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Progress of Theoretical Physics

Dynamics of the Ising Model near the Transition Point. II

Ryuzo Abe, Akira Hatano

Published Apr 1, 1969• 29 citations
Publisher

Abstract

The Isin spin system is discussed in which the interaction of spins with a heat bath causes the spontaneous flip of spins. A treatment discussed in a previous paper for the square lattice is extended to the honeycomb and triangular lattices. In the case of honeycomb lattice, an additional term to the one studied in the previous paper is calculated explicitly at the transition point and an exact expansion is derived for the dynamic susceptibililty χ(ω) up to the term of the order of 1/ω2. Some discussions are given of the low frequency behavior of χ(ω) on the basis of variational principle.

DOI
10.1143/ptp.41.941

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Statics and Dynamics of Antiferromagnetic Ising System on a Triangular Lattice. II

和子 川崎, Tatsuya Uezu

The Ising model on a triangular lattice with antiferromagnetic nearest neighbor and ferromagnetic next nearest neighbor interactions is investigated by treating the master equation. To terminate hierarchy structure of the equation of motion, the cluster decoupling approximation (CDA) is applied in a similar way as in the previous paper. But here, we retain not only single cumulants but also second ones. The fluctuation effect in the occurrence of the successive phase transition and “turn round” phenomena in a relaxation process are discussed. The formula of the frequency dependent susceptibility is derived within the present approximation, and the response of the system to the external oscillating field is studied. Finally, the fluctuation-dissipation theorem is derived for this spin system.

Journal of the Physical Society of Japan•Oct 15, 1988•2 citations
The Isin spin system is discussed in which the interaction of spins with a heat bath causes the spontaneous flip of spins. A treatment discussed in a previous paper for the square lattice is extended to the honeycomb and triangular lattices. In the case of honeycomb lattice, an additional term to the one studied in the previous paper is calculated explicitly at the transition point and an exact expansion is derived for the dynamic susceptibililty χ(ω) up to the term of the order of 1/ω2. Some discussions are given of the low frequency behavior of χ(ω) on the basis of variational principle.
The Ising spin system is discussed in which the interaction of spins with heat bath causes the spontaneous flip of spins. It is assumed that the system is in thermal equilibrium up to some instant under a uniform and sufficiently small magnetic field and the magnetic field is switched off at that instant. The initial decay of magnetization in such a situation is studied. In the neighborhood of transition point, it is proved that a relaxation time associated with the initial decay is proportional to the susceptibility, and an exact proportionality constant is calculated for the two-dimensional square lattice with nearest-neighbor interaction.
The method of developing exact power-series expansions for the partition function ZN and related thermodynamic functions for the Ising model valid below the critical point is generalized to include exchange interactions between first-, second-, and third-neighbor pairs. Expansions of the spontaneous magnetization M0(T) and zero field susceptibility χ0(T) are derived through to sixth order of perturbation for the s.q. lattice, and through to fifth order of perturbation for the Δ′r, b.c.c., s.c., and f.c.c. lattices, when interactions J1SizSjz and J2SkzSlz are present between first- and second-neighbor spins, respectively (second-neighbor model). These expansions have also been obtained for the case where interactions of equal magnitude (J1 = J2 = J3) are present between first-, second-, and third-neighbor pairs (third-equivalent-neighbor model); here expansions through to fifth order of perturbation are obtained for the s.q., Δ′r, b.c.c., and s.c. lattices and through to fourth order for the f.c.c. lattice. The Padé approximant·procedure is employed to discuss the effects of an extended but finite range of interaction on the behavior of M0(T) and χ0(T) for T→Tc− as characterized by the critical exponents β and γ′, respectively. For the second-equivalent-neighbor model lattices, it is found that 0.122 ≤ β ≤ 0.134 in two dimensions, and that 0.308 ≤ β ≤ 0.328 in three dimensions; from which it is concluded that β remains unchanged from its value in the nearest-neighbor model. The corresponding limits for γ′ in three dimensions are 1.18 ≤ γ′ ≤ 1.28; from this and the results for the b.c.c. lattice in particular, it is concluded that γ′ is probably 54 and hence the transition in χ0 is symmetrical about Tc (γ′ = γ). A repetition of this analysis for the third-equivalent-neighbor model three-dimensional lattices shows a marked shift in the estimated range of β and γ′; the results are 0.345 ≤ β ≤ 0.365, and 1.01 ≤ γ′ ≤ 1.14. In each of the above cases, the corresponding high-temperature (T > Tc) expansions of χ0(T) obtained previously have been analyzed to yield estimates of the critical exponent γ. The over-all results and in particular the estimates of γ for the s.q. and b.c.c. lattices suggest that this index is unaffected by extending the range of interaction, and that if γ is a rational fraction then it is the same fraction for the n.n. model and second- and third-equivalent neighbor models. Finally the high-temperature expansions of ZN in zero field, and of χ0(T) for the second-neighbor model are used to examine the dependence of the critical temperature Tc, the critical energy (E∞ − Ec)/kTc, and the critical entropy (S∞ − Sc)/k on the relative strengths of J1 and J2 for values of J2/J1 in the range 0 to 1. It is found that the variation of the critical point is well represented by Tc(α)=Tc(0)[1+m1α],where α = J2/J1 and lies in the range 0 ≤ α ≤ 1; and Tc(0) is the critical temperature of the nearest-neighbor model. The values of m1 are 0.61, 2.47, 0.84, 1.45, and 1.35 for the f.c.c., s.c., b.c.c., s.q., and Δ′r lattices, respectively. All these calculations are compared with the corresponding results for the Heisenberg model.
Earlier studies of the simple Ising model have been extended and the techniques applied to a variety of multistate and multispin models. Briefly presented is the authors' research results in which they have developed methods for the derivation of series developments for the Potts model, for which one of the most interesting questions is the order of the transition, for Ising models with four-spin interactions (including the eight-vertex model) and the three-spin triangular Ising model. For the three-spin triangular Ising model it has been found possible to conjecture the exact form of the spontaneous magnetisation. They have also continued to study the general cluster expansion for the high temperature specific heat of the Ising model of a ferromagnet. High field expansions have been derived for the Ising model on the hydrogen peroxide lattice. The exploration of the size and shape of polymer chains using insight gained from critical phenomena has continued.
A method to calculate the dynamical as well as the static properties of the Ising ferromagnet, developed in the previous paper for the two-dimensional square lattice, is extended to cubic lattices. The temperature dependence of the frequency-dependent magnetic susceptibility, approach to equilibrium of the magnetization of the system, the fluctuation-dissipation theorem, and some static properties are calculated. It is proved that all these properties are essentially the same for cubic lattices and for two-dimensional model. It is further demonstrated by explicit calculation that the non-linear effect is very important for the critical slowing-down near the Curie point, especially in the ferromagnetic region.
The Ising model on a triangular lattice with antiferromagnetic nearest neighbor and ferromagnetic next nearest neighbor interactions is investigated by treating the master equation. To terminate hierarchy structure of the equation of motion, the cluster decoupling approximation (CDA) is applied in a similar way as in the previous paper. But here, we retain not only single cumulants but also second ones. The fluctuation effect in the occurrence of the successive phase transition and “turn round” phenomena in a relaxation process are discussed. The formula of the frequency dependent susceptibility is derived within the present approximation, and the response of the system to the external oscillating field is studied. Finally, the fluctuation-dissipation theorem is derived for this spin system.