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

Exactly solved dynamics for an infinite-range spin system. II. Antiferromagnetic interactions

E. Milotti

Published Jan 18, 2002• 4 citations

Abstract

In a previous paper [E. Milotti, Phys. Rev. E 63, 026116 (2001)]. I have shown how to derive both thermodynamical and dynamical properties of an infinite-range Ising spin system with binary ferromagnetic interactions from the master equation for magnetization obtained from a simple spin dynamics. The same method can be adapted to different spin interactions: here I discuss the case of antiferromagnetic interactions. This model permits a study of the static properties of the antiferromagnetic lattice, and it displays very clearly the differences between the antiferromagnetic and the ferromagnetic case with long-range interactions. The dynamical behavior of the antiferromagnetic system is simpler, and the magnetization always relaxes exponentially.

DOI: 10.1103/physreve.65.027102

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In a previous paper [E. Milotti, Phys. Rev. E 63, 026116 (2001)]. I have shown how to derive both thermodynamical and dynamical properties of an infinite-range Ising spin system with binary ferromagnetic interactions from the master equation for magnetization obtained from a simple spin dynamics. The same method can be adapted to different spin interactions: here I discuss the case of antiferromagnetic interactions. This model permits a study of the static properties of the antiferromagnetic lattice, and it displays very clearly the differences between the antiferromagnetic and the ferromagnetic case with long-range interactions. The dynamical behavior of the antiferromagnetic system is simpler, and the magnetization always relaxes exponentially.

A new one-dimensional antiferromagnetic Ising model with exponential pair interactions is proposed. This model is composed of 2N Ising spins, half of which belong to an α-sublattice and the others to a β-sublattice. The interaction between spins in the same sublattice is long-rang ferromagnetic and that between spins in the different sublattices is long-rang antiferromagnetic. Thermodynamic properties of the model are shown to be the same as those obtained by using the molecular field theory for the Ising model with nearest-neighbor interactions.

The one-dimensional Ising model in an external magnetic field with uniform long-range interactions and random short-range interactions satisfying bimodal annealed distributions is studied. This generalizes the random model discussed by Paladin et al. (J. Phys. I France 4, 1994, p. 1597). Exact results are obtained for the thermodynamic functions at arbitrary temperatures, and special attention is given to the induced and spontaneous magnetization. At low temperatures the system can exist in a “ferrimagnetic” phase with magnetization 0 < σ < 1, in addition to the usual paramagnetic, ferromagnetic and antiferromagnetic phases. For a fixed distribution of the random variables the system presents up to three tricritical points for different intensities of the long-range interactions. Field-temperature diagrams can present up to four critical points. 1

The d=1 Ising ferromagnet and spin glass with long-range power-law interactions J r^-a is studied for all interaction range exponents a by a renormalization-group transformation that simultaneously projects local ferromagnetism and antiferromagnetism. In the ferromagnetic case, J>0, a finite-temperature ferromagnetic phase occurs for interaction range 0.742, the phase transition temperature discontinuously drops to zero and for a>2 there is no ordered phase above zero temperature, also as predicted by rigorous results. At the other end, on approaching a=0.74 from above, namely increasing the range of the interaction, the phase transition temperature diverges to infinity, meaning that, at all non-infinite temperatures, the system is ferromagnetically ordered. Thus, the equivalent-neighbor interactions regime is entered before (a > 0) the neighbors become equivalent, namely before the interactions become equal for all separations. The critical exponents alpha,beta, gamma,delta,eta,nu are calculated, from a large recursion matrix, varying as function of a. For antiferromagnetic J<0, all triplets of spins at all ranges have competing interactions and this highly frustrated system does not have an ordered phase. In the spin-glass system, where all couplings for all separations are randomly ferromagnetic or antiferromagnetic (with probability p), a finite-temperatures spin-glass phase is obtained in the absence of antiferromagnetic phase. In the spin-glass phase, the signature chaotic behavior under scale change occurs in a richer version than previously: In the long-range interaction of this system, the interactions at every separation become chaotic, yielding a piecewise chaotic interaction function.