Unveiling the different scaling regimes of the one-dimensional Kardar-Parisi-Zhang--Burgers equation using the functional renormalisation group

TL;DR

This paper uses the functional renormalisation group to analyze the one-dimensional KPZ equation, revealing a new scaling regime with dynamical exponent z=1 linked to the inviscid Burgers fixed point, and explores the crossover between regimes.

Contribution

It introduces an advanced numerical method to solve the full FRG equations, enabling a unified analysis of the correlation function across all momentum scales and fixed points.

Findings

Confirmed the z=1 scaling at the inviscid Burgers fixed point.

Quantified the crossover between KPZ and IB regimes.

Provided a comprehensive correlation function across momenta.

Abstract

The Kardar-Parisi-Zhang (KPZ) equation is a celebrated non-linear stochastic equation featuring non-equilibrium scaling. Although in one dimension, its statistical properties are very well understood, a new scaling regime has been reported in recent numerical simulations. This new regime is characterised by a dynamical exponent $z=1$, markedly different from the expected one $z=3/2$ for the KPZ universality class, and it emerges when approaching the inviscid limit. The origin of this scaling has been traced down to the existence of a new fixed point, termed the inviscid Burgers (IB) fixed point, which was uncovered using the functional renormalisation group (FRG). The FRG equations can be solved analytically in the asymptotic regime of vanishing viscosity and large momenta, showing that indeed $z=1$ exactly at the IB fixed point. In this work, we set up an advanced method to numerically solve the full FRG flow equations in a certain approximation, which allows us to determine in a unified way the correlation function over the whole range of momenta, not restricted to some particular regime. We analyse the crossover between the different fixed points, and quantitatively determine the extent of the IB regime.