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

Abstract

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 $\ensuremath{\propto}1/[1+{(r/{R}_{c})}^{\ensuremath{\alpha}}]$ $(\ensuremath{\alpha}\ensuremath{\ge}d)$, 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 $\ensuremath{\propto}exp(\ensuremath{-}k{t}^{2})$ with a constant $k$ and follows a stretched-exponential law at long time with an exponent $\ensuremath{\beta}=d/\ensuremath{\alpha}$. 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 ${l}_{\ensuremath{\rho}}/{R}_{c}$, with ${l}_{\ensuremath{\rho}}$ the mean interspin distance and the magnetization dynamics is investigated numerically. In the limit of ${l}_{\ensuremath{\rho}}/{R}_{c}\ensuremath{\ll}1$, a coherent many-body dynamics is recovered for the total magnetization despite the position disorder of spins. In the opposite limit of ${l}_{\ensuremath{\rho}}/{R}_{c}\ensuremath{\gg}1$, 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 $\ensuremath{\beta}\ensuremath{\approx}0.18$ for the asymptotic evolution with $d=3,\ensuremath{\alpha}=6$, which is different from that in the homogeneous case $(\ensuremath{\beta}=0.5)$.