Strong-Coupling Fixed Point of the Kardar-Parisi-Zhang Equation

Abstract

{\em NOTE: This paper presented the first attempt to tackle the Kardar-Parisi-Zhang (KPZ) equation using non-perturbative renormalisation group (NPRG) methods. It exploited the most natural and frequently used approximation scheme within the NPRG framework, namely the derivative expansion (DE). However, the latter approximation turned out to yield unphysical critical exponents in dimensions $d\ge 2$ and, furthermore, hinted at very poor convergence properties of the DE. The author has since realized that in fact, this approximation may not be valid for the KPZ problem, because of the very nature of the KPZ interaction, which is not {\em potential} but {\em derivative}. The probable failure of the DE is a very unusual -- and instructive -- feature within the NPRG framework. As such, the original work, unpublished, is left available on the arXiv and can be found below. Added note: the key to deal with the KPZ problem using NPRG lies in not truncating the momentum dependence of the correlation functions, which is investigated in a recent work {\em arXiv:0905.1025}.} We present a new approach to the Kardar-Parisi-Zhang (KPZ) equation based on the non-perturbative renormalisation group (NPRG). The NPRG flow equations derived here, at the lowest order of the derivative expansion, provide a stable strong-coupling fixed point in all dimensions $d$, embedding in particular the exact results in $d=0$ and $d=1$. However, it yields at this order unreliable dynamical and roughness exponents $z$ and $\chi$ in higher dimensions, which suggests that a richer approximation is needed to investigate the property of the rough phase in $d \ge 2$.