Fixing the fixed-point system - Applying Dynamic Renormalization Group to systems with long-range interactions

TL;DR

This paper introduces a Dynamic Renormalization Group approach tailored for nonlocal one-dimensional models, successfully resolving inconsistencies and aligning predictions with known exact results.

Contribution

It proposes a novel DRG method that correctly identifies fixed points in nonlocal dynamical systems, improving analysis accuracy.

Findings

Eliminates problematic predictions in nonlocal KPZ models

Recovers existing exact analytic results

Provides a consistent framework for dynamical systems with long-range interactions

Abstract

In this paper a mode of using the Dynamic Renormalization Group (DRG) method is suggested in order to cope with inconsistent results obtained when applying it to a continuous family of one-dimensional nonlocal models. The key observation is that the correct fixed-point dynamical system has to be identified during the analysis in order to account for all the relevant terms that are generated under renormalization. This is well established for static problems, however poorly implemented in dynamical ones. An application of this approach to a nonlocal extension of the Kardar-Parisi-Zhang equation resolves certain problems in one-dimension. Namely, obviously problematic predictions are eliminated and the existing exact analytic results are recovered.