Action at a distance in classical uniaxial ferromagnetic arrays

Abstract

We examine in detail the theoretical foundations of striking long-range couplings emerging in arrays of fluid cells connected by narrow channels by using a lattice gas (Ising model) description of a system. We present a reexamination of the well known exact determination of the two-point correlation function along the edge of a channel using the transfer matrix technique and a new interpretation is provided. The explicit form of the correlation length is found to grow exponentially with the cross section of the channels at the bulk two-phase coexistence. The aforementioned result is recaptured by a refined version of the Fisher-Privman theory of first order phase transitions in which the Boltzmann factor for a domain wall is decorated with a contribution stemming from the point tension originated at its endpoints. The Boltzmann factor for a domain wall together with the point tension is then identified exactly thanks to two independent analytical techniques, providing a critical test of the Fisher-Privman theory. We then illustrate how to build up the network model from its elementary constituents, the cells and the channels. Moreover, we are able to extract the strength of the coupling between cells and express them in terms of the length and width and coarse grained quantities such as surface and point tensions. We then support our theoretical investigation with a series of corroborating results based on Monte Carlo simulations. We illustrate how the long range ordering occurs and how the latter is signaled by the thermodynamic quantities corresponding to both planar and three-dimensional Ising arrays.

Ising Models with Holes: Crossover Behavior

“…suggesting the possible additional scaling variable to be (n − 1)/ξ + s . This seems to agree with the observation given in [8]. Another problems concerns the distribution of the roots for A(θ) and B(θ).…”

“…model is more simple than the one considered by Abraham et al [8]. However, due to the relative simplicity, exact calculations can be done to understand the impact of these lines connecting the strips.…”

Multipoint correlation functions at phase separation. Exact results from field theory

“…More interestingly, thanks to calculations based on the exact solution of the planar Ising model, the relation Σ ab ξ b = 1/2 has been proved to be valid for all subcritical temperatures. This result follows as an application of duality [9,38,39]; see also [40].…”

Multipoint correlation functions at phase separation. Exact results from field theory

“…More interestingly, thanks to calculations based on the exact solution of the planar Ising model, the relation Σ ab ξ b = 1/2 has been proved to be valid for all subcritical temperatures. This result follows as an application of duality [9,42,43]; see also [44].…”