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Physical review. B, Condensed matter

Phase transitions in the one-dimensional Ising model with ferromagnetic and antiferromagnetic interactions

Dong Huang, Barbara J. Gabryś

Published Apr 1, 1990• 0 citations
Publisher

Abstract

A special case of the one-dimensional Ising model with short-range antiferromagnetic and long-range ferromagnetic interactions is presented. Conditions of the occurrence of phase transitions as well as the behavior of magnetization and free energy are discussed.
DOI
10.1103/physrevb.41.7103

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We study an Ising model in one dimension with short-range ferromagnetic and long-range (power law) antiferromagnetic interactions. We show that the zero temperature phase diagram in a (longitudinal) field HHH involves a sequence of up and down domains whose size varies continuously with HHH, between $\ensuremath{-}{H}_{c}$ and Hc{H}_{c}Hc​, which represent the edges of the ferromagnetic up and down phases. The implications of long-range interaction in many body systems are discussed.
Using the integral equation method, we have got the analytic expressions of the internal energy and specific heat for the one-dimensional Ising model with nearest-neighbour antiferromagnetic repulsion Js and infinitely long-range exponential (Kae-Helfand-type) ferromagnetic attraction J1 acting only on even-numbered neighbours. Under the conditions of J1=1 and Js being 0.05, 0.1 and 0.3 respectively, we have drawn the curves of the internal energy and specific heat varying with temperature in zero external magnetic field, and pointed out its critical phase transition point. Under the conditions of J1=1,Js=0.1, we have drawn the curves of influence of the external magnetic field upon the internal energy and specific heat. Because there exist double solutions for the system under low temperatures, zero or weak external magnetic field and weaker antiferromagnetic interactions, it is inferred that there should exist another quasi-stable state of the model under the given conditions.
The Ising linear chain with nearest-neighbor interactions of strength J s and infinite long-range ferromagnetic exponential interactions of strength J l was solved exactly in the presence of the external field. The spontaneous magnetization and the susceptibility are obtained. The transition temperature depends on J s and J l . For J s >0, the model shows the second order phase transion. The specific heat shows both a broad peak and a sharp peak. The value of sharp peak of the specific heat depends on J s and J l . For J s <0, the properties of the transition agree with those of Nagle model with long-range equivalent-neighbor ferromagnetic interactions. The exact free energy and the long-range order of our model are identical with those of the variational theory of a quasi-one-dimensional model.