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

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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 < n < 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.