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

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.

Quantum Field Theory of Many-Body Systems

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“…Much of this progress has occurred in three areas of research: (1) the study of topological phases in condensed matter systems such as FQH systems [Wen and Niu (1990); Blok and Wen (1990); Read (1990); Fröhlich and Kerler (1991)], quantum dimer models [Rokhsar and Kivelson (1988); Read and Chakraborty (1989); Moessner and Sondhi (2001); Ardonne et al (2004)], quantum spin models [Kalmeyer and Laughlin (1987) ;Wen et al (1989); Wen (1990); Read and Sachdev (1991); Wen (1991a); Senthil and Fisher (2000); Wen (2002b); Sachdev and Park (2002) ;Balents et al (2002)], or even superconducting states [Wen (1991b); Hansson et al (2004)], (2) the study of lattice gauge theory [Wegner (1971); Banks et al (1977); Kogut and Susskind (1975); Kogut (1979)], and (3) the study of quantum computing by anyons [Kitaev (2003); Ioffe et al (2002); Freedman et al (2002)]. The phenomenon of string condensation is important in all of these fields, though the string picture is often de-emphasized.…”

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