On the Perturbation Expansion of the KPZ Equation

We present a simple argument to show that the β-function of the d-dimensional KPZ equation (d≥2) is to all orders in perturbation theory given by $\beta (g_R ) = (d - 2)g_R - [2/(8\pi )^{d/2} ]{\text{ }}\Gamma (2 - d/2)g_R^2 $ Neither the dynamical exponent z nor the roughness exponent ζ have any correction in any order of perturbation theory. This shows that standard perturbation theory cannot attain the strong-coupling regime and in addition breaks down at d = 4. We also calculate a class of correlation functions exactly.     

Zero Tension Kardar-Parisi-Zhang Equation in (d + 1)–Dimensions

The joint probability distribution function (PDF) of the height and its gradients is derived for a zero tension d + 1-dimensional Kardar-Parisi-Zhang (KPZ) equation. It is proved that the height's PDF of zero tension KPZ equation shows lack of positivity after a finite time t _c . The properties of zero tension KPZ equation and its differences with the case that it possess an infinitesimal surface tension is discussed. Also potential relation between the time scale t _c and the singularity time scale t _c.v→0 of the KPZ equation with an infinitesimal surface tension is investigated.     

A phase cell cluster expansion for Euclidean field theories

We adapt the cluster expansion first used to treat infrared problems for lattice models (a mass zero cluster expansion) to the usual field theory situation. The field is expanded in terms of special block spin functions and the cluster expansion given in terms of the expansion coefficients (phase cell variables); the cluster expansion expresses correlation functions in terms of contributions from finite coupled subsets of these variables. Most of the present work is carried through in d space time dimensions (for φ₂⁴ the details of the cluster expansion are pursued and convergence is proven). Thus most of the results in the present work will apply to a treatment of φ₃⁴ to which we hope to return in a succeeding paper. Of particular interest in this paper is a substitute for the stability of the vacuum bound appropriate to this cluster expansion (for d = 2 and d = 3), and a new method for performing estimates with tree graphs. The phase cell cluster expansions have the renormalization group incorporated intimately into their structure. We hope they will be useful ultimately in treating four dimensional field theories.     

含有广义守恒律的生长方程标度奇异性的直接标度分析

利用直接标度分析方法研究一个含有广义守恒律生长方程的标度奇异性,得到强弱耦合区域的奇异标度指数.作为其特殊情况,这个方程包含Kardar-Parisi-Zhang（KPZ）方程、 Sun-Guo-Grant（SGG）方程以及分子束外延（MBE）生长方程,并能对其进行统一的研究.研究发现, KPZ方程和SGG方程,无论在弱耦合还是在强耦合区域内都遵从自仿射Family -Vicsek正常标度规律;而MBE 方程在弱耦合区域内服从正常标度,在强耦合区域内能呈现内禀奇异标度行为.这里所得到生长方程的奇异标度性质与利用重正化群理论、数值模拟以及实验相符很好. 关键词： 标度奇异性 强耦合 弱耦合     

Two-parameter expansion in the renormalization-group analysis of turbulence

The renormalization of the solution of the Navier-Stokes equation for randomly stirred fluid with long-range correlations of the driving force is analysed near two dimensions. It is shown that a local term must be added to the correlation function of the random force for the correct renormalization of the model at two dimensions. The interplay of the short-range and long-range terms in the large-scale behaviour of the model is analysed near two dimensions by the field-theoretic renormalization group. A regular expansion in 2ε=d-2 and δ=2-λ is constructed, whered is the space dimension and λ the exponent of the powerlike correlation function of the driving force. It is shown that in spite of the additional divergences, the asymptotic behaviour of the model near two dimensions is the same as in higher dimensions, contrary to recent conjectures based on an incorrect renormalization procedure.     

Gaussian Multiplicative Chaos and KPZ Duality

This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter γ ² beyond the transition phase (i.e. γ ² > 2d) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a simplified proof of the classical KPZ formula as well as the dual KPZ formula for atomic Gaussian multiplicative chaos. In particular, this framework allows to construct singular Liouville measures and to understand the duality relation in Liouville quantum gravity.     

ϵ-expansion of the parametric equation of state near the critical point

Starting from an equation of Brezin, Wallace and Wilson, we obtain a parametric form of the critical equation of state of a generalised Heisenberg ferromagnet in an expansion in ? = 4 ? d, where d is the dimensionality, up to terms in ?².     

Flow equation approach to the sine-Gordon model

A continuous sequence of infinitesimal unitary transformations is used to diagonalize the quantum sine-Gordon model for β²∈(2π,∞). This approach can be understood as an extension of perturbative scaling theory since it links weak- to strong-coupling behavior in a systematic expansion: a small expansion parameter is identified and this parameter remains small throughout the entire flow unlike the diverging running coupling constant of perturbative scaling. Our approximation consists in neglecting higher orders in this small parameter. We find very accurate results for the single-particle/hole spectrum in the strong-coupling phase and can describe the full crossover from weak to strong-coupling. The integrable structure of the sine-Gordon model is not used in our approach. Our new method should be of interest for the investigation of nonintegrable perturbations and for other strong-coupling problems.     

Effects of memory on scaling behaviour of Kardar--Parisi--Zhang equation

In order to describe the time delay in the surface roughing process the Kardar-Parisis-Zhang (KPZ) equation with memory effects is constructed and analysed using the dynamic renormalization group and the power counting mode coupling approach by Chattopadhyay [2009 Phys. Rev. E 80 011144]. In this paper, the scaling analysis and the classical self-consistent mode-coupling approximation are utilized to investigate the dynamic scaling behaviour of the KPZ equation with memory effects. The values of the scaling exponents depending on the memory parameter are calculated for the substrate dimensions being 1 and 2, respectively. The more detailed relationship between the scaling exponent and memory parameter reveals the significant influence of memory effects on the scaling properties of the KPZ equation.     

Exact results for the Kardar-Parisi-Zhang equation with spatially correlated noise

We investigate the Kardar-Parisi-Zhang (KPZ) equation in d spatial dimensions with Gaussian spatially long-range correlated noise -- characterized by its second moment -- by means of dynamic field theory and the renormalization group. Using a stochastic Cole-Hopf transformation we derive exact exponents and scaling functions for the roughening transition and the smooth phase above the lower critical dimension . Below the lower critical dimension, there is a line marking the stability boundary between the short-range and long-range noise fixed points. For , the general structure of the renormalization-group equations fixes the values of the dynamic and roughness exponents exactly, whereas above , one has to rely on some perturbational techniques. We discuss the location of this stability boundary in light of the exact results derived in this paper, and from results known in the literature. In particular, we conjecture that there might be two qualitatively different strong-coupling phases above and below the lower critical dimension, respectively. Received 5 August 1998     

Effective potential for the massless KPZ equation

In previous work we have developed a general method for casting a classical field theory subject to Gaussian noise (that is, a stochastic partial differential equation (SPDE)) into a functional integral formalism that exhibits many of the properties more commonly associated with quantum field theories (QFTs). In particular, we demonstrated how to derive the one-loop effective potential. In this paper we apply the formalism to a specific field theory of considerable interest, the massless KPZ equation (massless noisy Burgers equation), and analyze its behavior in the ultraviolet (short-distance) regime. When this field theory is subject to white noise we can calculate the one-loop effective potential and show that it is one-loop ultraviolet renormalizable in 1, 2, and 3 space dimensions, and fails to be ultraviolet renormalizable in higher dimensions. We show that the one-loop effective potential for the massless KPZ equation is closely related to that for λφ⁴ QFT. In particular, we prove that the massless KPZ equation exhibits one-loop dynamical symmetry breaking (via an analog of the Coleman–Weinberg mechanism) in 1 and 2 space dimensions, and that this behavior does not persist in 3 space dimensions.     

Critical dynamics of the sine-Gordon model ind=2−ε dimensions

The renormalization scheme of Amit, Goldschmidt and Grinstein is extended tod=2?ε dimensions. The exponent ν of the correlation lengthv ^?1=2ε+O(ε²) is in agreement with the result of Kosterlitz for the Coulomb gas. The exponent η of the correlation function of the sine-Gordon field is η=ε+O(ε²). The scaling form of the dynamical structure factorS(q,ω) of the dynamic sine-Gordon ModelA is studied ind=2?ε dimensions. The dynamic exponentz is found to bez=2+(b?1)ε+O(ε²) for ε≧0. The constantb is given by the integral $$b = \int\limits_0^\infty {dss^{ - 2} \exp } \left( { - 2\int\limits_s^\infty {dxx^{ - 1} e^{ - x} } } \right) = 2,371544...$$      

Dynamic critical exponent of a Bose system to O(1n)

The dynamic critical exponent of an n-component Bose system is calculated by the 1/n expansion technique to order 1/n in 2 < d < 4 dimensions.     

On the dynamics of the Ashkin-Teller-Potts model

The dynamical critical exponent of the s-states Ashkin-Teller-Potts model is found with the help of the linear response theory in the ? = d_c - d expansion (d_c = 4 and 6).     

Dynamic renormalization-group analysis of the d+1 dimensional Kuramoto-Sivashinsky equation with both conservative and nonconservative noises

The long-wavelength properties of the (d + 1)-dimensional Kuramoto-Sivashinsky (KS) equation with both conservative and nonconservative noises are investigated by use of the dynamic renormalization-group (DRG) theory. The dynamic exponent z and roughness exponent α are calculated for substrate dimensions d = 1 and d = 2, respectively. In the case of d = 1, we arrive at the critical exponents z = 1.5 and α = 0.5 , which are consistent with the results obtained by Ueno et al. in the discussion of the same noisy KS equation in 1+1 dimensions [Phys. Rev. E 71, 046138 (2005)] and are believed to be identical with the dynamic scaling of the Kardar-Parisi-Zhang (KPZ) in 1+1 dimensions. In the case of d = 2, we find a fixed point with the dynamic exponents z = 2.866 and α = -0.866 , which show that, as in the 1 + 1 dimensions situation, the existence of the conservative noise in 2 + 1 or higher dimensional KS equation can also lead to new fixed points with different dynamic scaling exponents. In addition, since a higher order approximation is adopted, our calculations in this paper have improved the results obtained previously by Cuerno and Lauritsen [Phys. Rev. E 52, 4853 (1995)] in the DRG analysis of the noisy KS equation, where the conservative noise is not taken into account.     

Exact solutions of Dyson–Schwinger equations for iterated one-loop integrals and propagator-coupling duality

The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar φ³ theory, from all nestings and chainings of a primitive self-energy subdivergence. Here we formulate the nonperturbative problems which these resummations approximate. For Yukawa theory, at spacetime dimension d=4, we obtain an integrodifferential Dyson–Schwinger equation and solve it parametrically in terms of the complementary error function. For the scalar theory, at d=6, the nonperturbative problem is more severe; we transform it to a nonlinear fourth-order differential equation. After intensive use of symbolic computation we find an algorithm that extends both perturbation series to 500 loops in 7 minutes. Finally, we establish the propagator–coupling duality underlying these achievements making use of the Hopf structure of Feynman diagrams.     

Thermodynamic quantum critical behavior of the anisotropic Kondo necklace model

The Ising-like anisotropy parameter δ in the Kondo necklace model is analyzed using the bond-operator method at zero and finite temperatures for arbitrary d dimensions. A decoupling scheme on the double time Green's functions is used to find the dispersion relation for the excitations of the system. At zero temperature and in the paramagnetic side of the phase diagram, we determine the spin gap exponent νz≈0.5 in three dimensions and anisotropy between 0?δ?1, a result consistent with the dynamic exponent z=1 for the Gaussian character of the bond-operator treatment. On the other hand, in the antiferromagnetic phase at low but finite temperatures, the line of Neel transitions is calculated for δ?1. For d>2 it is only re-normalized by the anisotropy parameter and varies with the distance to the quantum critical point (QCP) |g| as, T_N∝|g|^ψ where the shift exponent ψ=1/(d-1). Nevertheless, in two dimensions, a long-range magnetic order occurs only at T=0 for any δ?1. In the paramagnetic phase, we also find a power law temperature dependence on the specific heat at the quantum critical trajectoryJ/t=(J/t)_c, T→0. It behaves as C_V∝T^d for δ?1 and ≈1, in concordance with the scaling theory for z=1.     

Universality classes of the Kardar-Parisi-Zhang equation

We reexamine mode-coupling theory for the Kardar-Parisi-Zhang equation in the strong-coupling limit and show that there exist two branches of solutions. One branch (or universality class) exists only for dimensionalities dor=2.     

Hydrodynamics of an n-component phonon system

The dynamic properties of an n-component phonon system in d dimensions, which serves as a model for a structural phase transition of second order, are investigated. The symmetry group of the hamiltonian is the group of orthogonal transformations $\text{O}$(n). For n ≥ 2 a continuous symmetry is broken for T<T_c, where T_c is the transition temperature. We derive the hydrodynamic equations for the generators of this group, the $\text{1}\text{2}\text{n (n ? 1)}$ angular-momentum variables. Besides the usual hydrodynamics of a phonon system, there are $\text{1}\text{2}\text{n (n ? 1)}$ additional independent diffusive modes for T > T_c. In the ordered phase we find 2 (n ? 1) propagating modes with linear dispersion and quadratic damping. Formally the hydrodynamics is similar in the isotropic Heisenberg ferromagnet (n = 2) or the isotropic antiferromagnet (n ≥ 3). The relaxing modes for T < T_c require special care. We study the critical dynamics by means of the dynamical scaling hypothesis and by a mode-coupling calculation, both of which give the critical dynamical exponent $\text{z =}\text{1}\text{2}\text{d}$. The results are compared with the 1/n expansion. It is shown that for large n there is a non-asymptotic region characterized by an effective exponent $\text{z}\text{?}\text{= φ/2ν}$, where φ is the crossover exponent for a uniaxial perturbation, and ν the critical exponent of the correlation length.     

Single-Point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equations in Planar Domains

Consider the diffusive Hamilton-Jacobi equation u _t = Δu + |?u| ^p , p > 2, on a bounded domain Ω with zero-Dirichlet boundary conditions, which arises in the KPZ model of growing interfaces. It is known that u remains bounded and that ?u may blow up only on the boundary ?Ω. In this paper, under suitable assumptions on ${\Omega\subset \mathbb{R}^2}Consider the diffusive Hamilton-Jacobi equation u _t = Δu + |∇u| ^p , p > 2, on a bounded domain Ω with zero-Dirichlet boundary conditions, which arises in the KPZ model of growing interfaces. It is known that u remains bounded and that ∇u may blow up only on the boundary ∂Ω. In this paper, under suitable assumptions on W ì \mathbbR²{\Omega\subset \mathbb{R}^2} and on the initial data, we show that the gradient blow-up singularity occurs only at a single point x₀ ? ?W{x_0\in\partial\Omega}. This is the first result of this kind in the study of problems involving gradient blow-up phenomena. In general domains of \mathbbRⁿ{\mathbb{R}^n}, we also obtain results on nondegeneracy and localization of blow-up points.