Two-loop renormalization-group analysis of the Burgers–Kardar-Parisi-Zhang equation

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 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(ε3), 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(ε3). For the smooth phase, some aspects of
the crossover from Gaussian to critical behavior are discussed.