Finite-size scaling in the ageing dynamics of the $1D$ Glauber-Ising model

TL;DR

The paper investigates ageing dynamics in the 1D Glauber-Ising model following a zero-temperature quench on a finite periodic chain. By applying an analytic continuation technique, the authors derive exact expressions for both the single-time and two-time spin-spin correlators in finite systems and connect them to their infinite-system counterparts. They confirm finite-size scaling for ageing, and, crucially, verify that the plateau height of the two-time auto-correlator scales as $C_^{(2)} \,\sim\, s^{1/2}/N$ for $s\ll N^2$ (and as $N^{-\lambda}$ for fixed $s$), consistent with the exponents $z=2$ and $\lambda=1$. These results provide a solid exact benchmark for ageing in finite systems and illustrate how finite-size effects modify coarsening dynamics, with potential extensions to higher dimensions and non-integrable dynamics.

Abstract

Single-time and two-time correlators are computed exactly in the $1D$ Glauber-Ising model after a quench to zero temperature and on a periodic chain of finite length $N$, using a simple analytical continuation technique. Besides the general confirmation of finite-size scaling in non-equilibrium dynamics, this allows to test the scaling behaviour of the plateau height $C_{\infty}^{(2)}$ to which the two-time auto-correlator converges, when deep into the finite-size regime.

Figures (5)

• Figure 1: Qualitative dependence of the scaled two-time auto-correlator $C(t,s)$ on the time ratio $y=t/s$ for (i) a spatially infinite system (dashed line) with the power-law behaviour $\sim y^{-\lambda/z}$ and (ii) in a fully finite system (full line) which converges to a characteristic plateau $C_{\infty}^{(2)}$.
• Figure 2: Properties of the scaled two-time correlator (\ref{['1.2']}) in the $1D$ Glauber-Ising model. Left panel: universal decay of the correlator $F_C(y,\xi)$ for large $y$, with $\xi=[0.0,0.2,0.5,1.0,2.0]$ from top to bottom. The inset shows the expected universal power-law decay (\ref{['2.19']}) for large values of $y$. Right panel: decay of the correlator $F_C(y,\xi)$ as a function of $\xi$ for for $y=[1.0,1.5,3.0,4.5]$ from bottom to top on the right of the figure. The inset highlights the expected gaussian decay for large $\xi$ and the dashed lines indicate the leading decay behaviour (\ref{['2.20']}).
• Figure 3: Periodic ring with $N$ sites. Starting from an arbitrary site labelled $0$, the property $C(t;x)=C(t;N-x)$ becomes intuitive for $x\geq 1$.
• Figure 4: (a) Analytically continued function $C(t;x)$ as computed in appendix C, for $t=5$ and $N=[5,10,20,30]$ from top to bottom, in the interval $0\leq x\leq N$. It satisfies the periodicity conditions (\ref{['3.2']}). The full black line gives the initial function $C(0;x)$, for a completely disordered initial lattice. The thin horizontal lines indicate the values $C(t;x)=0$ and $C(t;x)=1$, respectively. (b) Physical scaling function $F_C(1,\xi)$ of eq. (\ref{['1.2']}), for the same values of $t$ and $N$. The full black line corresponds to a completely disordered initial state.
• Figure 5: Two-time scaled auto-correlator $C(ys,s)=F_C(y,0)$ in the $1D$ Glauber-Ising model quenched to $T=0$, as a function of $y=t/s$ for (a) finite systems of sizes $N=[10,20,30,40]$ from top to bottom and for a waiting time $s=5$ and (b) the waiting times $s=[5,20,80]$ from bottom to top and the finite size $N=30$. The full black line is the scaled infinite-size auto-correlator (\ref{['autoC']}).