Finite-sizing of the Glauber model of critical dynamics

Abstract

We obtain the exact critical relaxation time TL(E), where E is the bulk correlation length, for the Glauber kinetic Ising model of spins on a one-dimensional lattice of finite length L for both periodic and free boundary conditions (BC’s). We show that, independent of the BC’s, the dynamic critical exponent has the well-known value z=2, and we comment on a recent claim that z=1 for this model. The ratio TL(E)/Too(E), in the double limit L,E→oo for fixed x=L/E, approaches a limiting functional form, f T(L/E), the finite-size scaling function. For free BC’s we derive the exact scaling function f T(x)=[ 1+(w(x)/x)2 ]-1, where w(x) is the smallest root of the transcendental equation w tan(w/2)=x. We provide expansions of w(x) in powers of x and x-1 for the regimes of small and large x, respectively, and establish their radii of convergence. The scaling function shows anomalous behavior at small x, f T(x)=x, instead of the usual f T(x)=xz, as x→0. This is because, even for finite L, the lifetime of the slowest dynamical mode diverges for T→ 0 K. For periodic BC’s, with the exception of one system, TL is independent of L, and hence f T=1. The exceptional system, that with an odd number of spins and antiferromagnetic couplings, exhibits frustration at T=0 K, and the scaling function is given by f T(x)=[ 1+(rr/x)2 ]-1. [S1063-651X(96)07306-0]