Existence of long-range order in random-field Ising model on Dyson hierarchical lattice

TL;DR

This paper proves the existence of long-range order in the random-field Ising model on Dyson hierarchical lattices for certain decay exponents and low temperatures, extending known results to include random fields.

Contribution

It provides a rigorous proof of long-range order in the RFIM on Dyson lattices for 1<α<3/2, incorporating the effects of random fields at low temperatures.

Findings

Long-range order exists in RFIM for 1<α<3/2 at low temperatures.

The proof extends Dyson's method to include random fields.

Order persists even at zero temperature with small random fields.

Abstract

We study the random-field Ising model on a Dyson hierarchical lattice, where the interactions decay in a power-law-like form, $J(r)\sim r^{-\alpha}$, with respect to the distance. Without a random field, the Ising model on the Dyson hierarchical lattice has a long-range order at finite low temperatures when $1<\alpha<2$. In this study, for $1<\alpha<3/2$, we rigorously prove that there is a long-range order in the random-field Ising model on the Dyson hierarchical lattice at finite low temperatures, including zero temperature, when the strength of the random field is sufficiently small but nonzero. Our proof is based on Dyson's method for the case without a random field, and the concentration inequalities in probability theory enable us to evaluate the effect of a random field.