with those obtained by the conformal bootstrap. Additionally, this work gives a strong justification to the success of a widely used criterion for fixing the appropriate renormalization scheme, namely the principle of minimal sensitivity.
We provide analytical arguments showing that the "nonperturbative" approximation scheme to Wilson's renormalization group known as the derivative expansion has a finite radius of convergence. We also provide guidelines for choosing the regulator function at the heart of the procedure and propose empirical rules for selecting an optimal one, without prior knowledge of the problem at stake. Using the Ising model in three dimensions as a testing ground and the derivative expansion at order six, we find fast convergence of critical exponents to their exact values, irrespective of the well-behaved regulator used, in full agreement with our general arguments. We hope these findings will put an end to disputes regarding this type of nonperturbative methods.
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 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