Fully developed isotropic turbulence: Nonperturbative renormalization group formalism and fixed-point solution

Inviscid fixed point of the multidimensional Burgers-Kardar-Parisi-Zhang equation

A new scaling regime characterized by a z=1 dynamical critical exponent has been reported in several numerical simulations of the one-dimensional Kardar-Parisi-Zhang and noisy Burgers equations. In these works, this scaling, differing from the well-known KPZ one z=3/2, was found to emerge in the tensionless limit for the interface and in the inviscid limit for the fluid. Based on functional renormalization group, the origin of this scaling has been elucidated. It was shown to be controlled by a yet unpredicted fixed point of the one-dimensional Burgers-KPZ equation, termed inviscid Burgers (IB) fixed point. The associated universal properties, including the scaling function, were calculated. All these findings were restricted to d=1, and it raises the intriguing question of the fate of this new IB fixed point in higher dimensions. In this work, we address this issue and analyze the multidimensional Burgers-KPZ equation using functional renormalization group. We show that the IB fixed point exists in all dimensions d≥0, and that it controls the large momentum behavior of the correlation functions in the inviscid limit. It turns out that it yields in all d the same super-universal value z=1 for the dynamical exponent.

Unpredicted Scaling of the One-Dimensional Kardar-Parisi-Zhang Equation

The celebrated Kardar-Parisi-Zhang (KPZ) equation describes the kinetic roughening of stochastically growing interfaces. In one dimension, the KPZ equation is exactly solvable and its statistical properties are known to an exquisite degree. Yet recent numerical simulations in the tensionless (or inviscid) limit of the KPZ equation [C. Cartes et al., The Galerkin-truncated Burgers equation: Crossover from inviscid-thermalized to Kardar-Parisi-Zhang scaling, Phil. Trans. R. Soc. A 380, 20210090 (2022).PTRMAD1364-503X10.1098/rsta.2021.0090; E. Rodríguez-Fernández et al., Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation, Phys. Rev. E 106, 024802 (2022).PRESCM2470-004510.1103/PhysRevE.106.024802] unveiled a new scaling, with a critical dynamical exponent z=1 different from the KPZ one z=3/2. In this Letter, we show that this scaling is controlled by a fixed point which had been missed so far and which corresponds to an infinite nonlinear coupling. Using the functional renormalization group (FRG), we demonstrate the existence of this fixed point and show that it yields z=1. We calculate the correlation function and associated scaling function at this fixed point, providing both a numerical solution of the FRG equations within a reliable approximation, and an exact asymptotic form obtained in the limit of large wave numbers. Both scaling functions accurately match the one from the numerical simulations.

One Fixed Point Can Hide Another One: Nonperturbative Behavior of the Tetracritical Fixed Point of O(N) Models at Large N

We show that at N=∞ and below its upper critical dimension, d<d_{up}, the critical and tetracritical behaviors of the O(N) models are associated with the same renormalization group fixed point (FP) potential. Only their derivatives make them different with the subtleties that taking their N→∞ limit and deriving them do not commute and that two relevant eigenperturbations show singularities. This invalidates both the ε-and the 1/N-expansions. We also show how the Bardeen-Moshe-Bander line of tetracritical FPs at N=∞ and d=d_{up} can be understood from a finite-N analysis.

Gravity as a Quantum Field Theory

Classical gravity is understood as the geometry of spacetime, and it seems very different from the other known interactions. In this review, I will instead stress the analogies: Like strong interactions, the low energy effective field theory of gravity is related to a nonlinearly realized symmetry, and like electroweak interactions, it is a gauge theory in Higgs phase, with a massive connection. I will also discuss the possibility of finding a UV complete quantum field theoretic description of all interactions.

Critical Probability Distributions of the Order Parameter from the Functional Renormalization Group

We show that the functional renormalization group (FRG) allows for the calculation of the probability distribution function of the sum of strongly correlated random variables. On the example of the three-dimensional Ising model at criticality and using the simplest implementation of the FRG, we compute the probability distribution functions of the order parameter or, equivalently, its logarithm, called the rate functions in large deviation theory. We compute the entire family of universal scaling functions, obtained in the limit where the system size L and the correlation length of the infinite system ξ_{∞} diverge, with the ratio ζ=L/ξ_{∞} held fixed. It compares very accurately with numerical simulations.

Incompleteness of the large-N analysis of the O(N) models: Nonperturbative cuspy fixed points and their nontrivial homotopy at finite N

We summarize the usual implementations of the large-N limit of O(N) models and show in detail why and how they can miss some physically important fixed points when they become singular in the limit N→∞. Using Wilson's renormalization group in its functional nonperturbative versions, we show how the singularities build up as N increases. In the Wilson-Polchinski version of the nonperturbative renormalization group, we show that the singularities are cusps, which become boundary layers for finite but large values of N. The corresponding fixed points being never close to the Gaussian, are out of reach of the usual perturbative approaches. We find four new fixed points and study them in all dimensions and for all N>0 and show that they play an important role for the tricritical physics of O(N) models. Finally, we show that some of these fixed points are bivalued when they are considered as functions of d and N thus revealing important and nontrivial homotopy structures. The Bardeen-Moshe-Bander phenomenon that occurs at N=∞ and d=3 is shown to play a crucial role for the internal consistency of all our results.

Ensemble model of turbulence based on states of constant flux in wavenumber space

An ensemble model of turbulence is proposed. The ensemble consists of flow fields in which the flux of an inviscid conserved quantity, such as energy (or enstrophy in two-dimensional flow fields), across the wave number k is a constant independent of k in an appropriate range. Two-dimensional flow fields of constant enstrophy flux are sampled randomly by a Monte Carlo method. The energy spectra E_{k} of the flow fields are consistent with the scaling E_{k}∝k^{-3}[ln(k/k_{b})]^{-1/3} where k_{b} is the bottom wave number of the constant-flux range.

Kardar-Parisi-Zhang equation with temporally correlated noise: A nonperturbative renormalization group approach

We investigate the universal behavior of the Kardar-Parisi-Zhang (KPZ) equation with temporally correlated noise. The presence of time correlations in the microscopic noise breaks the statistical tilt symmetry, or Galilean invariance, of the original KPZ equation with δ-correlated noise (denoted SR-KPZ). Thus, it is not clear whether the KPZ universality class is preserved in this case. Conflicting results exist in the literature, some advocating that it is destroyed even in the limit of infinitesimal temporal correlations, while others find that it persists up to a critical range of such correlations. Using nonperturbative and functional renormalization group techniques, we study the influence of two types of temporal correlators of the noise: a short-range one with a typical timescale τ, and a power-law one with a varying exponent θ. We show that for the short-range noise with any finite τ, the symmetries (the Galilean symmetry, and the time-reversal one in 1+1 dimension) are dynamically restored at large scales, such that the long-distance and long-time properties are governed by the SR-KPZ fixed point. In the presence of a power-law noise, we find that the SR-KPZ fixed point is still stable for θ below a critical value θ_{th}, in accordance with previous renormalization group results, while a long-range fixed point controls the critical scaling for θ>θ_{th}, and we evaluate the θ-dependent critical exponents at this long-range fixed point, in both 1+1 and 2+1 dimensions. While the results in 1+1 dimension can be compared with previous studies, no other prediction was available in 2+1 dimension. We finally report in 1+1 dimension the emergence of anomalous scaling in the long-range phase.

Morphology of renormalization-group flow for the de Almeida-Thouless-Gardner universality class

A replica-symmetry-breaking phase transition is predicted in a host of disordered media. The criticality of the transition has, however, long been questioned below its upper critical dimension, six, due to the absence of a critical fixed point in the renormalization-group flows at one-loop order. A recent two-loop analysis revealed a possible strong-coupling fixed point, but given the uncontrolled nature of perturbative analysis in the strong-coupling regime, debate persists. Here we examine the nature of the transition as a function of spatial dimension and show that the strong-coupling fixed point can go through a Hopf bifurcation, resulting in a critical limit cycle and a concomitant discrete scale invariance. We further investigate a different renormalization scheme and argue that the basin of attraction of the strong-coupling fixed point (or limit cycle) may stay finite for all dimensions.

Why Might the Standard Large N Analysis Fail in the O(N) Model: The Role of Cusps in Fixed Point Potentials

The large N expansion plays a fundamental role in quantum and statistical field theory. We show on the example of the O(N) model that at N=∞ its standard implementation misses some fixed points of the renormalization group in all dimensions smaller than four. These new fixed points show singularities under the form of cusps at N=∞ in their effective potential that become a boundary layer at finite N. We show that they have a physical impact on the multicritical physics of the O(N) model at finite N. We also show that the mechanism at play holds also for the O(N)⊗O(2) model and is thus probably generic.

Sinc noise for the Kardar-Parisi-Zhang equation

In this paper we study the one-dimensional Kardar-Parisi-Zhang (KPZ) equation with correlated noise by field-theoretic dynamic renormalization-group techniques. We focus on spatially correlated noise where the correlations are characterized by a sinc profile in Fourier space with a certain correlation length ξ. The influence of this correlation length on the dynamics of the KPZ equation is analyzed. It is found that its large-scale behavior is controlled by the standard KPZ fixed point, i.e., in this limit the KPZ system forced by sinc noise with arbitrarily large but finite correlation length ξ behaves as if it were excited by pure white noise. A similar result has been found by Mathey et al. [S. Mathey et al., Phys. Rev. E 95, 032117 (2017)2470-004510.1103/PhysRevE.95.032117] for a spatial noise correlation of Gaussian type (∼e^{-x^{2}/2ξ^{2}}), using a different method. These two findings together suggest that the KPZ dynamics is universal with respect to the exact noise structure, provided the noise correlation length ξ is finite.

Glassy phase in quenched disordered crystalline membranes

We investigate the flat phase of D-dimensional crystalline membranes embedded in a d-dimensional space and submitted to both metric and curvature quenched disorders using a nonperturbative renormalization group approach. We identify a second-order phase transition controlled by a finite-temperature, finite-disorder fixed point unreachable within the leading order of ε=4-D and 1/d expansions. This critical point divides the flow diagram into two basins of attraction: that associated with the finite-temperature fixed point controlling the long-distance behavior of disorder-free membranes and that associated with the zero-temperature, finite-disorder fixed point. Our work thus strongly suggests the existence of a whole low-temperature glassy phase for quenched disordered crystalline membranes and, possibly, for graphene and graphene-like compounds.

Influence of turbulent mixing on critical behavior of directed percolation process: Effect of compressibility

Universal behavior is a typical emergent feature of critical systems. A paramount model of the nonequilibrium critical behavior is the directed bond percolation process that exhibits an active-to-absorbing state phase transition in the vicinity of a percolation threshold. Fluctuations of the ambient environment might affect or destroy the universality properties completely. In this work, we assume that the random environment can be described by means of compressible velocity fluctuations. Using field-theoretic models and renormalization group methods, we investigate large-scale and long-time behavior. Altogether, 11 universality classes are found, out of which 4 are stable in the infrared limit and thus macroscopically accessible. In contrast to the model without velocity fluctuations, a possible candidate for a realistic three-dimensional case, a regime with relevant short-range noise, is identified. Depending on the dimensionality of space and the structure of the turbulent flow, we calculate critical exponents of the directed percolation process. In the limit of the purely transversal random force, critical exponents comply with the incompressible results obtained by previous authors. We have found intriguing nonuniversal behavior related to the mutual effect of compressibility and advection.

Surprises in O(N) Models: Nonperturbative Fixed Points, Large N Limits, and Multicriticality

We find that the multicritical fixed point structure of the O(N) models is much more complicated than widely believed. In particular, we find new nonperturbative fixed points in three dimensions (d=3) as well as at N=∞. These fixed points come together with an intricate double-valued structure when they are considered as functions of d and N. Many features found for the O(N) models are shared by the O(N)⊗O(2) models relevant to frustrated magnetic systems.

Nonuniversality in the erosion of tilted landscapes

The anisotropic model for landscapes erosion proposed by Pastor-Satorras and Rothman [R. Pastor-Satorras and D. H. Rothman, Phys. Rev. Lett. 80, 4349 (1998)PRLTAO0031-900710.1103/PhysRevLett.80.4349] is believed to capture the physics of erosion at intermediate length scale (≲3 km), and to account for the large value of the roughness exponent α observed in real data at this scale. Our study of this model-conducted using the nonperturbative renormalization group-concludes on the nonuniversality of this exponent because of the existence of a line of fixed points. Thus the roughness exponent depends (weakly) on the details of the soil and the erosion mechanisms. We conjecture that this feature, while preserving the generic scaling observed in real data, could explain the wide spectrum of values of α measured for natural landscapes.

Discontinuous Transition from Direct to Inverse Cascade in Three-Dimensional Turbulence

Inviscid invariants of flow equations are crucial in determining the direction of the turbulent energy cascade. In this work we investigate a variant of the three-dimensional Navier-Stokes equations that shares exactly the same ideal invariants (energy and helicity) and the same symmetries (under rotations, reflections, and scale transforms) as the original equations. It is demonstrated that the examined system displays a change in the direction of the energy cascade when varying the value of a free parameter which controls the relative weights of the triadic interactions between different helical Fourier modes. The transition from a forward to inverse cascade is shown to occur at a critical point in a discontinuous manner with diverging fluctuations close to criticality. Our work thus supports the observation that purely isotropic and three-dimensional flow configurations can support inverse energy transfer when interactions are altered and that inside all turbulent flows there is a competition among forward and backward transfer mechanisms which might lead to multiple energy-containing turbulent states.

Kardar-Parisi-Zhang equation with short-range correlated noise: Emergent symmetries and nonuniversal observables

We investigate the stationary-state fluctuations of a growing one-dimensional interface described by the Kardar-Parisi-Zhang (KPZ) dynamics with a noise featuring smooth spatial correlations of characteristic range ξ. We employ nonperturbative functional renormalization group methods to resolve the properties of the system at all scales. We show that the physics of the standard (uncorrelated) KPZ equation emerges on large scales independently of ξ. Moreover, the renormalization group flow is followed from the initial condition to the fixed point, that is, from the microscopic dynamics to the large-distance properties. This provides access to the small-scale features (and their dependence on the details of the noise correlations) as well as to the universal large-scale physics. In particular, we compute the kinetic energy spectrum of the stationary state as well as its nonuniversal amplitude. The latter is experimentally accessible by measurements at large scales and retains a signature of the microscopic noise correlations. Our results are compared to previous analytical and numerical results from independent approaches. They are in agreement with direct numerical simulations for the kinetic energy spectrum as well as with the prediction, obtained with the replica trick by Gaussian variational method, of a crossover in ξ of the nonuniversal amplitude of this spectrum.

Spatiotemporal velocity-velocity correlation function in fully developed turbulence

Turbulence is a ubiquitous phenomenon in natural and industrial flows. Since the celebrated work of Kolmogorov in 1941, understanding the statistical properties of fully developed turbulence has remained a major quest. In particular, deriving the properties of turbulent flows from a mesoscopic description, that is, from the Navier-Stokes equation, has eluded most theoretical attempts. Here, we provide a theoretical prediction for the functional space and time dependence of the velocity-velocity correlation function of homogeneous and isotropic turbulence from the field theory associated to the Navier-Stokes equation with stochastic forcing. This prediction, which goes beyond Kolmogorov theory, is the analytical fixed point solution of nonperturbative renormalization group flow equations, which are exact in the limit of large wave numbers. This solution is compared to two-point two-times correlation functions computed in direct numerical simulations. We obtain a remarkable agreement both in the inertial and in the dissipative ranges.

Multilocality and fusion rules on the generalized structure functions in two-dimensional and three-dimensional Navier-Stokes turbulence

Using the fusion-rules hypothesis for three-dimensional and two-dimensional Navier-Stokes turbulence, we generalize a previous nonperturbative locality proof to multiple applications of the nonlinear interactions operator on generalized structure functions of velocity differences. We call this generalization of nonperturbative locality to multiple applications of the nonlinear interactions operator "multilocality." The resulting cross terms pose a new challenge requiring a new argument and the introduction of a new fusion rule that takes advantage of rotational symmetry. Our main result is that the fusion-rules hypothesis implies both locality and multilocality in both the IR and UV limits for the downscale energy cascade of three-dimensional Navier-Stokes turbulence and the downscale enstrophy cascade and inverse energy cascade of two-dimensional Navier-Stokes turbulence. We stress that these claims relate to nonperturbative locality of generalized structure functions on all orders and not the term-by-term perturbative locality of diagrammatic theories or closure models that involve only two-point correlation and response functions.