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Modern Physics Letters B

COHERENT-ANOMALY APPROACH TO BLUME-EMERY-GRIFFITHS MODEL

M. Kolesík

Published Jan 20, 1994• 5 citations
Publisher

Abstract

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.

DOI
10.1142/s021798499400011x

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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.
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.
A new scheme of the coherent-anomaly method (CAM) is proposed to study critical phenomena in the models for which a mean-field description gives spurious first-order phase transition. A canonical series of mean-field-type approximations are constructed so that the spurious discontinuity should vanish asymptotically as the approximate critical temperature approachs the true value. The true value of the critical exponents β and γ are related to the coherent-anomaly exponents defined among the classical approximations. The formulation is demonstrated in the two-dimensional q -state Potts models for q =3 and 4. The result shows that the present method enables us to estimate the critical exponents with high accuracy by using the date of the cluster-mean-field approximations.
We investigate the thermodynamic quantities of Blume-Emery-Griffiths model with long-range interactions and observe negative specific heat and susceptibility near the first order line, where the temperature, specific heat and susceptibility are discontinuity. When order-disorder phase transition takes place, the critical exponent of specific heat (α=α′=0) and the critical exponent of susceptibility (γ=γ′=1) are consistent with the prediction of mean field theory. However, when coupling constant corresponds to the value of canonical tri-critical point, the critical exponent of specific heat and susceptibility take the values of 0=α≠α′=1 and 1=γ≠γ′=2. The critical exponents violate the scaling homogeneity equalities and α′+β+γ′>2 breaks down the Widom scaling law.
A general procedure to study critical phenomena of magnetic systems is discussed. It consists of systematic series of Landau-like approximations (extended variational method) and the coherent-anomaly method (CAM). As for susceptibility, the present method is equivalent to the power-series CAM theory. On the other hand, the EVM gives a set of new approximants for other physical quantities. Applications to d-dimensional Ising ferromagnets are also described. The critical points and exponents are estimated with high accuracy.
Introduction Among pseudospin systems studied theoretically, a great attention is paid to Ising-type models, that is, to systems with the Hamiltonian containing only z-component of pseudospin. That is so due to relative simplicity with which approximate calculations for these models can be carried out and tested and possibility of their application to a wide class of real objects. Thus, the Blume-Emery-Griffiths model (BEG) corresponding to the Isingtype model with j = 2 (hereafter S z = \\Gammaj; \\Gammaj +2; : : : ; j \\Gamma 2; j) was proposed for the investigation of phase transitions (PT) in He 3 \\Gamma He 4 mixture [1]. A general form of the Ising-type model with j = 2 contains extra terms, as compared to the BEG model, like S z i (S z j<
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.