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Physical review. B, Condensed matter

Hamiltonian studies of the Blume-Emery-Griffiths model

Francisco Castilho Alcaraz, J. R. Drugowich de Felício, R. Köberle, Jürgen F. Stilck

Published Dec 1, 1985• 48 citations
Publisher

Abstract

Finite-size scaling methods are used to obtain the phase diagram of the Blume-Emery-Griffiths model in its time-continuous Hamiltonian version. In particular, we locate the tricritical point, where a first-order transition changes to one of second order, and evaluate its exponents. The exponents are in complete agreement with Nienhuis' conjecture. We also discuss a recent conjecture concerning the universality of the ratio of mass-gap amplitudes. Our results suggest the validity of this conjecture even at the tricritical point.

DOI
10.1103/physrevb.32.7469

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Physical review. B, Condensed matter•Jan 1, 1985•34 citations
Finite-size scaling methods are used to obtain the phase diagram of the Blume-Emery-Griffiths model in its time-continuous Hamiltonian version. In particular, we locate the tricritical point, where a first-order transition changes to one of second order, and evaluate its exponents. The exponents are in complete agreement with Nienhuis' conjecture. We also discuss a recent conjecture concerning the universality of the ratio of mass-gap amplitudes. Our results suggest the validity of this conjecture even at the tricritical point.
The phase diagram of the two-dimensional Blume-Capel model is investigated by using the technique of phenomenological finite-size scaling. The location of the tricritical point and the values of the critical and tricritical exponents are determined. The location of the tricritical point (Tt{T}_{t}Tt​=0.610\ifmmode\pm\else\textpm\fi{}0.005, Dt{D}_{t}Dt​=1.9655\ifmmode\pm\else\textpm\fi{}0.0010) is well outside the error bars for the value quoted in previous Monte Carlo simulations but in excellent agreement with more recent Monte Carlo renormalization-group results. The values of the critical and tricritical exponents, with the exception of the leading thermal tricritical exponent, are in excellent agreement with previous calculations, conjectured values, and Monte Carlo renormalization-group studies.
We investigate the tricritical scaling behavior of the two-dimensional spin-1 Blume-Capel model by using the Wang-Landau method of measuring the joint density of states for lattice sizes up to 48×48 sites. We find that the specific heat deep in the first-order area of the phase diagram exhibits a double-peak structure of the Schottky-like anomaly appearing with the transition peak. The first-order transition curve is systematically determined by employing the method of field mixing in conjunction with finite-size scaling, showing a significant deviation from the previous data points. At the tricritical point, we characterize the tricritical exponents through finite-size-scaling analysis including the phenomenological finite-size scaling with thermodynamic variables. Our estimation of the tricritical eigenvalue exponents, yt=1.804(5), yg=0.80(1), and yh=1.925(3), provides the first Wang-Landau verification of the conjectured exact values, demonstrating the effectiveness of the density-of-states-based approach in finite-size scaling study of multicritical phenomena.
The Blume-Emery-Griffiths model is investigated using the coherent-anomaly method applied to the series of approximations which are based on variational series expansion. Universality of the model is checked directly by calculating the critical exponent γ along second-order-phase-transition lines. It is shown that the coherent-anomaly approach works also in the immediate neighborhood of tricritical points, and that it can differentiate between different universality classes (Potts subspace in the BEG model) without exhibiting a disturbing crossover effect.
We present extensive Monte Carlo spin-dynamics simulations of the vector Blume-Emery-Griffiths model with three-dimensional spins on a simple cubic lattice and periodic boundary conditions. For a range of Hamiltonian parameters, this model reproduces the phase diagram topology of the one experimentally observed for bulk mixtures of 3He-4He . Highly efficient decomposition time integration algorithms have been used to obtain the evolution of the equations of motion from which we have computed the dynamic structure factor S(q,omega) close to critical and tricritical points. The dispersion relation and the corresponding dynamic exponents have been obtained in the critical and multicritical transitions. We find that the dynamic critical exponents for in-plane and out-of-plane fluctuations are the same at the tricritical point but are in fact different along the critical line.
A model for the critical relaxation in the one-dimensional Ising-type S=1 spin system is presented. This model is equivalent to the Blume-Emery-Griffiths model and exhibits two simple critical points and one tricritical point. The kinetic behavior is studied using the real-space renormalization-group approach. In the two critical points we find that the critical slowing down is described by the dynamic exponent z, z=2. In each point this exponent belongs to the critical order parameter, while the second order parameter relaxes faster, with z=1 or 0. At the critical point the two order parameters relax with the same z, z=1.