Field Theory of Disordered Elastic Interfaces at 3-Loop Order: The $\beta$-Function

TL;DR

This paper computes the third-order renormalization-group $eta$-function for disordered elastic interfaces in equilibrium, advancing theoretical understanding of their critical behavior near the upper critical dimension.

Contribution

It provides a detailed 3-loop calculation of the $eta$-function using exact RG techniques, addressing issues related to the cusp in the disorder distribution.

Findings

Third-order $eta$-function derived for disordered elastic interfaces.

Resolution of cusp-related problems in the renormalized disorder.

Enhanced theoretical framework for critical phenomena in disordered systems.

Abstract

We calculate the effective action for disordered elastic manifolds in the ground state (equilibrium) up to 3-loop order. This yields the renormalization-group $\beta$-function to third order in $\epsilon=4-d$, in an expansion in the dimension $d$ around the upper critical dimension $d=4$. The calculations are performed using exact RG, and several other techniques, which allow us to resolve consistently the problems associated with the cusp of the renormalized disorder.