We study renormalization group (RG) flows in far-from-equilibrium states. The study is made tractable by focusing on states that are spatially homogeneous, time independent, and scale invariant. Such states, in which mode k has occupation numbers n k ∼ k − γ , are well known in nonlinear physics. RG flow in such states is qualitatively different from that in the vacuum—a positive γ decreases the dimension of an operator, turning marginal interactions into relevant interactions. We compute one-loop beta functions. Depending on the sign of the beta function, backreaction may either cause a minor shift of the state in the IR or completely change the nature of the state. Focusing on nearly marginal interactions, we construct an analog of the epsilon expansion and IR fixed points, with epsilon now set by the scaling of the interaction rather than the spacetime dimension. In the language of RG flow, critical balance scaling—which has applications in fields as varied as astrophysics and ocean waves—corresponds to the state dynamically adjusting itself along the RG flow until the interaction becomes marginal.
We study renormalization group (RG) flows in far-from-equilibrium states. The study is made tractable by focusing on states that are spatially homogeneous, time independent, and scale invariant. Such states, in which mode k has occupation numbers n k ∼ k − γ , are well known in nonlinear physics. RG flow in such states is qualitatively different from that in the vacuum—a positive γ decreases the dimension of an operator, turning marginal interactions into relevant interactions. We compute one-loop beta functions. Depending on the sign of the beta function, backreaction may either cause a minor shift of the state in the IR or completely change the nature of the state. Focusing on nearly marginal interactions, we construct an analog of the epsilon expansion and IR fixed points, with epsilon now set by the scaling of the interaction rather than the spacetime dimension. In the language of RG flow, critical balance scaling—which has applications in fields as varied as astrophysics and ocean waves—corresponds to the state dynamically adjusting itself along the RG flow until the interaction becomes marginal.
Abstract Chapter 7 is devoted to a discussion of the renormalization group (RG) flow when the effective field theory that describes universal properties of critical phenomena depends on several coupling constants. The universal properties of a large class of macroscopic phase transitions with short range interactions can be described by statistical field theories involving scalar fields with quartic interactions. The simplest critical systems have an O(N) orthogonal symmetry and, therefore, the corresponding field theory has only one quartic interaction. However, in more general physical systems, the flow of quartic interactions is more complicated. This chapter examines these systems from the RG viewpoint. RG beta functions are shown to generate a gradient flow. Some examples illustrate the notion of emergent symmetry. The local stability of fixed points is related to the value of the scaling field dimension.
In the successful framework of the Renormalization Group (RG) approach to Quantum Field Theories (QFT), establishing general constraints on the RG flow has become a central problem. This is formally addressed by the monotonicity theorems, such as Zamolodchikov’s c-theorem in d=2, the a-theorem in d=4, and the F-theorem in d=3. Essentially, these theorems state that if a RG trajectory leads from a UV fixed point to a IR one, each described by a Conformal Field Theory (CFT), then a suitable positive quantity decreases. In both even and odd dimensions, this quantity can be extracted, at fixed points, from the free energy of the CFT on a sphere, leading to the conjecture of a generalized F-theorem valid in continuous dimension d. In this thesis we investigate this framework in a multicritical multiscalar theory with a generic sextic interaction for N massless scalar fields below its upper critical dimension d=3. This model captures, among many others, the critical features of the Tricritical Ising universality class. A first step is to establish whether the RG flow admits a gradient structure, i.e. that there exists a monotonic function A whose gradient yields the beta function up to a positive-definite metric in the space of couplings. This is one of the strongest realizations of monotonicity, as it guarantees the irreversibility of the RG flow and gives an intrinsic geometric structure to the theory space. Given the known beta function up to next-to-leading order, we solve the gradient flow equation perturbatively, finding a parametric family of solutions for the monotonic A-function together with the metric at and below the upper critical dimension, so that the RG flow admits a gradient structure at this order. As this procedure leaves some parameters undetermined, to turn it into a physical statement these functions must be matched to suitable observables of the CFTs describing the fixed points. We therefore compute the CFT candidates suggested by the generalized F-theorem, namely the sphere free energy of the CFT and the Zamolodchikov metric. Specializing to the O(N)-symmetric fixed point, we consistently match the RG and CFT results, fixing all the free parameters in a unique way. This matching is the central result of this work: it establishes that the RG flow admits a gradient structure perturbatively, with F as the monotonic function and Zamolodchikov's metric as the metric on the theory space, recovered from the geometric construction for a specific and unique choice of the free parameters. Importantly, this result provides a nontrivial, perturbative confirmation of the F-theorem in its strongest form, at and below the upper critical dimension, for renormalizable three-dimensional multiscalar theories.
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 concept of renormalization group (RG) flow is one of the most novel and broad-reaching aspects of quantum field theory (QFT). The RG flow is implemented by constructing effective descriptions of a QFT at decreasing energy scales. One reason that RG flow is useful is that often one is interested in low-energy properties of theories with complicated short-distance structures. RG flows are subject to C theorems in relativistic QFT. The C theorems order the space of Lorentz-invariant QFTs. RG flows generically begin at scale-invariant fixed points known as conformal field theories (CFTs) and end in trivial massive theories. With tuning, the RG flows may end at non-trivial CFTs. Each CFT has an associated dimensionless C value. The C theorem states that under RG flow from a UV to an IR fixed point, the C value decreases. In this Dissertation I present the F-theorem, which is a C theorem in three spacetime dimensions. I show that the correct quantity to consider is the Euclidean free energy of the CFT conformally mapped to the three-sphere, known as the F value. After motivating the F-theorem, I develop tools for calculating the F value in a variety of CFTs, with and without supersymmetry, including free field theories and gauge theories with large numbers of flavors. I also show that the F value is itself a useful quantity for probing the gauge/gravity duality and understanding other aspects of CFT, such as the scaling dimensions of monopole operators. The F theorem is closely related to quantum entanglement entropy. At conformal fixed points, the F value is equal to minus the renormalized entanglement entropy (REE) in flat Minkowski space across a circle. Away from the fixed points, the REE is a monotonically decreasing function along the RG flow. I compute the REE in a variety of holographic and non-holographic theories. I conclude the Dissertation by discussing a somewhat surprising result: the REE is not stationary at conformal fixed points.
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.
Renormalization group (RG) flows for systems subject to an external field can be derived either by fixing the field or by fixing the conjugate order parameter. We show how to implement these boundary conditions within the functional renormalization group approach. We derive and compare the corresponding flow equations for ϕ4 theory and show that some scaling variables and scaling exponents associated with the linearized flow of the relevant three-point vertex in the vicinity of the Ising fixed point depend on the boundary conditions. We also discuss the effect of elasticity on Ising criticality. In this case two of the fixed points (namely the renormalized Ising fixed point and the spherical fixed point) obtained for constant strain are shifted to physically inaccessible regimes of parameter space when we perform the RG at constant stress.