Fixed $d$ Renormalization Group Analysis of Conserved Surface Roughening

Abstract

Conserved surface roughening represents a special case of interface dynamics where the total height of the interface is conserved. Recently, it was suggested [F. Caballero et al., Phys. Rev. Lett. 121, 020601 (2018)] that the original continuum model known as `Conserved Kardar-Parisi-Zhang'(CKPZ) equation is incomplete, as additional non-linearity is not forbidden by any symmetry in $d > 1$. In this work, we perform detailed field-theoretic renormalization group (RG) analysis of a general stochastic model describing conserved surface roughening. Systematic power counting reveals additional marginal interaction at the upper critical dimension, which appears also in the context of molecular beam epitaxy. Depending on the origin of the surface particle's mobility, the resulting model shows two different scaling regimes; If the particles move mainly due to the gravity, the leading dispersion law is $\omega \sim k^2$, and the mean-field approximation describing a flat interface is exact in any spatial dimension. On the other hand, if the particles move mainly due to the surface curvature, the interface becomes rough with the mean-field dispersion law $\omega \sim k^4$ , and the corrections to scaling exponents must be taken into account. We show, that the latter model consist of two sub-class of models that are decoupled in all orders of perturbation theory. Moreover, our RG analysis of the general model reveals that the universal scaling is described by a rougher interface than CKPZ universality class. The universal exponents are derived within the one-loop approximation in both fixed $d$ and $\varepsilon$-expansion schemes, and their relation is discussed. We point out all important details behind these two schemes which are often overlooked in the literature, and their misinterpretation might lead to inconsistent results.