Dynamics of position-disordered Ising spins with a soft-core potential

We theoretically study the magnetization relaxation of Ising spins distributed randomly in a d-dimension homogeneous and Gaussian profile under a soft-core two-body interaction potential proportional to 1/[1 (r/R-c)(alpha)] (alpha >= d), where r is the interspin distance and R-c is the soft-core radius. The dynamics starts with all spins polarized in the transverse direction. In the homogeneous case, an analytic expression is derived at the thermodynamic limit, which starts as proportional to exp(-kt(2)) with a constant k and follows a stretched-exponential law at long time with an exponent beta = d/alpha. In between an oscillating behavior is observed with a damping amplitude. For Gaussian samples, the degree of disorder in the system can be controlled by the ratio l(rho)/R-c, with l(rho) the mean interspin distance and the magnetization dynamics is investigated numerically. In the limit of l(rho)/R-c << 1, a coherent manybody dynamics is recovered for the total magnetization despite the position disorder of spins. In the opposite limit of l(rho)/R-c >> 1, a similar dynamics as that in the homogeneous case emerges at a later time after a initial fast decay of the magnetization. We obtain a stretched exponent of beta approximate to 0.18 for the asymptotic evolution with d = 3, alpha = 6, which is different from that in the homogeneous case (beta = 0.5).