Universality Classes with Strong Coupling in Conserved Surface Roughening: Explicit vs Emergent Symmetries

TL;DR

This paper investigates strong coupling behavior in conserved surface roughening models, revealing how nonlinear scaling exponents and symmetries depend on a parameter n, with numerical and analytical methods showing vertex renormalization and non-Gaussian fluctuations.

Contribution

It introduces a family of conserved stochastic equations generalizing the stochastic Burgers equation, analyzing their strong coupling regimes and symmetries through renormalization group and numerical simulations.

Findings

Strong coupling exponents depend on parameter n.

Numerical evidence suggests vertex renormalization for odd n.

Height fluctuation distributions are non-Gaussian with zero skewness.

Abstract

The occurrence of strong coupling or nonlinear scaling behavior for kinetically rough interfaces whose dynamics are conserved, but not necessarily variational, remains to be fully understood. Here we formulate and study a family of conserved stochastic evolution equations for one-dimensional interfaces, whose nonlinearity depends on a parameter n, thus generalizing that of the stochastic Burgers equation, whose behavior is retrieved for n=0. This family of equations includes as particular instances a stochastic porous medium equation and other continuum models relevant to various hard and soft condensed matter systems. We perform a one-loop dynamical renormalization group analysis of the equations, which contemplates strong coupling scaling exponents that depend on the value of $n$ and may or may not imply vertex renormalization. These analytical expectations are contrasted with explicit numerical simulations of the equations with n=1,2, and 3. For odd n, numerical stability issues have required us to generalize the scheme originally proposed for n=0 by T. Sasamoto and H. Spohn. Precisely for n=1 and 3, and at variance with the n=0 and 2 cases (whose numerical exponents are consistent with non-renormalization of the vertex), numerical strong coupling exponent values are obtained which suggest vertex renormalization, akin to that reported for the celebrated conserved KPZ equation. We also study numerically the statistics of height fluctuations, whose probability distribution function turns out (at variance with cKPZ) to have zero skewness for long times and at saturation, irrespective of the value of n. However, the kurtosis is non-Gaussian, further supporting the conclusion on strong coupling asymptotic behavior. The zero skewness seems related with space symmetries of the n=0 and 2 equations, and with an emergent symmetry at the strong coupling fixed point for odd values of n.