Logarithmic or algebraic: roughening of an active Kardar-Parisi-Zhang surface

TL;DR

This work extends the KPZ framework by incorporating symmetry-allowed nonlocal nonlinearities, producing the active-KPZ equation that supports stable 2D steady states with generalized quasi-long-ranged order and nonuniversal logarithmic roughness. A dynamic RG treatment shows no one-loop renormalization of the nonlinearities themselves, while D and ν acquire logarithmic corrections that drive a nonuniversal scaling of fluctuations via μ and κ; in 2D, the system can exhibit SQLRO or WQLRO depending on parameter values, and in d > 2 a nonuniversal roughening transition emerges. The results highlight how nonlocal, active interactions compete with local KPZ nonlinearity to yield rich scaling behavior distinct from both KPZ and EW universality, providing a paradigmatic nonlocal growth equation with potential relevance to nonlocal transport, active membranes, and soft matter interfaces.

Abstract

The Kardar-Parisi-Zhang (KPZ) equation sets the universality class for growing and roughening of nonequilibrium surfaces without any conservation law and nonlocal effects. We argue here that the KPZ equation can be generalized by including a symmetry-permitted nonlocal nonlinear term of active origin that is of the same order as the one included in the KPZ equation. Including this term, the 2D active KPZ equation is stable in some parameter regimes, in which the interface conformation fluctuations exhibit sublogarithmic or superlogarithmic roughness, with nonuniversal exponents, giving positional generalised quasi-long-ranged order. For other parameter choices, the model is unstable, suggesting a perturbatively inaccessible algebraically rough interface or positional short-ranged order. Our model should serve as a paradigmatic nonlocal growth equation.

Figures (6)

• Figure 1: (a) RG flow diagram in the $g$-$\gamma_1$ plane in the achiral limit ($\gamma_2=0$). Arrows indicate RG flows. Flow in the stable (unstable), i.e., toward (away from), $g=0$ region are marked. (b) Variations of $\mu$ and $\kappa$ as functions of $\gamma_1$ in the stable region for the achiral case. (c) RG flow diagram in the space spanned by $\gamma_1$-$\gamma_2$-$g$ in the full a-KPZ equation. RG flow lines in the stable and unstable regions are shown by the arrows. (d) Phase diagram in the $\gamma_1$-$\gamma_2$ plane for the a-KPZ equation. The central gray region containing the origin is unstable. Regions with SQLRO and WQLRO are marked (see text).
• Figure 2: (a) Variation of $z$ and $\chi$ with $\gamma_1$ on the fixed line $g_c^*=-1/ {\cal \tilde{A}}(\gamma_1)$ for $\epsilon=1.$ (b) RG flow diagram in the $g$-$\gamma_1$ plane for $d>2$. The black dashed line is the fixed line $g_c^*=-1/ {\cal \tilde{A}}(\gamma_1)$ for $\epsilon=1$, bounded by lines $\gamma_1=\gamma_-,\gamma_+$ (blue dashed lines). Stable (unstable) flow lines are the arrows pointing toward (away from) $g=0$ (see text).
• Figure 3: Diagrammatic representation of two point functions.
• Figure 4: One-loop Feynman diagrams that contribute to the renormalisation of $D$
• Figure 5: One-loop Feynman diagrams that contribute to the renormalisation of $\nu$