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Journal of Applied Physics

Monte Carlo simulation of the spin-glass transition of a dilute Ising model on a triangular lattice

Julio F. Fernández, Carmen Z. Andérico, Thomas S. J. Streit

Published Nov 1, 1982• 6 citations
Publisher

Abstract

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.

DOI
10.1063/1.330250

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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 ($T=\frac{0.3J}{{k}_{B}}$) 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 $x=1, 0.9, \mathrm{and} 0.74$, 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 $x=0.74 \mathrm{and} 0.9$. ${\ensuremath{\chi}}_{\mathrm{SG}}$, 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 ${T}_{0}=0$ yields $c\ensuremath{\approx}1$ for $x=1$ but $c\ensuremath{\approx}2$ for $x=0.74 \mathrm{and} 0.9$.

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Journal of the Physical Society of Japan•May 1, 1983•61 citations
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 xxx 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 xxx at low temperature (T=0.3JkBT=\frac{0.3J}{{k}_{B}}T=kB​0.3J​) 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,and0.74x=1, 0.9, \mathrm{and} 0.74x=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=0.74and0.9x=0.74 \mathrm{and} 0.9x=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{T}_{0}=0T0​=0 yields $c\ensuremath{\approx}1$ for x=1x=1x=1 but $c\ensuremath{\approx}2$ for x=0.74and0.9x=0.74 \mathrm{and} 0.9x=0.74and0.9.
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.
Exact analytical and numerical results for the site-diluted antiferromagnetic Ising model on the triangular lattice (AFIT) are presented. For infinitesimal dilution the change in the free energy of the system is related to the distribution of local fields, and it is shown that for a frustrated system such as the AFIT, dilution lowers the entropy per spin. For lattices of finite size and dilution the transfer matrix for the partition function is evaluated numerically. The entropy per spin shows a marked minimum near a concentration of spins x=0.70, in some disagreement with earlier transfer-matrix results.
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.