Finite-size scaling of the Glauber model of critical dynamics

We obtain the exact critical relaxation time tau(L)(xi), where xi 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 tau(L)(xi)/tau(infinity)(xi), in the double limit L,xi-->infinity for fixed x=L/xi, approaches a limiting functional form, f(tau)(L/xi), the finite-size scaling function. For free BC's we derive the exact scaling function f(tau)(x) = [1+(omega(x)/(x)(2)](-1), where omega(x) is the smallest root of the transcendental equation omega tan(omega/2)=x. We provide expansions of omega(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(tau)(x)approximate to x, instead of the usual f(tau)(x)approximate to x(z), as x-->O. 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, sigma(L) is independent of L, and hence f(tau)=1. The exceptional system, that with an odd number of spins and antiferromagnetic couplings, exhibits frustration at T=O K, and the scaling function is given by f(tau)(x)=[1+(pi/x)(2)](-1).