Functional Renormalization Description of the Roughening Transition

TL;DR

This paper uses a perturbative Functional Renormalization Group approach to analyze the static thermal roughening transition of an elastic manifold at the critical dimension, providing analytical and numerical insights into the effective potential and surface tension.

Contribution

It introduces a new flow equation for the potential and surface tension valid at all temperatures, and characterizes the large-scale effective potential structure.

Findings

Renormalized potential consists of quasi parabolic wells at large scales.

Numerical step energy as a function of temperature matches experimental data.

Scenario outlined for dimensions below two and for non-local elasticity cases.

Abstract

We reconsider the problem of the static thermal roughening of an elastic manifold at the critical dimension $d=2$ in a periodic potential, using a perturbative Functional Renormalization Group approach. Our aim is to describe the effective potential seen by the manifold below the roughening temperature on large length scales. We obtain analytically a flow equation for the potential and surface tension of the manifold, valid at all temperatures. On a length scale $L$, the renormalized potential is made up of a succession of quasi parabolic wells, matching onto one another in a singular region of width $\sim L^{-6/5}$ for large $L$. We also obtain numerically the step energy as a function of temperature, and relate our results to the existing experimental data on $^4$He. Finally, we sketch the scenario expected for an arbitrary dimension $d<2$ and examine the case of a non local elasticity which is realized physically for the contact line.