Renormalization group analysis of the anisotropic Kardar-Parasi-Zhang equation with spatially correlated noise

TL;DR

This paper uses renormalization group analysis to study the anisotropic KPZ equation with spatially correlated noise, revealing a novel stable fixed point and critical exponents depending on the substrate dimension and noise correlation.

Contribution

It identifies a new finite stable fixed point for the anisotropic KPZ equation with correlated noise in certain dimensions, extending understanding of its critical behavior.

Findings

Discovery of a stable fixed point for $d' < 2+2 ho$

Explicit formulas for roughening and dynamic exponents

Flow to weak-coupling fixed point for higher dimensions

Abstract

We analyze the anisotropic Kardar-Parisi-Zhang equation in general substrate dimensions $d'$ with spatially correlated noise, $\langle\tilde \eta({\bf{k}},\omega)\rangle=0$ and $\langle\tilde \eta({\bf{k}},\omega) \tilde \eta({\bf{k}}',\omega')\rangle$ $=2D(k)\delta^{d'}({\bf{k}}+{\bf{k}}')\delta(\omega+\omega')$ where $D(k)=D_0+D_{\rho}k^{-2\rho}$, using the dynamic renormalization group (RG) method. When the signs of the nonlinear terms in parallel and perpendicular directions are opposite, a novel finite stable fixed point is found for $d'< d'_c \equiv 2+2\rho$ within one-loop order. The roughening exponent $\alpha$ and the dynamic exponent $z$ associated with the stable fixed point are determined as $\alpha={2 \over 3}\big(\rho-{{d'-2}\over 2}\big)$, and $z=2-\alpha$. For $d' > d'_c$, the RG transformations flow to the fixed point of the weak-coupling limit, so that the dynamic exponent becomes $z=2$.