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

TL;DR

This work analyzes the random-field Ising model on a Dyson hierarchical lattice with power-law interactions $J(r) \sim r^{-\ alpha}$, focusing on the regime $1<\alpha<3/2$. By extending Dyson's hierarchical, recursion-based approach and employing concentration inequalities to control random-field fluctuations, the authors derive a recurrence for $f_N=2^{-2N}\mathbb{E}[\langle S_{N,1}^2\rangle_N]$ and establish a nontrivial lower bound on the long-range order parameter $m^2=\liminf f_N$. They show that for sufficiently small random-field strength $h$ (and large inverse temperature $\beta$), $m^2>0$, implying long-range order at finite temperatures, including $T=0$. The results hold for Gaussian and Bernoulli random fields and extend to other symmetric i.i.d. distributions via a modified logarithmic Sobolev inequality; these findings support the existence of a phase transition in the corresponding one-dimensional long-range RFIM for this $\alpha$-range, while highlighting limitations near the upper bound where GK-S inequalities fail.

Abstract

We study the random-field Ising model on a Dyson hierarchical lattice, where the interactions decay in a power-law-like form, $J(r)\sim r^{-α}$, 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 $1<α<2$. In this study, for $1<α<3/2$, 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.

Figures (1)

• Figure 1: (Color online) Regions where $m^2$ is greater than 0 examined numerically based on Eq. (\ref{['m-bound']}). The horizontal and vertical axes denote the strength of the random field and the inverse temperature, respectively. Left panel: For $\alpha=1.1$ and $c=10$, the blue region guarantees $m^2>0$. Right pane: For $\alpha=1.49$ and $c=10$, the blue region guarantees $m^2>0$.

Theorems & Definitions (9)

• Theorem 1
• Corollary 2
• Remark 3
• Lemma 4
• Remark 5
• proof : Proof of Lemma \ref{['lemma3']}
• Lemma 6
• proof : Proof of Lemma \ref{['lemma5']}
• proof : Proof of Theorem \ref{['theorem1']}