Finite-range-scaling analysis of metastability in an Ising model with long-range interactions

We apply both a scalar field theory and a recently developed transfer-matrix method to study the stationary properties of metastability in a two-state model with weak, long-range interactions: the N×∞ quasi-one-dimensional Ising model. Using the field theory, we find the analytic continuation f̃ of the free energy across the first-order transition, assuming that the system escapes the metastable state by the nucleation of noninteracting droplets. We find that corrections to the field dependence are substantial, and, by solving the Euler-Lagrange equation for the model numerically, we have verified the form of the free-energy cost of nucleation, including the first correction. In the transfer-matrix method, we associate with the subdominant eigenvectors of the transfer matrix a complex-valued ‘‘constrained’’ free-energy density ${\mathit{f}}_{\mathrm{\alpha }}$ computed directly from the matrix. For the eigenvector with an associated magnetization most strongly opposed to the applied magnetic field, ${\mathit{f}}_{\mathrm{\alpha }}$ exhibits finite-range scaling behavior in agreement with f̃ over a wide range of temperatures and fields, extending nearly to the classical spinodal. Some implications of these results for numerical studies of metastability are discussed.