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Physical Review E

Critical dynamics of the kinetic Glauber-Ising model on hierarchical lattices

Xiang-Mu Kong, Zhao‐Di Yang

Published Jan 13, 2004• 4 citations
Publisher

Abstract

The critical dynamics of the kinetic Glauber-Ising model is studied on a family of the diamond-type hierarchical lattices with various branches. By carrying out the time-dependent real-space renormalization-group transformation to the master equation of the systems considered, the dynamic exponent is calculated. We find that the dynamic exponent depends on fractal dimension d(f) or the branch number m in a generator, and that it increases with the increase of d(f) or m. We notice that for the case of m=1 (one-dimensional spin chain, d(f)=1) our result z=2 is the same as the exact result obtained by Glauber, and for the case of m=2 (the simplest one in the diamond-type hierarchical lattices, d(f)=2) the exponent z=2.626 is higher than those of the two-dimensional regular lattice and the triangular lattice.

DOI
10.1103/physreve.69.016101

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The critical dynamics of the kinetic Glauber-Ising model is studied on a family of the diamond-type hierarchical lattices with various branches. By carrying out the time-dependent real-space renormalization-group transformation to the master equation of the systems considered, the dynamic exponent is calculated. We find that the dynamic exponent depends on fractal dimension d(f) or the branch number m in a generator, and that it increases with the increase of d(f) or m. We notice that for the case of m=1 (one-dimensional spin chain, d(f)=1) our result z=2 is the same as the exact result obtained by Glauber, and for the case of m=2 (the simplest one in the diamond-type hierarchical lattices, d(f)=2) the exponent z=2.626 is higher than those of the two-dimensional regular lattice and the triangular lattice.
The critical slowing down of the kinetic Gaussian model on hierarchical lattices is studied by means of a real-space time-dependent renormalization-group transformation. The dynamic critical exponent z and the exponent Delta are calculated. For hierarchical lattices with reducible generators, both the dynamic critical exponent z and the exponent Delta are independent of the fractal dimension Df of the lattice, the number of branches m, and the number of bonds per branch b of the generator--i.e., z = 2 and Delta = 1. For hierarchical lattices with irreducible generators, the exponent Delta is the same--i.e., Delta = 1; however, the dynamic critical exponent z is dependent on the concrete geometrical structure of these lattices. In addition, it was found that the lattice dependence of the correlation-length critical exponent nu is the same as that of the dynamic critical exponent z. Finally we give a brief discussion about universality for critical dynamics.
Based on the single-spin transition critical dynamics,we have investigated the critical slowing down of the Gaussian spin model situated on the fractal family of diamond-type hierarchical lattices.We calculate the dynamical critical exponent z and the correlation-length critical exponent v using the dynamical decimation renormalization-group technique.The result,together with some earlier ones,suggests us to conclude that on a wide range of geometries,zv=1 is the general relationship,while the two exponents depend on the specific structure,However,we have investigated for various lattices in an earlier paper,the system studied in this paper shows highly universal z=1/v=2 independent of the structure and the dimensionality.
Based on the single-spin transition critical dynamics, we have investigated the critical slowing down of the Gaussian spin model situated on the fractal family of diamond-type hierarchical lattices. We calculate the dynamical critical exponent z and the correlation-length critical exponent ν using the dynamical decimation renormalization-group technique. The result, together with some earlier ones, suggests us to conclude that on a wide range of geometries, zν = 1 is the general relationship, while the two exponents depend on the specific structure. However, we have investigated for various lattices in an earlier paper, the system studied in this paper shows highly universal z = 1/ν = 2 independent of the structure and the dimensionality.
The phase transitions of the anisotropic Ashkin-Teller model on a family of diamond-type hierarchical lattices is studied by means of the transfer-matrix method and the real-space renormalization-group transformation. We find that the phase diagram, for the ferromagnetic case, consists of five phases, i.e., the fully disordered paramagnetic phase P, the fully ordered ferromagnetic phase F, and three partially ordered ferromagnetic phases F(s), F(sigma), and F(s sigma), as well as ten nontrivial fixed points. The correlation length critical exponents and the crossover exponents are also calculated. In addition, we also investigate the variations of the critical exponents with the fractal dimension d(f), the number of branches m, and the number of bonds per branch b of the generator of the family of diamond-type hierarchical lattices. Finally we give a brief discussion about universality.
The form of the universal scaling law of the critical dynamic exponent, , is found on a family of regular fractals by the exact TDRG method. Here, we generate a regular fractal by an anisotropic growing process. Identifying the growing probabilities as the interactions between Ising spins on the fractals, we map the growing probability clouds as a group of the anisotropic Ising Hamiltonians. Applying the RG transformations, we find that the systems of this group of Ising Hamiltonians can be described by two universal static correlation exponents and . So, the growing processes proposed by us capture the essential features in the directed DLA simulations. The studies about their critical dynamic behaviours reveal that unlike the one-dimensional chain the critical dynamics of the kinetic Ising model on the regular fractals is universal. The further discussions show that there is a universal scaling law form of the critical dynamic exponent of the kinetic Ising model, , on the site models of the regular fractals with . Meanwhile, we discuss Daniel Kandal's correction to the formula of the, critical dynamic exponent in the TDRG method and show that our TDRG calculations are exact.