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Journal of the Physical Society of Japan

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

和子 川崎, Tatsuya Uezu

Published Oct 15, 1988• 2 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.57.3532

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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.
The phase transitions (PTs) and critical properties of the antiferromagnetic Ising model on a layered (stacked) triangular lattice have been studied by the Monte Carlo method using a replica algorithm with allowance for the next-nearest-neighbor interactions. The character of PTs is analyzed using the histogram technique and the method of Binder cumulants. It is established that the transition from the disordered to paramagnetic phase in the adopted model is a second-order PT. Static critical exponents of the heat capacity (α), susceptibility (γ), order parameter (β), and correlation radius (ν) and the Fischer exponent η are calculated using the finite-size scaling theory. It is shown that (i) the antiferromagnetic Ising model on a layered triangular lattice belongs to the XY universality class of critical behavior and (ii) allowance for the intralayer interactions of next-nearest neighbors in the adopted model leads to a change in the universality class of critical behavior.
The authors study the Ising model on the triangular lattice with nearest-neighbor couplings K(nn) , next-nearest-neighbor couplings K(nnn) >0 , and a magnetic field H . This work is done by means of finite-size scaling of numerical results of transfer matrix calculations, and Monte Carlo simulations. We determine the phase diagram and confirm the character of the critical manifolds. The emphasis of this work is on the antiferromagnetic case K(nn) <0 , but we also explore the ferromagnetic regime K(nn) >/=0 for H=0 . For K(nn) <0 and H=0 we locate a critical phase presumably covering the whole range -infinity< K(nn) <0 . For K(nn) <0 , H not equal 0 we locate a plane of phase transitions containing a line of tricritical three-state Potts transitions. In the limit H-->infinity this line leads to a tricritical model of hard hexagons with an attractive next-nearest-neighbor potential.
The phase diagrams of the ferromagnetic two-dimensional mixed spin-1 and spin-1/2 Ising model with four-spin interaction J4 and next-nearest couplings J’ are investigated by the use of the finite cluster approximation based on single-site cluster theory. The state equations are derived for the two-dimensional square lattice. The phase diagram is qualitatively and quantitatively different from that obtained when the interaction J’ is ignored. In fact, the system exhibits a variety of interesting features resulting from the introduction of the next-nearest neighbor interaction (NNN). It undergoes two kinds of behavior according to the negative and the positive value of J’. In particular, for J4/J2=-4, the system keeps the coexistence of the two ground state up to J’-dependent finite temperature.
Abstract The static critical behavior of a 2D decorated ferromagnetic Ising model on a triangular lattice has been investigated by the methods of computational physics. For nodal spins, ferromagnetic exchange interaction between the first nearest neighbors and antiferromagnetic interaction between the next nearest neighbors have been taken into account. Exchange interaction Jd between nodal and decorated spins varies within [–3.0, 3.0]. For different values of Jd, the ground state of the model has been determined, critical temperatures have been calculated, and a phase diagram has been constructed. It has been shown that the magnitude of exchange interaction with decorated spins influences the type of phase transition to the paramagnetic state.
A two-dimensional triangular Ising lattice with the nearest-neighbor antiferromagnetic interaction J 1 and the next-nearest-neighbor ferromagnetic interaction - J 2 is studied by the Monte Carlo simulation. It is shown that the system with a finite J 2 undergoes the successive phase transition, paramagnetic→Kosterlitz-Thouless (KT)-like→ferrimagnetic phases, as temperature decreases. The sub-lattice switching phenomenon observed in our previous work is shown to be an intrinsic property inherent to the present system which has an XY character. The marginal exponent of the order-parameter correlation function at the KT-like transition point is seen to increase from 1/4 as the ratio J 2 / J 1 decreases, namely, it reveals a deviation from the universal behavior of a genuine XY model.
We study the effect of antiferromagnetic interactions on the single spin-flip Glauber dynamics of two different one-dimensional Ising models with spin $\ifmmode\pm\else\textpm\fi{}1$. The first model is an Ising chain with antiferromagnetic exchange interaction limited to nearest neighbors and subject to an oscillating magnetic field. The system of master equations describing the time evolution of sublattice magnetizations can easily be solved within a linear field approximation and a long time limit. Resonant behavior of the magnetization as a function of temperature (stochastic resonance) is found, at low frequency, only when spins on opposite sublattices are uncompensated owing to different gyromagnetic factors (i.e., in the presence of a ferrimagnetic short range order). The second model is the axial next-nearest-neighbor Ising (ANNNI) chain, where an antiferromagnetic exchange between next-nearest neighbors is assumed to compete with a nearest-neighbor exchange interaction of either sign. The long time response of the model to a weak, oscillating magnetic field is investigated in the framework of a decoupling approximation for three-spin correlation functions, which is required to close the system of master equations. The calculation, within such an approximate theoretical scheme, of the dynamic critical exponent zzz, defined as $1∕\ensuremath{\tau}\ensuremath{\approx}{(1∕\ensuremath{\xi})}^{z}$ (where $\ensuremath{\tau}$ is the longest relaxation time and $\ensuremath{\xi}$ is the correlation length of the chain), suggests that the T=0T=0T=0 single spin-flip Glauber dynamics of the ANNNI chain is in a different universality class than that of the unfrustrated Ising chain.