Nonuniversal critical dynamics of the alternating-bond Ising chain: Relaxational and diffusive kinetics

Nonuniversal dynamic critical exponents are obtained for both the Glauber and Kawasaki dynamics of the Ising chain with alternating near-neighbor interactions, ${J}_{1}$, ${J}_{2}$, 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 (${J}_{1}$>0,${J}_{2}$>0). For Glauber (spin-flip) dynamics we obtain the exact exponent z=1+\ensuremath{\rho} (\ensuremath{\rho}\ensuremath{\equiv}\ensuremath{\Vert}${J}_{1}$/${J}_{2}$\ensuremath{\Vert}) which generalizes the result of Droz et al. and is valid irrespective of the signs of ${J}_{1}$ and ${J}_{2}$, where it is assumed that \ensuremath{\Vert}${J}_{1}$\ensuremath{\Vert}>\ensuremath{\Vert}${J}_{2}$\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 ${J}_{1}$>0,${J}_{2}$0 we derive the conventional exponent z=1+\ensuremath{\rho}, whereas for ${J}_{1}$0 we find z=2 irrespective of ${J}_{2}$. A key aspect of this system is the narrowing of the dynamic critical region as compared with the isotropic system (${J}_{1}$=${J}_{2}$). 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.