Non-perturbative renormalisation group for the Kardar-Parisi-Zhang equation: general framework and first applications

TL;DR

This paper develops a non-perturbative renormalization group method to analytically compute critical exponents and correlation functions for the KPZ equation across all regimes, including strong coupling, with results matching exact solutions.

Contribution

It introduces a symmetry-preserving approximation scheme within the non-perturbative RG framework for the KPZ equation, providing a comprehensive phase diagram and accurate correlation functions.

Findings

Qualitative phase diagram in all dimensions

Accurate critical exponents in physical dimensions

Excellent agreement with exact 1D correlation functions

Abstract

We present an analytical method, rooted in the non-perturbative renormalization group, that allows one to calculate the critical exponents and the correlation and response functions of the Kardar-Parisi-Zhang (KPZ) growth equation in all its different regimes, including the strong-coupling one. We analyze the symmetries of the KPZ problem and derive an approximation scheme that satisfies the linearly realized ones. We implement this scheme at the minimal order in the response field, and show that it yields a complete, qualitatively correct phase diagram in all dimensions, with reasonable values for the critical exponents in physical dimensions. We also compute in one dimension the full (momentum and frequency dependent) correlation function, and the associated universal scaling functions. We find an excellent quantitative agreement with the exact results from Praehofer and Spohn (J. Stat. Phys. 115 (2004)). We emphasize that all these results, which can be systematically improved, are obtained with sole input the bare action and its symmetries, without further assumptions on the existence of scaling or on the form of the scaling function.