Strong-coupling phases of the anisotropic Kardar-Parisi-Zhang equation

We study the anisotropic Kardar-Parisi-Zhang equation using nonperturbative renormalization group methods. In contrast to a previous analysis in the weak-coupling regime we find the strong coupling fixed point corresponding to the isotropic rough phase to be always locally stable and unaffected by the anisotropy even at non-integer dimensions. Apart from the well-known weak coupling and the now well established isotropic strong coupling behavior, we find an anisotropic strong coupling fixed point for nonlinear couplings of opposite signs at non-integer dimensions.Comment: 18 pages, 7 figures, enlarged figures + minor changes, final versio

Breaking of scale invariance in the time dependence of correlation functions in isotropic and homogeneous turbulence

In this paper, we present theoretical results on the statistical properties of stationary, homogeneous and isotropic turbulence in incompressible flows in three dimensions. Within the framework of the Non-Perturbative Renormalization Group, we derive a closed renormalization flow equation for a generic $n$-point correlation (and response) function for large wave-numbers with respect to the inverse integral scale. The closure is obtained from a controlled expansion and relies on extended symmetries of the Navier-Stokes field theory. It yields the exact leading behavior of the flow equation at large wave-numbers $|\vec p_i|$, and for arbitrary time differences $t_i$ in the stationary state. Furthermore, we obtain the form of the general solution of the corresponding fixed point equation, which yields the analytical form of the leading wave-number and time dependence of $n$-point correlation functions, for large wave-numbers and both for small $t_i$ and in the limit $t_i\to \infty$. At small $t_i$, the leading contribution at large wave-number is logarithmically equivalent to $-\alpha (\epsilon L)^{2/3}|\sum t_i \vec p_i|^2$, where $\alpha$ is a nonuniversal constant, $L$ the integral scale and $\varepsilon$ the mean energy injection rate. For the 2-point function, the $(t p)^2$ dependence is known to originate from the sweeping effect. The derived formula embodies the generalization of the effect of sweeping to $n-$point correlation functions. At large wave-number and large $t_i$, we show that the $t_i^2$ dependence in the leading order contribution crosses over to a $|t_i|$ dependence. The expression of the correlation functions in this regime was not derived before, even for the 2-point function. Both predictions can be tested in direct numerical simulations and in experiments.Comment: 23 pages, minor typos correcte

Non-perturbative Approach to Critical Dynamics

This paper is devoted to a non-perturbative renormalization group (NPRG) analysis of Model A, which stands as a paradigm for the study of critical dynamics. The NPRG formalism has appeared as a valuable theoretical tool to investigate non-equilibrium critical phenomena, yet the simplest -- and nontrivial -- models for critical dynamics have never been studied using NPRG techniques. In this paper we focus on Model A taking this opportunity to provide a pedagological introduction to NPRG methods for dynamical problems in statistical physics. The dynamical exponent $z$ is computed in $d=3$ and $d=2$ and is found in close agreement with results from other methods.Comment: 13 page

Nonperturbative renormalization group for the stationary Kardar-Parisi-Zhang equation: scaling functions and amplitude ratios in 1+1, 2+1 and 3+1 dimensions

We investigate the strong-coupling regime of the stationary Kardar-Parisi-Zhang equation for interfaces growing on a substrate of dimension d=1, 2, and 3 using a nonperturbative renormalization group (NPRG) approach. We compute critical exponents, correlation and response functions, extract the related scaling functions and calculate universal amplitude ratios. We work with a simplified implementation of the second-order (in the response field) approximation proposed in a previous work [PRE 84, 061128 (2011) and Erratum 86, 019904 (2012)], which greatly simplifies the frequency sector of the NPRG flow equations, while keeping a nontrivial frequency dependence for the 2-point functions. The one-dimensional scaling function obtained within this approach compares very accurately with the scaling function obtained from the full second-order NPRG equations and with the exact scaling function. Furthermore, the approach is easily applicable to higher dimensions and we provide scaling functions and amplitude ratios in d=2 and d=3. We argue that our ansatz is reliable up to d \simeq 3.5.Comment: 21 pages, 7 figures, minor corrections prior to publicatio

General framework of the non-perturbative renormalization group for non-equilibrium steady states

This paper is devoted to presenting in detail the non-perturbative renormalization group (NPRG) formalism to investigate out-of-equilibrium systems and critical dynamics in statistical physics. The general NPRG framework for studying non-equilibrium steady states in stochastic models is expounded and fundamental technicalities are stressed, mainly regarding the role of causality and of Ito's discretization. We analyze the consequences of Ito's prescription in the NPRG framework and eventually provide an adequate regularization to encode them automatically. Besides, we show how to build a supersymmetric NPRG formalism with emphasis on time-reversal symmetric problems, whose supersymmetric structure allows for a particularly simple implementation of NPRG in which causality issues are transparent. We illustrate the two approaches on the example of Model A within the derivative expansion approximation at order two, and check that they yield identical results.Comment: 28 pages, 1 figure, minor corrections prior to publicatio

Reaction-diffusion processes and non-perturbative renormalisation group

This paper is devoted to investigating non-equilibrium phase transitions to an absorbing state, which are generically encountered in reaction-diffusion processes. It is a review, based on [Phys. Rev. Lett. 92, 195703; Phys. Rev. Lett. 92, 255703; Phys. Rev. Lett. 95, 100601], of recent progress in this field that has been allowed by a non-perturbative renormalisation group approach. We mainly focus on branching and annihilating random walks and show that their critical properties strongly rely on non-perturbative features and that hence the use of a non-perturbative method turns out to be crucial to get a correct picture of the physics of these models.Comment: 14 pages, submitted to J. Phys. A for the proceedings of the conference 'Renormalization Group 2005', Helsink

Microscopics of disordered two-dimensional electron gases under high magnetic fields: Equilibrium properties and dissipation in the hydrodynamic regime

We develop in detail a new formalism [as a sequel to the work of T. Champel and S. Florens, Phys. Rev. B 75, 245326 (2007)] that is well-suited for treating quantum problems involving slowly-varying potentials at high magnetic fields in two-dimensional electron gases. For an arbitrary smooth potential we show that electronic Green's function is fully determined by closed recursive expressions that take the form of a high magnetic field expansion in powers of the magnetic length l_B. For illustration we determine entirely Green's function at order l_B^3, which is then used to obtain quantum expressions for the local charge and current electronic densities at equilibrium. Such results are valid at high but finite magnetic fields and for arbitrary temperatures, as they take into account Landau level mixing processes and wave function broadening. We also check the accuracy of our general functionals against the exact solution of a one-dimensional parabolic confining potential, demonstrating the controlled character of the theory to get equilibrium properties. Finally, we show that transport in high magnetic fields can be described hydrodynamically by a local equilibrium regime and that dissipation mechanisms and quantum tunneling processes are intrinsically included at the microscopic level in our high magnetic field theory. We calculate microscopic expressions for the local conductivity tensor, which possesses both transverse and longitudinal components, providing a microscopic basis for the understanding of dissipative features in quantum Hall systems.Comment: small typos corrected; published versio

Single-site approximation for reaction-diffusion processes

We consider the branching and annihilating random walk $A\to 2A$ and $2A\to 0$ with reaction rates $\sigma$ and $\lambda$, respectively, and hopping rate $D$, and study the phase diagram in the $(\lambda/D,\sigma/D)$ plane. According to standard mean-field theory, this system is in an active state for all $\sigma/D>0$, and perturbative renormalization suggests that this mean-field result is valid for $d >2$; however, nonperturbative renormalization predicts that for all $d$ there is a phase transition line to an absorbing state in the $(\lambda/D,\sigma/D)$ plane. We show here that a simple single-site approximation reproduces with minimal effort the nonperturbative phase diagram both qualitatively and quantitatively for all dimensions $d>2$. We expect the approach to be useful for other reaction-diffusion processes involving absorbing state transitions.Comment: 15 pages, 2 figures, published versio

Nonperturbative renormalization group approach to the Ising model: a derivative expansion at order ∂4\partial^4

On the example of the three-dimensional Ising model, we show that nonperturbative renormalization group equations allow one to obtain very accurate critical exponents. Implementing the order $\partial^4$ of the derivative expansion leads to $\nu=0.632$ and to an anomalous dimension $\eta=0.033$ which is significantly improved compared with lower orders calculations.Comment: 4 pages, 3 figure