Abstract
We study the stochastic Kardar-Parisi-Zhang equation for kinetic roughening where the time-independent (columnar or spatially quenched) Gaussian random noise f(t,x) is specified by the pair correlation function ⟨f(t,x)f(t′,x′)⟩∝δ(d)(x−x′), d being the dimension of space. The field-theoretic renormalization group analysis shows that the effect of turbulent motion of the environment (modelled by the coupling with the velocity field described by the Kazantsev-Kraichnan statistical ensemble for an incompressible fluid) gives rise to a new nonlinear term, quadratic in the velocity field. It turns out that this “induced” nonlinearity strongly affects the scaling behaviour in several universality classes (types of long-time, large-scale asymptotic regimes) even when the turbulent advection appears irrelevant in itself. Practical calculation of the critical exponents (that determine the universality classes) is performed to the first order of the double expansion in ε=4−d and the velocity exponent ξ (one-loop approximation). As is the case with most “descendants” of the Kardar-Parisi-Zhang model, some relevant fixed points of the renormalization group equations lie in “forbidden zones”, i.e., in those corresponding to negative kinetic coefficients or complex couplings. This persistent phenomenon in stochastic non-equilibrium models requires careful and inventive physical interpretation.
Highlights
If an appropriate fixed point does exist, it seems to be inaccessible within any kind of perturbative treatment. This point would correspond to the rough phase, or to be precise, to the non-trivial asymptotic behaviour of the interface in the IR range, i.e., to kinetic roughening
We found out that coupling with the turbulent velocity field leads to an emergence of a new nonlinearity that must be included in the model to make it renormalizable
All critical exponents for the point FP4 coincide with the ones calculated for nontrivial regime of a system described by the original KPZ model and Kazantsev-Kraichnan ensemble, see equation (6.4) in [67]
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. If an appropriate fixed point does exist, it seems to be inaccessible within any kind of perturbative treatment This point would correspond to the rough phase, or to be precise, to the non-trivial asymptotic behaviour of the interface in the IR range (which implies that times and distances are large in comparison with the characteristic microscopic scales), i.e., to kinetic roughening (or critical scaling). Other open questions include the random noise interpretation [32,33], the value of the upper critical dimension [3,19,34–38] and its very existence [20,39–43] 3 All of these facts suggest that instead of a more sophisticated analysis, the KPZ model may need modifications or adjustments that might lead to a drastic change in the RG analysis.
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