Disorder Averaging and Finite Size Scaling

TL;DR

This paper introduces a novel RG approach for disordered systems that emphasizes analyzing individual sample trajectories before averaging, leading to improved understanding of finite-size scaling in disordered models.

Contribution

It proposes a new perspective on the RG in disordered systems by focusing on sample-specific trajectories rather than averaged quantities, and demonstrates its effectiveness on the 2D random-bond Ising model.

Findings

Finite-size corrections are mainly due to sample-to-sample critical temperature fluctuations.

Scaling is more accurate using the new variables derived from the sample-specific approach.

The new method improves the understanding of finite-size effects in disordered systems.

Abstract

We propose a new picture of the renormalization group (RG) approach in the presence of disorder, which considers the RG trajectories of each random sample (realization) separately instead of the usual renormalization of the averaged free energy. The main consequence of the theory is that the average over randomness has to be taken after finding the critical point of each realization. To demonstrate these concepts, we study the finite-size scaling properties of the two-dimensional random-bond Ising model. We find that most of the previously observed finite-size corrections are due to the sample-to-sample fluctuation of the critical temperature and scaling is more adequate in terms of the new scaling variables.