The non-perturbative sides of the Kardar-Parisi-Zhang equation

TL;DR

The paper surveys non-perturbative aspects of the KPZ equation using functional renormalization group (FRG) methods. It identifies and characterizes the KPZ fixed point across dimensions, constrained by extended symmetries and Ward identities, and reveals a novel inviscid Burgers (IB) fixed point governing large-momentum behaviour in the inviscid limit. Starting from the KPZ action, the FRG framework yields a phase diagram with EW, KPZ, and RT fixed points, and, through progressively refined approximations, accurately reproduces KPZ scaling functions in one dimension and provides quantitative insights in higher dimensions. The IB fixed point, accessible via non-perturbative FRG, exhibits dynamical exponent $z\approx 1$ and connects to broader contexts such as complex Ginzburg-Landau and Kuramoto-Sivashinsky systems, with implications for experiments in driven-dissipative condensates and cold-atom setups. Overall, the work highlights how non-perturbative FRG approaches uncover deep fixed-point structures and universal scaling in non-equilibrium growth models.

Abstract

The Kardar-Parisi-Zhang (KPZ) equation is a celebrated non-linear stochastic dynamical equation yielding non-equilibrium universal scaling. It exhibits notorious non-perturbative aspects. The KPZ fixed point is strong-coupling, all the more in $d>1$. Strikingly, another, even stronger-coupling fixed point of the KPZ equation, called inviscid Burgers fixed point, has been recently unveiled. These non-pertubative features can be theoretically accessed and studied in a controlled way in all dimensions using the functional renormalisation group. We propose an overview of the related results, which provide a unified picture of the fixed-point structure and associated scaling regimes of the KPZ equation in $d=1$ and in higher dimensions.

Figures (8)

• Figure 1: Flows of $\sqrt{\tilde{g}_s}$ as a function of the RG time $s=\ln(\kappa/\Lambda)$ for different initial values of $\tilde{g}_0$ in dimension (a) $d=1$ and (b) $d=3$. The (IR) flow runs from the microscopic ($s=0$) to the macroscopic ($s\to-\infty$) scale from right to left.
• Figure 2: Flow diagram of the KPZ equation: fixed point values $\sqrt{\tilde{g}_*}$ while varying the dimension $d$ in the vertical axis. The dots represent the fixed points, and the grey lines with arrows indicate the RG flows (from $s=0$ to $s\to-\infty$) which show whether these fixed points are stable are unstable (in the IR flow).
• Figure 3: Scaling functions in $d=1$ associated with the correlation function (a) in Fourier space $C(t,p)$ and (b) in real space $C(t,x)$, from the FRG calculation Canet2011kpz and from the exact result of Ref. Praehofer04. There are no adjustable parameters.
• Figure 4: Scaling functions from the FRG calculation in $d=1,2,3$: (a) $\mathring{F}_{\rm KPZ}$ associated with the correlation function and (b) $\mathring{H}_{\rm KPZ}$ associated with the response function Kloss2012.
• Figure 5: Same as Fig. \ref{['fig:phasediag']}, but featuring the additional IB fixed point, which is a UV fixed point, ie always repulsive.