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

TL;DR

This paper combines scalar field theory and transfer-matrix methods to analyze metastability in a long-range interacting Ising model, revealing significant corrections to nucleation free energy and finite-range scaling behavior.

Contribution

It introduces a novel combined approach using field theory and transfer-matrix analysis to study metastability in long-range Ising models, including corrections to nucleation free energy.

Findings

Analytic continuation of free energy across first-order transition.

Verification of nucleation free-energy form with corrections.

Finite-range scaling behavior observed near spinodal.

Abstract

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$$\times$$\infty$ quasi-one-dimensional Ising model. Using the field theory, we find the analytic continuation $\tilde f$ of the free energy across the first-order transition, assuming that the system escapes the metastable state by 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 subdominant eigenvectors of the transfer matrix a complex-valued ``constrained'' free-energy density $f_\alpha$ computed directly from the matrix. For the eigenvector with an associated magnetization most strongly opposed to the applied magnetic field, $f_\alpha$ exhibits finite-range scaling behavior in agreement with $\tilde f$ 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.