Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters

TLDR

The authors introduce a family of d‑dimensional Ising models Mdn, indexed by n, whose Hamiltonians involve n‑spin interactions and are connected by two duality transformations, including a simple‑cubic dual model with four‑spin terms and a star‑square transformation that produces a competing‑interaction model. They demonstrate that for any positive‑coupling Ising model, duality relates it to another model, and that models with 1 < n < d undergo a phase transition without a local order parameter, evidenced by a specific‑heat singularity and a qualitative change in spin‑correlation functions while lacking long‑range spin order.

Abstract

It is shown that any Ising model with positive coupling constants is related to another Ising model by a duality transformation. We define a class of Ising models Mdn on d-dimensional lattices characterized by a number n = 1, 2, … , d (n = 1 corresponds to the Ising model with two-spin interaction). These models are related by two duality transformations. The models with 1 &amp;lt; n &amp;lt; d exhibit a phase transition without local order parameter. A nonanalyticity in the specific heat and a different qualitative behavior of certain spin correlation functions in the low and the high temperature phases indicate the existence of a phase transition. The Hamiltonian of the simple cubic dual model contains products of four Ising spin operators. Applying a star square transformation, one obtains an Ising model with competing interactions exhibiting a singularity in the specific heat but no long-range order of the spins in the low temperature phase.