Monte Carlo Study of the Critical Dynamics at the Surface of an Ising Model

Abstract

The model studied is the ferromagnetic 3D-Ising model on a simple cubic lattice with a free surface, i.e., the hamiltonian

$$ - \beta H = K\mathop \Sigma \limits_{ < ij > _{\text{b}} } {\text{s}}_{\text{i}} s_{\text{j}} + {\text{K}}_{\text{s}} \mathop \Sigma \limits_{ < {\text{ij}} > _{\text{s}} } {\text{s}}_{\text{i}} {\text{s}}_{\text{j}} ,{\text{ s}}_{\text{i}} = \pm 1,{\text{ }}(1)$$

(1)

where the second sum runs over spin pairs where both spins are in the surface, the first sum runs over all other nearest-neighbor spin pairs, and β = 1/kT, with k the Boltzmann constant and T temperature. The static properties of this model are well known [1–4]. For temperatures below the bulk critical temperature, K_c ≈ 0.221654 [5], both the surface and the bulk are ferromagnetically ordered. For strong surface coupling, i.e., ratios of the surface to bulk coupling, R = K_s/K, above a tricritical value, R_t ≈ 1.52 [2], the surface orders at a higher temperature than the bulk. We have studied the critical dynamics of the surface layer in this model at the ordinary transition (R = 1.0) and at the tricritical point (R = 1.52, also called the special transition).