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Journal of the Physical Society of Japan

Antiferromagnetic Triangular Ising Model with Ferromagnetic Next Nearest Neighbor Interactions –Transfer Matrix Method–

Hidetsugu Kitatani, Takehiko Oguchi

Published Apr 15, 1988• 18 citations
Publisher

Abstract

An antiferromagnetic triangular Ising model with ferromagnetic next nearest neighbor interactions is studied by means of a transfer matrix method and a finite size scaling. Using the Roomany-Wyld approximation, it cannot be concluded definitely whether the system has only one critical point or not. By studying decays of the correlation functions of each sublattice, it is found that there is a finite temperature region where the correlation functions of each sublattice show a power law decay, namely, in this region every temperature is a critical temperature. The results by means of the present transfer matrix method are in good agreement with that by Monte Carlo methods, namely, the discrepancy between both results, reported recently, can be got rid of.

DOI
10.1143/jpsj.57.1344

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Journal of the Physical Society of Japan•May 1, 1983•61 citations
An antiferromagnetic triangular Ising model with ferromagnetic next nearest neighbor interactions is studied by means of a transfer matrix method and a finite size scaling. Using the Roomany-Wyld approximation, it cannot be concluded definitely whether the system has only one critical point or not. By studying decays of the correlation functions of each sublattice, it is found that there is a finite temperature region where the correlation functions of each sublattice show a power law decay, namely, in this region every temperature is a critical temperature. The results by means of the present transfer matrix method are in good agreement with that by Monte Carlo methods, namely, the discrepancy between both results, reported recently, can be got rid of.
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.
Positions of critical temperatures of antiferromagnetic Ising model with ferromagnetic next nearest neighbour interactions on the triangular lattice are investigated by making use of their properties of the Kosterlitz-Thouless type of phase transition. In particular, finite-size scaling method and Monte Carlo Renormalization Group methods (MCRG) with flow diagrams of the anomalous dimension η show the fixed line structure where the correlation length diverges. Another MCRG with flow diagram of the stiffness constant also shows a critical property clearly. The temperature dependence of the long range order is also discussed in order to locate the lower critical point.
The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.
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.
Monte Carlo simulations of the Ising model on the triangular lattice with the antiferromagnetic nn interaction J and the ferromagnetic nnn interaction J ' are carried out. Several physical quantities (magnetizations, energies, susceptibilities, specific heats of sublattices and of the total system) are calculated. A phase which resembles the partially disordered phase proposed by Mekata is observed in the finite temperature region. But it is shown in this region that the sublattice magnetizations vanish at the thermodynamic limit and the Kosterlitz-Thouless phase transition is predicted by the finite size analysis.