Critical behavior of the random-field Ising model with long-range interactions in one dimension

TL;DR

This paper investigates the critical behavior of the one-dimensional random-field Ising model with long-range interactions using the nonperturbative functional renormalization group, revealing two regimes separated by a critical value of the interaction decay parameter.

Contribution

It identifies two distinct critical regimes in the 1D RFIM with long-range interactions, characterized by the presence or absence of a cusp-like nonanalyticity in the renormalized cumulants.

Findings

Two regimes of critical behavior separated by a critical $\sigma_c$.

Presence or absence of a cusp-like nonanalyticity in the fixed point.

Predictions for lattice simulations to verify the theoretical results.

Abstract

We study the critical behavior of the one-dimensional random field Ising model (RFIM) with long-range interactions ($\propto r^{-(d+\sigma)}$) by the nonperturbative functional renormalization group. We find two distinct regimes of critical behavior as a function of $\sigma$, separated by a critical value $\sigma_c$. What distinguishes these two regimes is the presence or not of a cusp-like nonanalyticity in the functional dependence of the renormalized cumulants of the random field at the fixed point. This change of behavior can be associated to the characteristics of the large-scale avalanches present in the system at zero temperature. We propose ways to check these predictions through lattice simulations. We also discuss the difference with the RFIM on the Dyson hierarchical lattice.