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

Abstract

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 $\propto 1/[1+(r/R_c{)}^{\alpha }]$ ($\alpha \ge d$), 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 $\beta =d/\alpha$. 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 $l_{\rho }/R_c$ with $l_{\rho }$ the mean inter-spin distance and the magnetization dynamics is investigated numerically. In the limit of $l_{\rho }/R_c\ll 1$, a coherent many-body dynamics is recovered for the total magnetization despite of the position disorder of spins. In the opposite limit of $l_{\rho }/R_c\gg 1$, 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 $\beta \approx 0.18$ for the asymptotic evolution with $d=3,\alpha =6$, which is different from that in the homogeneous case ($\beta =0.5$).

arXiv comment7 pages, 5 figures, comments welcome

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