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.
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.
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.
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.
We study an Ising model with nearest-neighbor antiferromagnetic interactions. It is placed on a triangular lattice, where each site is occupied by a spin with
x probability. There is no applied magnetic field. Randomness and frustration, two essential ingredients of spin-glasses, are present in this model. We study its critical properties here. The entropy is obtained by a transfer-matrix calculation as a function of
x at low temperature (
T=kB0.3J) for systems on a lattice of 10 \ifmmode\times\else\texttimes\fi{} 20 sites. A fairly shallow minimum appears near $x\ensuremath{\simeq}0.9$, which suggests that this case is the one most likely to show a transition into an ordered state at low temperature. We study the cases
x=1,0.9,and0.74, which is about half-way to the critical percolation. We simulate systems on lattices of 50 \ifmmode\times\else\texttimes\fi{} 50 sites and 30 \ifmmode\times\else\texttimes\fi{} 30 sites by the Monte Carlo method. The specific heat has a broad maximum at $T\ensuremath{\simeq}0.9$ for
x=0.74and0.9. ${\ensuremath{\chi}}_{\mathrm{SG}}$, defined by ${\ensuremath{\chi}}_{\mathrm{SG}}={N}^{\ensuremath{-}1}\ensuremath{\Sigma}{i,j}^{}{〈{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{j}〉}^{2}$, and the relaxation time ($\ensuremath{\tau}$) are obtained for $T\ensuremath{\ge}\frac{0.6J}{{k}_{B}}$. Both quantities, $\ensuremath{\tau}$ and ${\ensuremath{\chi}}_{\mathrm{SG}}$, turn out to be proportional to $\mathrm{exp}[\frac{A}{{(T\ensuremath{-}{T}_{0})}^{c}}]$ and $0\ensuremath{\le}{T}_{0}\ensuremath{\lesssim}0.4$; a fit with
T0=0 yields $c\ensuremath{\approx}1$ for
x=1 but $c\ensuremath{\approx}2$ for
x=0.74and0.9.
We apply the phenomenological scaling approach of Nightingale to the triangular Ising model with nearest-neighbor interactions in the presence of a magnetic field. In order to do so, we calculate the largest eigenvalues of the transfer matrix of semi-infinite strips with widths 3, 6, 9,..., etc. The technique is first applied to two exactly soluble cases, the ferromagnetic and the hard hexagon model. Employing maximum strip widths of 9 and 15, respectively, in these two cases, we obtain results for the thermal exponent
yT within 0.3% of exact values. Applied to the antiferromagnet with maximum strip width of 9, we obtain values of the exponent which vary with the applied field. This variation is only appreciable, however, near zero field where crossover to different critical behavior is anticipated. We formulate a criterion to determine the best values of the thermal exponent and obtain
yT=1.2002, in excellent agreement with the value 1.2 expected on the basis of symmetry.
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.