One Dimensional Phase-Ordering in the Ising Model with Space Decaying Interactions

The study of the phase ordering kinetics of the ferromagnetic one-dimensional Ising model dates back to 1963 (R. J. Glauber, J. Math. Phys. 4, 294) for non conserved order parameter (NCOP) and to 1991 (S. J. Cornell, K. Kaski and R.B. Stinchcombe, Phys. Rev. B 44, 12263) for conserved order parameter (COP). The case of long range interactions J(r) has been widely studied at equilibrium but their effect on relaxation is a much less investigated field. Here we make a detailed numerical and analytical study of both cases, NCOP and COP. Many results are valid for any positive, decreasing coupling J(r), but we focus specifically on the exponential case, Jexp(r) = e-r/R with varying R> 0 , and on the integrable power law case, Jpow(r) = 1 / r1+σ with σ> 0. We find that the asymptotic growth law L(t) is the usual algebraic one, L(t) ∼ t1/z, of the corresponding model with nearest neighbouring interaction (zNCOP= 2 and zCOP= 3) for all models except Jpow for small σ: in the non conserved case when σ≤ 1 (zNCOP= σ+ 1) and in the conserved case when σ→ 0 + (zCOP= 4 β+ 3 , where β= 1 / T is the inverse of the absolute temperature). The models with space decaying interactions also differ markedly from the ones with nearest neighbors due to the presence of many long-lasting preasymptotic regimes, such as an exponential mean-field behavior with L(t) ∼ et, a ballistic one with L(t) ∼ t, a slow (logarithmic) behavior L(t) ∼ ln t and one with L(t) ∼ t1/σ+1. All these regimes and their validity ranges have been found analytically and verified in numerical simulations. Our results show that the main effect of the conservation law is a strong slowdown of COP dynamics if interactions have an extended range. Finally, by comparing the Ising model at hand with continuum approaches based on a Ginzburg–Landau free energy, we discuss when and to which extent the latter represent a faithful description of the former.