Two-Loop Renormalization-Group Analysis of the Burgers-Kardar-Parisi-Zhang Equation

Abstract

A systematic analysis of the Burgers—Kardar–Parisi–Zhang equation in d+ 1 dimensions by dynamic renormalization group theory is described. The fixed points and exponents are calculated to two–loop order. We use the dimensional regularization scheme, carefully keeping the full d dependence originating from the angular parts of the loop integrals. For dimensions less than dc = 2 we find a strong–coupling fixed point, which diverges at d = 2, indicating that there is non–perturbative strong– coupling behavior for all d ≥ 2. At d = 1 our method yields the identical fixed point as in the one–loop approximation, and the two–loop contributions to the scaling functions are non–singular. For d > 2 dimensions, there is no finite strong–coupling fixed point. In the framework of a 2 + ϵ expansion, we find the dynamic exponent corresponding to the unstable fixed point, which describes the non–equilibrium roughening transition, to be z = 2 + O (ϵ3), in agreement with a recent scaling argument by Doty and Kosterlitz. Similarly, our result for the correlation length exponent at the transition is 1/ν = ϵ + O (ϵ3). For the smooth phase, some aspects of the crossover from Gaussian to critical behavior are discussed.