Reexamining the renormalization group: Period doubling onset of chaos

TL;DR

This paper re-examines the renormalization group in the context of the period doubling route to chaos, revealing multiple fixed points and clarifying corrections to scaling, with implications for critical phenomena.

Contribution

It demonstrates the non-uniqueness of the RG fixed point in one-humped maps and introduces a framework to analyze singular and gauge corrections to scaling.

Findings

Multiple fixed points exist for the RG in period doubling.

A systematic distinction between singular and gauge corrections is established.

Numerical results from literature are explained through this framework.

Abstract

We explore fundamental questions about the renormalization group through a detailed re-examination of Feigenbaum's period doubling route to chaos. In the space of one-humped maps, the renormalization group characterizes the behavior near any critical point by the behavior near the fixed point. We show that this fixed point is far from unique, and characterize a submanifold of fixed points of alternative RG transformations. We build on this framework to systematically distinguish and analyze the allowed singular and `gauge' (analytic and redundant) corrections to scaling, explaining numerical results from the literature. Our analysis inspires several conjectures for critical phenomena in statistical mechanics.