Avalanches and perturbation theory in the random-field Ising model

TL;DR

This paper explores the limitations of perturbation theory in the random-field Ising model, emphasizing the role of avalanches and nonanalyticities, and clarifies the conditions under which dimensional reduction holds.

Contribution

It introduces a method to incorporate avalanche effects into perturbation theory, refining understanding of critical behavior in the RFIM near the upper critical dimension.

Findings

Dimensional reduction applies above a dimension close to 5.

Avalanches induce nonanalyticities affecting critical scaling.

Perturbation theory's validity is limited below certain dimensions.

Abstract

Perturbation theory for the random-field Ising model (RFIM) has the infamous attribute that it predicts at all orders a dimensional-reduction property for the critical behavior that turns out to be wrong in low dimension. Guided by our previous work based on the nonperturbative functional renormalization group (NP-FRG), we show that one can still make some use of the perturbation theory for a finite range of dimension below the upper critical dimension, d=6. The new twist is to account for the influence of large-scale zero-temperature events known as avalanches. These avalanches induce nonanalyticities in the field dependence of the correlation functions and renormalized vertices, and we compute in a loop expansion the eigenvalue associated with the corresponding anomalous operator. The outcome confirms the NP-FRG prediction that the dimensional-reduction fixed point correctly describes the dominant critical scaling of the RFIM above some dimension close to 5 but not below.