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Minority spin dynamics in non-homogeneous Ising model: diverging timescales and exponents

We investigate the dynamical behaviour of the Ising model under a zero temperature quench with the initial fraction of up spins $0\leq x\leq 1$. In one dimension, the known results for persistence probability are verified; it shows algebraic decay for both up and down spins asymptotically with different exponents. It is found that the conventional finite size scaling is valid here. In two dimensions however, the persistence probabilities are no longer algebraic; in particular for $x\leq 0.5$, persistence for the up (minority) spins shows the behaviour $P_{min}(t) \sim t^{-γ}\exp(-(t/τ)^δ)$ with time $t$, while for the down (majority) spins, $P_{maj}(t)$ approaches a finite value. We find that the timescale $τ$ diverges as $(x_c-x)^{- λ}$, where $x_c=0.5$ and $λ\simeq2.31$. The exponent $γ$ varies as $θ_{2d}+c_0(x_c-x)^β$ where $θ_{2d}\simeq0.215$ is very close to the persistence exponent in two dimensions; $β\simeq1$. The results in two dimensions can be understood qualitatively by studying the exit probability, which for different system size is found to have the form $E(x) = f\big[(\frac{x-x_c}{x_c})L^{1/ν}\big]$, with $ν\approx 1.47$. This result suggests that $τ\sim L^{\tilde{z}}$, where $\tilde{z} = \fracλν = 1.57 \pm 0.11$ is an exponent not explored earlier.