Nonuniversal dynamic critical exponents are obtained for both the Glauber and Kawasaki dynamics of the Ising chain with alternating near-neighbor interactions,
J1,
J2, with the exponents related to the ratio of the two interaction strengths. We expand upon the recent work of Droz et al. [Phys. Lett. 115A, 448 (1986)] for the ferromagnetic system (
J1>0,
J2>0). For Glauber (spin-flip) dynamics we obtain the exact exponent z=1+\ensuremath{\rho} (\ensuremath{\rho}\ensuremath{\equiv}\ensuremath{\Vert}
J1/
J2\ensuremath{\Vert}) which generalizes the result of Droz et al. and is valid irrespective of the signs of
J1 and
J2, where it is assumed that \ensuremath{\Vert}
J1\ensuremath{\Vert}>\ensuremath{\Vert}
J2\ensuremath{\Vert} (\ensuremath{\rho}\ensuremath{\ge}1). For Kawasaki (spin-exchange) dynamics, we obtain dynamic critical exponents from conventional theory which provides a rigorous lower bound for the exponent z. For the case of the conserved (ferromagnetic) order parameter, however, we present arguments that the conventional exponent is exact. We obtain z=4+\ensuremath{\rho} in this case. For
J1>0,${J}_{2}$0 we derive the conventional exponent z=1+\ensuremath{\rho}, whereas for ${J}_{1}$0 we find z=2 irrespective of
J2. A key aspect of this system is the narrowing of the dynamic critical region as compared with the isotropic system (
J1=
J2). The extra bond periodicity splits the isotropic order parameter into components such that only comparatively closer to criticality does the order parameter become the dominant slow mode. The nonuniversal critical dynamics is intrinsically linked to the nonuniform bond distribution, being shown to arise from a kinetic coefficient which vanishes with a nonuniversal critical exponent. Nonuniversality of the dynamic exponent for Ising systems with inhomogeneous couplings is argued to be specific to zero-temperature critical points.
The one-dimensional kinetic Ising models with order-parameter conservation are studied for different transition rates. The effect on critical dynamics of global and local conservation laws is considered. Rigorous lower bounds for the dynamical exponent (z) are obtained from the initial response rate and the scaling hypothesis for the relaxation time. It is shown for several dynamical models that z\ensuremath{\ge}5 even if the order-parameter conservation occurs only globally. For models in which the maximal separation between two exchanged spins is proportional to the equilibrium correlation length, the inequality z\ensuremath{\ge}3 is obtained. The d-dimensional case is discussed briefly.
One-dimensional kinetic Ising models with long-range exchange of two spins with opposite signs are considered. The exchange probability is a power-law function of the distance between exchanged spins. Rigorous lower bounds for the dynamical exponent z are obtained from the initial response and the scaling hypothesis of the relaxation time. The dependence of the z exponent on the maximal exchange distance and on the exchange probability is studied. Nonuniversal behavior in the ferromagnetic model with two different alternating coupling constants is found. The z exponent is also obtained from the domain-wall argument.
Recent results concerning non-universality in the dynamical critical behavior of the one-dimensional Ising model, are generalised to several formulations of the Potts model, using arguments based on domain wall motion. By applying real-space renormalisation group methods, the authors also study the dynamical Ising model, with single flips, both for a period distribution of nearest-neighbor exchanges and for the random chain (with nonzero couplings). They show that the critical dynamical exponent is z=1+J M /J m where J M (J m ) is the greatest (smallest) of the exchanges.
The one-dimensional kinetic Potts models are studied for both Glauber and Kawasaki dynamics with different transition rates. Rigorous lower bounds for the dynamical exponent (z) are obtained from the initial response rate and the scaling hypothesis for the relaxation time. It is shown that the z exponent is non-universal for several periodic models with a basic unit cell (J 1 ,J 2 ,. . .,JM) containing different coupling constants. The critical dynamics of the alternating bond model is also studied by a simple argument about the movement of domain walls.
The non-equilibrium critical dynamics of the Baxter model is studied through Monte Carlo simulations. The single-spin-flip stochastic dynamics is introduced on the basis of the formulation of the model as an Ising-type model on the square lattice with next-nearest-neighbor and four-spin interactions. At several points on the critical line, the dynamic critical exponent z and the critical exponent \bar λ c , which characterizes the non-equilibrium critical relaxation, are estimated for the two order parameters of the model, the magnetization and the polarization. For both order parameters, the estimated \bar λ c varies systematically along the critical line. On the other hand, the estimated z is almost constant. This suggests that while the exponent z is universal, the exponent \bar λ c is not.