Critical behavior of Ising spin systems

Abstract

Physical phenomena commonly observed in nature such as phase transitions, critical phenomena and metastability when studied froma mathematical point of view may give arise to a rich variety of behavior whose study becomes interesting in itself. In Chapter 1 we illustrate the phase transition phenomenon at low temperatures for one-dimensional long range Ising models with inhomogeneous external fields. More precisely, we consider Ising spins arranged on the one-dimensional integer lattice where such spins interact via ferromagnetic pairwise interactions whose strength is inversely proportional to their distance to the power ®; furthermore, the system is put under the influence of an external magnetic field that vanishes with polynomial power ± as the distance between the spin and the origin increases. In that case we show that a phase transition manifests itself in the form of the existence of two distinct infinite-volume Gibbs states, obtained by means of the application of the thermodynamic limit considering “plus” and “minus” boundary conditions respectively, whenever the system is subject at low temperatures and an inequality involving ® and ± holds. The proof of this result is done by means of the Peierls’ contour argument adapted to one-dimensional long range Ising models, first introduced by J. Fröhlich and T. Spencer in 1982 and later modified by M. Cassandro, P.A. Ferrari, I. Merola and E. Presutti in 2005. Our results improve the one obtained by the latter authors since we managed to avoid the assumption of large nearest-neighbor interactions and added the influence of an external field, showing an interplay between the constants ® and ± in order to guarantee the manifestation of the phase transition.