Thermal fluctuations in pinned elastic systems: field theory of rare events and droplets

TL;DR

This paper develops a non-perturbative field theory framework using the functional renormalization group to analyze thermal fluctuations and rare events in pinned elastic systems, connecting droplet physics with critical phenomena.

Contribution

It introduces a thermal boundary layer concept in the effective action to describe rare droplet events and resolves singularities, providing a systematic approach to study thermal fluctuations in disordered elastic systems.

Findings

The thermal boundary layer captures droplet physics and rare degeneracies.

Exact relations between TBL quantities and droplet probabilities are established.

The beta-function at zero temperature is computed up to four loops, removing ambiguities.

Abstract

Using the functional renormalization group (FRG) we study the thermal fluctuations of elastic objects, described by a displacement field u and internal dimension d, pinned by a random potential at low temperature T, as prototypes for glasses. A challenge is how the field theory can describe both typical (minimum energy T=0) configurations, as well as thermal averages which, at any non-zero T as in the phenomenological droplet picture, are dominated by rare degeneracies between low lying minima. We show that this occurs through an essentially non-perturbative *thermal boundary layer* (TBL) in the (running) effective action Gamma[u] at T>0 for which we find a consistent scaling ansatz to all orders. The TBL resolves the singularities of the T=0 theory and contains rare droplet physics. The formal structure of this TBL is explored around d=4 using a one loop Wilson RG. A more systematic Exact RG (ERG) method is employed and tested on d=0 models. There we obtain precise relations between TBL quantities and droplet probabilities which are checked against exact results. We illustrate how the TBL scaling remains consistent to all orders in higher d using the ERG and how droplet picture results can be retrieved. Finally, we solve for d=0,N=1 the formidable "matching problem" of how this T>0 TBL recovers a critical T=0 field theory. We thereby obtain the beta-function at T=0, *all ambiguities removed*, displayed here up to four loops. A discussion of d>4 case and an exact solution at large d are also provided.