The unpredicted scaling of the one-dimensional Kardar-Parisi-Zhang equation

Abstract

The celebrated Kardar-Parisi-Zhang (KPZ) equation describes the kinetic roughening of stochastically growing interfaces. In one dimension, the KPZ equation is exactly solvable and its statistical properties are known to an exquisite degree. Yet recent numerical simulations in the tensionless (or inviscid) limit of the KPZ equation [Phil. Trans. Roy. Soc. A 380, 20210090 (2022); Phys. Rev. E 106, 024802 (2022)] unveiled a new scaling, with a critical dynamical exponent $z=1$ different from the KPZ one $z=3/2$. In this Letter, we show that this scaling is controlled by a fixed point which had been missed so far and which corresponds to an infinite non-linear coupling. Using the functional renormalization group (FRG), we demonstrate the existence of this fixed point and show that it yields $z=1$. We calculate the correlation function and associated scaling function at this fixed point, providing both a numerical solution of the FRG equations within a reliable approximation, and an exact asymptotic form obtained in the limit of large wavenumbers. Both scaling functions accurately match the one from the numerical simulations.

revised version, 19 pages, 10 figures

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• The inviscid fixed point of the multi-dimensional Burgers-KPZ equation
• Scaling regimes of the one-dimensional phase turbulence in the deterministic complex Ginzburg-Landau equation
• Kardar-Parisi-Zhang scaling in time-crystalline matter
• Numerical Integration of the KPZ and Related Equations on Networks: The Case of the Cayley Tree
• Nonconserved critical dynamics of the two-dimensional Ising model as a surface kinetic roughening process
• The non-perturbative sides of the Kardar-Parisi-Zhang equation
• Quantum criticality and non-Fermi liquids: the Wilsonian renormalization group perspective
• Unveiling the different scaling regimes of the one-dimensional Kardar-Parisi-Zhang--Burgers equation using the functional renormalisation group