Phase Transition in the 1d Random Field Ising Model with Long Range Interaction

Abstract

We study one-dimensional Ising spin systems with ferromagnetic, long-range interaction decaying as n −2+α , α ∈ ( 1 2 , ln 3 ln 2 − 1), in the presence of external random fields. We assume that the random fields are given by a collection of symmetric, independent, identically distributed real random variables, gaussian or subgaussian. We show, for temperature and strength of the randomness (variance) small enough, with IP = 1 with respect to the random fields, that there are at least two distinct extremal Gibbs measures.

Exact ground states of one-dimensional long-range random-field Ising magnets

“…This is an interesting result, because with the Imry-Ma argument 19,23,24,28 only conclusions for the cases σ < 1/2 or σ > 1/2 are possible. Rigorous studies 25-28 do also not exclude σ = 1/2 as possible value of a finite-disorder phase transition at zero temperature.…”

“…In addition, mathematical proofs [25][26][27][28] do not exclude the possibility of h c > 0 for σ c = 1/2 at zero temperature. Recent work 29 , which was performed independently and in parallel to our work, support h c > 0.…”

“…also Refs. 19,23,24,28) is given and also calculations of exact ground states were carried out independently of our work, but it was restricted to the analysis of the Binder cumulant and few other observables. Nevertheless, all measured exponents agree with theory, if one assumes the theory (cf.…”

“…The result 19,23,24,28,29 that the lower critical dimension in short-range models (d c = 2) corresponds to the critical value σ c = 1/2 in long-range models is obtained by a scaling argument similar to the Imry-Ma argument. In this argument no long-range order exists for σ > 1/2, whereas for σ < 1/2 a phase transition at zero temperature should occur.…”

Proof of rounding by quenched disorder of first order transitions in low-dimensional quantum systems

“…The recent result of Cassandro, Orlandi and Picco [23] allows to conclude that at d = 1 the condition (A.2) does indeed provide the threshold decay rate for the validity of Theorem IV.1. They prove that the phase transition is stable under weak disorder in a family of one dimensional Ising spin systems with long range interactions with decay rates arbitrarily close to 3/2, more explicitly with Q {x,y} ≈ Const./|x − y| 3/2−γ at any γ ∈ (0, 0.08).…”