Zero-temperature coarsening in the two-dimensional long-range Ising model
Henrik Christiansen, Suman Majumder, Wolfhard Janke
Abstract
We investigate the nonequilibrium dynamics following a quench to zero temperature of the nonconserved Ising model with power-law decaying long-range interactions $\ensuremath{\propto}1/{r}^{d+\ensuremath{\sigma}}$ in $d=2$ spatial dimensions. The zero-temperature coarsening is always of special interest among nonequilibrium processes, because often peculiar behavior is observed. We provide estimates of the nonequilibrium exponents, viz., the growth exponent $\ensuremath{\alpha}$, the persistence exponent $\ensuremath{\theta}$, and the fractal dimension ${d}_{f}$. It is found that the growth exponent $\ensuremath{\alpha}\ensuremath{\approx}3/4$ is independent of $\ensuremath{\sigma}$ and different from $\ensuremath{\alpha}=1/2$, as expected for nearest-neighbor models. In the large $\ensuremath{\sigma}$ regime of the tunable interactions only the fractal dimension ${d}_{f}$ of the nearest-neighbor Ising model is recovered, while the other exponents differ significantly. For the persistence exponents $\ensuremath{\theta}$ this is a direct consequence of the different growth exponents $\ensuremath{\alpha}$ as can be understood from the relation $d\ensuremath{-}{d}_{f}=\ensuremath{\theta}/\ensuremath{\alpha}$; they just differ by the ratio of the growth exponents $\ensuremath{\approx}3/2$. This relation has been proposed for annihilation processes and later numerically tested for the $d=2$ nearest-neighbor Ising model. We confirm this relation for all $\ensuremath{\sigma}$ studied, reinforcing its general validity.