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

TL;DR

This paper performs a detailed two-loop renormalization group analysis of the Burgers--KPZ equation across different dimensions, revealing fixed points, critical exponents, and non-perturbative behaviors relevant to surface growth and turbulence.

Contribution

It provides a comprehensive two-loop RG calculation for the Burgers--KPZ equation, including fixed points and exponents in various dimensions, and clarifies the nature of strong-coupling behavior.

Findings

Strong-coupling fixed point for dimensions less than 2

No finite strong-coupling fixed point for dimensions greater than 2

Dynamic exponent at the non-equilibrium transition is approximately 2 + O(ε^3)

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 $d_c=2$ we find a strong--coupling fixed point, which diverges at $d=2$, indicating that there is non--perturbative strong--coupling behavior for all $d \geq 2$. 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 $2+\epsilon$ expansion, we find the dynamic exponent corresponding to the unstable fixed point, which describes the non--equilibrium roughening transition, to be $z = 2 + {\cal O} (\epsilon^3)$, in agreement with a recent scaling argument by Doty and Kosterlitz. Similarly, our result for the correlation length exponent at the transition is $1/\nu = \epsilon + {\cal O} (\epsilon^3)$. For the smooth phase, some aspects of the crossover from Gaussian to critical behavior are discussed.