Finite-size scaling of the Glauber model of critical dynamics

We obtain the exact critical relaxation time τL({ξ}), where {ξ} 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 τL({ξ})/τ_∞({ξ}), in the double limit L,{ξ}{→}{∞} for fixed x=L/{ξ}, approaches a limiting functional form, f_τ(L/{ξ}), the finite-size scaling function. For free BC's we derive the exact scaling function f_τ(x)=[1+({ω}(x)/x)²{]}^{{{-}}1}, where ω(x) is the smallest root of the transcendental equation ωtan(ω/2)=x. We provide expansions of ω(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, ${f}_{{{τ}}}(x)x, instead of the usual{f}_{{{τ}}}(x){x}ᶻ$, 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, τL is independent of L, and hence f_τ=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_τ(x)=[1+({π}/x)²{]}^{{{-}}1}$. {} 1996 The American Physical Society.