Universality classes of the Kardar-Parisi-Zhang equation

We re-examine mode-coupling theory for the Kardar-Parisi-Zhang (KPZ) equation in the strong coupling limit and show that there exists two branches of solutions. One branch (or universality class) only exists for dimensionalities $d<d_c=2$ and is similar to that found by a variety of analytic approaches, including replica symmetry breaking and Flory-Imry-Ma arguments. The second branch exists up to $d_c=4$ and gives values for the dynamical exponent $z$ similar to those of numerical studies for $d\ge2$.Comment: 4 pages, 1 figure, published versio

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

Quantitative Phase Diagrams of Branching and Annihilating Random Walks

We demonstrate the full power of nonperturbative renormalisation group methods for nonequilibrium situations by calculating the quantitative phase diagrams of simple branching and annihilating random walks and checking these results against careful numerical simulations. Specifically, we show, for the 2A->0, A -> 2A case, that an absorbing phase transition exists in dimensions d=1 to 6, and argue that mean field theory is restored not in d=3, as suggested by previous analyses, but only in the limit d -> $\infty$.Comment: 4 pages, 3 figures, published version (some typos corrected

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

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

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

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

Non perturbative renormalization group and momentum dependence of n-point functions (II)

In a companion paper (hep-th/0512317), we have presented an approximation scheme to solve the Non Perturbative Renormalization Group equations that allows the calculation of the $n$-point functions for arbitrary values of the external momenta. The method was applied in its leading order to the calculation of the self-energy of the O($N$) model in the critical regime. The purpose of the present paper is to extend this study to the next-to-leading order of the approximation scheme. This involves the calculation of the 4-point function at leading order, where new features arise, related to the occurrence of exceptional configurations of momenta in the flow equations. These require a special treatment, inviting us to improve the straightforward iteration scheme that we originally proposed. The final result for the self-energy at next-to-leading order exhibits a remarkable improvement as compared to the leading order calculation. This is demonstrated by the calculation of the shift $\Delta T_c$, caused by weak interactions, in the temperature of Bose-Einstein condensation. This quantity depends on the self-energy at all momentum scales and can be used as a benchmark of the approximation. The improved next-to-leading order calculation of the self-energy presented in this paper leads to excellent agreement with lattice data and is within 4% of the exact large $N$ result.Comment: 35 pages, 11 figure

Non-Perturbative Renormalization Group for Simple Fluids

We present a new non perturbative renormalization group for classical simple fluids. The theory is built in the Grand Canonical ensemble and in the framework of two equivalent scalar field theories as well. The exact mapping between the three renormalization flows is established rigorously. In the Grand Canonical ensemble the theory may be seen as an extension of the Hierarchical Reference Theory (L. Reatto and A. Parola, \textit{Adv. Phys.}, \textbf{44}, 211 (1995)) but however does not suffer from its shortcomings at subcritical temperatures. In the framework of a new canonical field theory of liquid state developed in that aim our construction identifies with the effective average action approach developed recently (J. Berges, N. Tetradis, and C. Wetterich, \textit{Phys. Rep.}, \textbf{363} (2002))

Non perturbative renormalisation group and momentum dependence of nn-point functions (I)

We present an approximation scheme to solve the Non Perturbative Renormalization Group equations and obtain the full momentum dependence of the $n$-point functions. It is based on an iterative procedure where, in a first step, an initial ansatz for the $n$-point functions is constructed by solving approximate flow equations derived from well motivated approximations. These approximations exploit the derivative expansion and the decoupling of high momentum modes. The method is applied to the O($N$) model. In leading order, the self energy is already accurate both in the perturbative and the scaling regimes. A stringent test is provided by the calculation of the shift $\Delta T_c$ in the transition temperature of the weakly repulsive Bose gas, a quantity which is particularly sensitive to all momentum scales. The leading order result is in agreement with lattice calculations, albeit with a theoretical uncertainty of about 25%.Comment: 48 pages, 15 figures A few minor corrections. A reference adde