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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Correlation transitions in the Ising chain with competing short-range and long-range mirror interactions

István Daruka, Zsolt Gulácsi

Published Nov 1, 1998• 7 citations
Publisher

Abstract

We present exact four-spin- and string-correlation functions derived for an Ising chain with competing geometrical nearest-neighbor short-range and mirror-image-type long-range interactions. Unusual $T\ensuremath{\ne}0$ correlation effects were found, indicating qualitative changes in the behavior of the system in different T domains. These domains are not separated by a sharp, traditional phase transition, but are delimited by a temperature ${T}_{i}$ at which the exact connected four-spin-correlation function vanishes. At the same ${T}_{i}$ the pair-correlation function changes its character: The functional form of the correlation length and the nature of the long-range decay are modified.

DOI
10.1103/physreve.58.5403

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We present exact four-spin- and string-correlation functions derived for an Ising chain with competing geometrical nearest-neighbor short-range and mirror-image-type long-range interactions. Unusual T\ensuremath≠0T\ensuremath{\ne}0T\ensuremath=0 correlation effects were found, indicating qualitative changes in the behavior of the system in different T domains. These domains are not separated by a sharp, traditional phase transition, but are delimited by a temperature Ti{T}_{i}Ti​ at which the exact connected four-spin-correlation function vanishes. At the same Ti{T}_{i}Ti​ the pair-correlation function changes its character: The functional form of the correlation length and the nature of the long-range decay are modified.
The exact partition function, free energy, pair-correlation function, and connected quantities like entropy, specific heat, and magnetic susceptibility were deduced for an Ising chain in which each spin interacts with its mirror-image spin and the neighbors of the image. A qualitatively different behavior was found for the pair-correlation function in low- and high-temperature limits, which was interpreted in terms of bound spin pairs emerging in the low-T regime. The physical implications of the model are analyzed, and applications in condensed matter and biology discussed. \textcopyright{} 1996 The American Physical Society.
Pair correlations for the Ising model on a triple-chain with a crossed second-neighbor interaction are obtained analytically and their asymptotic decay for the large spin separation is investigated. The change in the nature of the long-distance asymptotic decay is observed in small regions of the parameter space, where the interactions are competing. This change is caused by the interchange of the second largest eigenvalue of the transfer matrix in absolute value. The results are compared with the double-chain case to study the effect of the frustration in the triangle made of three nearest-neighbor spins on different chains when the interchain nearest-neighbor interaction is antiferromagnetic.
We examine correlation functions in the presence of competing long and short ranged interactions to find multiple correlation and modulation lengths. We calculate the ground state stripe width of an Ising ferromagnet, frustrated by an arbitrary long range interaction. In large nnn systems, we demonstrate that for a short range system frustrated by a general competing long range interaction, the crossover temperature T∗T^*T∗ veers towards the critical temperature of the unfrustrated short range system (i.e., that in which the frustrating long range interaction is removed). We also show that apart from certain special crossover points, the total number of correlation and modulation lengths remains conserved. We derive an expression for the change in modulation length with temperature for a general system near the ground state with a ferromagnetic interaction and an opposing long range interaction. We illustrate that the correlation functions associated with the exact dipolar interactions differ substantially from those in which a scalar product form between the dipoles is assumed.
Abstract In this paper we study an Ising spin chain with short-range competing interactions in the presence of long-range ferromagnetic interactions in the canonical ensemble. The simultaneous presence of the frustration induced by the short-range couplings together with their competition with the long-range term gives rise to a rich thermodynamic-phase diagram. We compare our results with the limit in which one of two local interactions is turned off, which was previously studied in the literature. Eight regions of parameters with qualitatively distinct properties are featured, with different first- and second-order phase transition lines and critical points.
Recently, it has been found that an effective long-range interaction is realized among local bistable variables (spins) in systems where the elastic interaction causes ordering of the spins. In such systems, we generally expect both long-range and short-range interactions to exist. In the short-range Ising model, the correlation length diverges at the critical point. In contrast, in the long-range interacting model the spin configuration is always uniform and the correlation length is zero. As long as a system has nonzero long-range interactions, it shows criticality in the mean-field universality class, and the spin configuration is uniform beyond a certain scale. Here we study the crossover from the pure short-range interacting model to the long-range interacting model. We investigate the infinite-range model (Husimi-Temperley model) as a prototype of this competition, and we study how the critical temperature changes as a function of the strength of the long-range interaction. This model can also be interpreted as an approximation for the Ising model on a small-world network. We derive a formula for the critical temperature as a function of the strength of the long-range interaction. We also propose a finite-size scaling form for the spin correlation length at the critical point, which is finite as long as the long-range interaction is included, though it diverges in the limit of the pure short-range model. These properties are confirmed by extensive Monte Carlo simulations.