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Physical review. B, Condensed matter

Finite-size scaling analysis of theS=1Ising model on the triangular lattice

Joseph B. Collins, Per Arne Rikvold, E. T. Gawlinski

Published Oct 1, 1988• 59 citations
Publisher

Abstract

We study the S=1 Ising model, equivalent to the three-state lattice-gas model, with nearest-neighbor, pairwise interactions on a two-dimensional, triangular lattice. We pay particular attention to the antiferromagnetic phase diagrams. We show its relation to other well-studied models (S=(1/2 Ising, Blume-Capel, Blume-Emery-Griffiths), classify the ground states, and calculate finite-temperature phase diagrams using transfer matrices and finite-size scaling for infinite strips of three and six sites width. The phase diagrams are quite complicated, with surfaces of first- and second-order transitions that intersect along lines of multicritical points of various kinds, providing a rich laboratory for studying a number of first-order phase transitions, critical and multicritical phenomena within the framework of one single model.

DOI
10.1103/physrevb.38.6741

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Physical review. B, Solid state•Dec 1, 1976•371 citations
We study the S=1 Ising model, equivalent to the three-state lattice-gas model, with nearest-neighbor, pairwise interactions on a two-dimensional, triangular lattice. We pay particular attention to the antiferromagnetic phase diagrams. We show its relation to other well-studied models (S=(1/2 Ising, Blume-Capel, Blume-Emery-Griffiths), classify the ground states, and calculate finite-temperature phase diagrams using transfer matrices and finite-size scaling for infinite strips of three and six sites width. The phase diagrams are quite complicated, with surfaces of first- and second-order transitions that intersect along lines of multicritical points of various kinds, providing a rich laboratory for studying a number of first-order phase transitions, critical and multicritical phenomena within the framework of one single model.
We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.
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.
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.
We study phase transitions of the Potts model on the centered-triangular lattice with two types of couplings, namely, K between neighboring triangular sites, and J between the centered and the triangular sites. Results are obtained by means of a finite-size analysis based on numerical transfer-matrix calculations and Monte Carlo simulations. Our investigation covers the whole (K,J) phase diagram, but we find that most of the interesting physics applies to the antiferromagnetic case K<0, where the model is geometrically frustrated. In particular, we find that there are, for all finite J, two transitions when K is varied. Their critical properties are explored. In the limits J→±∞ we find algebraic phases with infinite-order transitions to the ferromagnetic phase.
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (JJJ) and biquadratic (KKK) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in JJJ, KKK, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (HHH) and another odd interaction (LLL) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.