Potential Flow of Renormalization Group in Quasi-One-Dimensional Systems

TL;DR

This paper reveals that the complex RG flow equations in quasi-one-dimensional systems can be described by a potential function, simplifying the understanding of phase diagrams and ruling out chaotic behaviors.

Contribution

The authors explicitly construct a renormalization-group potential using Majorana fermions, providing a new analytical tool for analyzing phase diagrams in these systems.

Findings

RG flows are governed by a potential function

Existence of the potential prevents chaotic RG behaviors

Phase diagrams are determined by fixed-ray solutions

Abstract

We address the issue why the phase diagrams for quasi-one-dimensional systems are rather simple, while the renormalization group equations behind the scene are non-linear and messy looking. The puzzle is answered in two steps -- we first demonstrate that the complicated coupled flow equations are simply described by a potential $V(h_i)$, in an appropriate basis for the interaction couplings $h_i$. The renormalization-group potential is explicitly constructed by introducing the Majorana fermion representation. The existence of the potential prevents chaotic behaviors and other exotic possibilities such as limit cycles. Once the potential is obtained, the ultimate fate of the flows are described by a special set of fixed-ray solutions and the phase diagram is determined by Abelian bosonization. Extension to strong coupling regime and comparison with the Zamolodchikov c-theorem are discussed at the end.