Finite-Size Scaling in the Ageing Dynamics of the 1 D Glauber-Ising Model

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 for testing the scaling behaviour of the plateau height C∞(2), to which the two-time auto-correlator converges when deep in the finite-size regime.

Keywords: Glauber-Ising model; ageing dynamics; finite-size scaling.

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 ∼$y^{-\lambda /z}$ and (ii) in a fully finite system (full line) which converges to a characteristic plateau $C_{\infty }^{\left(2\right)}$.

Figure 2

Properties of the scaled two-time correlator (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 (24) 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 (25).

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 $x\ge 1$.

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 $0\le x\le N$. It satisfies the periodicity conditions (27). 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 Equation (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 of finite size $N=30$. The full black line is the scaled infinite-size auto-correlator $\left(23\right)$.