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Journal of the Physical Society of Japan

The 2×1 and 2×2 Structures on a Triangular Lattice

Yukio Saitō, Kazushi Furuta, Masato Hojou

Published Jan 15, 1987• 7 citations
Publisher

Abstract

Phase transitions of the triangular Ising model in a field with nearest and second nearest neighbor antiferromagnetic interactions with couplings, - J and - J /2, respectively, are studied by the interfacial free energy method, by the mean field approximation, by the transfer matrix and phenomenological renormalization methods and by the Monte Carlo simulation. At low temperatures and at fields weaker than the lower critical field h c1 the system takes the 2 × 1 structure, and at fields between h c1 and the upper critical field h c2 it takes the 2×2 structure. Between the 2×1 and 2×2 phases near the field h c1 , a partial disordered phase, in which a part of the sublattices are in disorder, is found. Specific heat shows no divergence, when the partial disordered phase changes to other phases.

DOI
10.1143/jpsj.56.178

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Phase transitions of the triangular Ising model in a field with nearest and second nearest neighbor antiferromagnetic interactions with couplings, - J and - J /2, respectively, are studied by the interfacial free energy method, by the mean field approximation, by the transfer matrix and phenomenological renormalization methods and by the Monte Carlo simulation. At low temperatures and at fields weaker than the lower critical field h c1 the system takes the 2 × 1 structure, and at fields between h c1 and the upper critical field h c2 it takes the 2×2 structure. Between the 2×1 and 2×2 phases near the field h c1 , a partial disordered phase, in which a part of the sublattices are in disorder, is found. Specific heat shows no divergence, when the partial disordered phase changes to other phases.
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.
An antiferromagnetic Ising system on a triangular lattice is studied by means of a transfer matrix and the phenomenological scaling method. System consists of a semi-infinite strip and helical boundary conditions are imposed at the end of each row. At external magnetic fields h =0 and 6 J , where J is a nearest neighbor coupling constant, a unite residual entropy is obtained at the zero temperature. With a small field, specific heat is found to have two peaks; a high-temperature peak corresponding to the enhanced short-range order, and a low-temperature peak to the appearance of a long-range order. A phase diagram in h - T space is obtained by means of the phenomenological scaling, and agrees fairly well with the one obtained by Monte Carlo simulation. The main deficiency is the finite transition temperature at h =0, which may be due to a crossover to the XY-like phase transition.
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 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 Ising model with nearest and next nearest neighbor antiferromagnetic interactions on the triangular lattice displays, for Jnnn/Jnn = 0.1, three phase transitions in different universality classes as the magnetic field is increased. We have studied this model using Monte Carlo and renormalization group techniques. The transition from the paramagnetic to the 2 × 1 phase (universality class of the Heisenberg model with cubic anisotropy) is found to be first order; the transition from the paramagnetic phase to the [Formula: see text] phase (universality class of the three state Potts model) is continuous; and the transition from the paramagnetic to the 2 × 2 phase (universality class of the four state Potts model) is found to change from first order to continuous as the field is increased. We have mapped out the phase diagram and determined the critical exponents for the continuous transitions. A novel technique, using a Landau-like free energy functional determined from Monte Carlo calculations, to distinguish between first order and continuous transitions, is described.
We study the critical properties of an Ising model with nearest neighbor antiferromagnetic interactions, placed on a triangular lattice, where each site is occupied by a spin, only with x probability. Randomness and frustration, two essential ingredients of spin-glasses, are present in this model. The entropy is obtained, by a transfer matrix calculation, as a function of x at low temperature (T=0.3 J/kB), for systems on a lattice of 10×19 sites. A fairly shallow minimum in the entropy appears near x=0.9, which suggests that this case is the most likely to show a transition into an ordered state at low temperature. We therefore simulate systems on lattices of 50×50 sites for x=0.9 and x=1 by the Monte Carlo method. The specific heat has a broad maximum at T=0.9 J/kB (1.2) for x=0.9 (1). The relaxation time (τ) obtained for T≥0.6 J/kB, turns out to be proportional to exp[A/(T−T0)c], and 0<T0≲0.4 J/kB; a fit with T0=0 yields c≂1 for x=1 but c≂2 for x=0.9.