Singularities of the renormalization group flow for random elastic manifolds

TL;DR

This paper investigates the singularities in the zero-temperature renormalization group flow of random elastic manifolds, identifying key points where perturbation theory breaks down and clarifying the relationship between these singularities.

Contribution

It demonstrates that under certain conditions, the singularity where the potential's fourth derivative diverges occurs before the potential's average value turns negative, clarifying the flow's behavior.

Findings

The point $l_c$ occurs before $l^*$, preventing the negative renormalization of $<U^2>$.

Perturbation theory breaks down at the scale $l_c$ due to metastable states.

The flow's singularities are characterized and their physical implications are clarified.

Abstract

We consider the singularities of the zero temperature renormalization group flow for random elastic manifolds. When starting from small scales, this flow goes through two particular points $l^{*}$ and $l_{c}$, where the average value of the random squared potential $<U^{2}>$ turnes negative ($l^{*}$) and where the fourth derivative of the potential correlator becomes infinite at the origin ($l_{c}$). The latter point sets the scale where simple perturbation theory breaks down as a consequence of the competition between many metastable states. We show that under physically well defined circumstances $l_{c}<l^{*}$ and thus the apparent renormalization of $<U^{2}>$ to negative values does not take place.