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Physical review. B, Condensed matter

Comment on ‘‘Critical relaxation of the one-dimensional Blume-Emery-Griffiths model’’

P. O. Weir, J. Michael Kosterlitz

Published Jan 1, 1986• 4 citations
Publisher

Abstract

In a recent paper Achiam [Phys. Rev. B 31, 260 (1985)] analyzes the dynamics of the one-dimensional Blume-Emery-Griffiths model with use of real-space renormalization-group (RG) techniques. We find that his results for the dynamical critical exponent are incorrect at the tricritical fixed point of the RG procedure.
DOI
10.1103/physrevb.33.622

Related papers

Reply to ‘‘Comment on ‘Critical relaxation of the one-dimensional Blume-Emery-Griffiths model’ ’’

Y Achiam

A previous value of the dynamic exponent z, which characterizes the critical relaxation of the one-dimensional Blume-Emery-Griffiths model, was found by Weir and Kosterlitz to be incorrect. The correct value, z=2, is found with use of a renormalization-group transformation in a different parameter space. In this parameter space, which is coupled to the three components of the S=1 spin, the master equation is invariant under an exact renormalization-group transformation.

Physical review. B, Condensed matter•Jan 1, 1986•2 citations

Critical relaxation of the one-dimensional Blume-Emery-Griffiths model

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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.

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Certain difficulties associated with using a parameter-dependent real-space renormalization group are illustrated by a study of the one-dimensional Blume-Emery-Griffiths (BEG) model. In particular, it is noted that renormalization trajectories which are determined by optimizing the parameter-dependent free energy do not necessarily preserve the symmetries of the initial Hamiltonian and therefore can lead to incorrect results. This difficulty arises in a variational treatment of the BEG model but can be resolved by using a transformation which is the product of two successive variational renormalization transformations.

Physical review. B, Condensed matter•Jul 1, 1978•5 citations
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.
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 prefaced renormalization-group study indicates that all three features of the Blume-Emery-Griffiths model with repulsive biquadratic interaction, namely (1) a critical-end-point structure occurring inside the ferromagnetic phase, (2) disordered-ferromagnetic-disordered reentrance, and (3) a ferrimagnetic phase sandwiched between the ferromagnetic and antiquadrupolar phases, survive fluctuations in three dimensions but not in two dimensions. The renormalization-group mechanism for a critical-end-point structure inside the ordered phase is shown to be the same as that originally found for a critical-end-point structure inside the disordered phase, namely a distinct hybrid fixed point, contrary to a previous claim.
Certain difficulties associated with using a parameter-dependent real-space renormalization group are illustrated by a study of the one-dimensional Blume-Emery-Griffiths (BEG) model. In particular, it is noted that renormalization trajectories which are determined by optimizing the parameter-dependent free energy do not necessarily preserve the symmetries of the initial Hamiltonian and therefore can lead to incorrect results. This difficulty arises in a variational treatment of the BEG model but can be resolved by using a transformation which is the product of two successive variational renormalization transformations.