Dynamics of position disordered Ising spins with a soft-core potential
We theoretically study magnetization relaxation of Ising spins distributed randomly in a $d$-dimension homogeneous and Gaussian profile under a soft-core two-body interaction potential $\propto1/[1+(r/R_c)^α]$ ($α\ge d$), where $r$ is the inter-spin 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 $\propto\exp(-t^2)$ and follows a stretched-exponential law asymptotically at long time with an exponent $β=d/α$. In between an oscillating behaviour is observed with a damping amplitude. For Gaussian samples, the degree of disorder in the system can be controlled by the ratio $l_ρ/R_c$ with $l_ρ$ the mean inter-spin distance and the magnetization dynamics is investigated numerically. In the limit of $l_ρ/R_c\ll1$, a coherent many-body dynamics is recovered for the total magnetization despite of the position disorder of spins. In the opposite limit of $l_ρ/R_c\gg1$, a similar dynamics as that in the homogeneous case emerges at later time after a initial fast decay of the magnetization. We obtain a stretched exponent of $β\approx0.18$ for the asymptotic evolution with $d=3, α=6$, which is different from that in the homogeneous case ($β=0.5$).