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 ${\mathit{d}}_{\mathit{c}}$=2 we find a strong-coupling fixed point, which diverges at d=2, indicating that there is nonperturbative 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 nonsingular. 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 described the nonequilibrium roughening transition, to be z=2+O(${\mathrm{\epsilon }}^3$), in agreement with a recent scaling argument by Doty and Kosterlitz [Phys. Rev. Lett. 69, 1979 (1992)]. Similarly, our result for the correlation length exponent at the transition is 1/ν=ε+O(${\mathrm{\epsilon }}^3$). For the smooth phase, some aspects of the crossover from Gaussian to critical behavior are discussed.