Morphology of renormalization-group flow for the de Almeida-Thouless-Gardner universality class

TL;DR

This paper investigates the renormalization-group flow for the de Almeida-Thouless-Gardner universality class, revealing complex dynamics such as limit cycles and bifurcations that influence the nature of the phase transition across dimensions.

Contribution

It introduces a novel analysis of the RG flow, showing the possibility of limit cycles and bifurcations, and discusses how different schemes affect the fixed point structure.

Findings

Strong-coupling fixed point can undergo Hopf bifurcation

Presence of a critical limit cycle indicating discrete scale invariance

The basin of attraction of the fixed point/limit cycle may remain finite across dimensions

Abstract

A replica-symmetry-breaking phase transition is predicted in a host of disordered media. The criticality of the transition has, however, long been questioned below its upper critical dimension, six, due to the absence of a critical fixed point in the renormalization-group flows at one-loop order. A recent two-loop analysis revealed a possible strong-coupling fixed point but, given the uncontrolled nature of perturbative analysis in the strong-coupling regime, debate persists. Here we examine the nature of the transition as a function of spatial dimension and show that the strong-coupling fixed point can go through a Hopf bifurcation, resulting in a critical limit cycle and a concomitant discrete scale invariance. We further investigate a different renormalization scheme and argue that the basin of attraction of the strong-coupling fixed point/limit cycle may thus stay finite for all dimensions.