Development of order in a symmetric unstable system

Abstract

We consider the problem of growth of order on arbitrary length scales in a thermodynamically unstable system. Starting from simple physical ideas about self-similarity, we develop a renormalization-group formalism to treat the case of an initially disordered square-lattice single spin-flip kinetic Ising model in zero field subjected to a quench to a final temperature $T_F$ lower than the critical temperature $T_c$. We derive and solve recursion relations for the structure factor $\tilde{C}(\overrightarrow{\mathrm{q}},t)$ and for the short-range order parameter $\epsilon \mathrm{}(1,0;t)\mathrm{}$. We find that $\tilde{C}(\overrightarrow{\mathrm{q}},t)$ has, as a function of wave number $q$, an approximately Gaussian peak at $q=0\mathrm{}$ that sharpens to a $\delta$ function as the time, $t$, after the quench goes to infinity. This peak is associated with the growth of domains. At intermediate times the width, $q_w\mathrm{}\left(t\right)\mathrm{}$, decreases as ${(t-t_0)}^{-\frac{1}{2}}$. The area under the peak increases with time logarithmically at first and tends asymptotically to the square of the equilibrium magnetization. The average magnetization is zero for all finite times. At long times and small $\overrightarrow{\mathrm{q}}$, scaling laws follow analytically from the recursion relations. As a function of $t$ and of $T_F$, $\tilde{C}(\overrightarrow{\mathrm{q}},t)$ exhibits a pulse and peak structure similar to, but richer than, that previously found for quenches within the disordered phase. We discuss the relation of our results to Monte Carlo, experimental, and previous theoretical work and conclude with suggestions for improvements.