Numerical Transfer-Matrix Study of Metastability in the d=2 Ising Model

TL;DR

This paper introduces a numerical transfer-matrix method to analyze metastability in the 2D Ising model, providing insights into free energies, critical clusters, and lifetimes that align with theoretical and simulation results.

Contribution

The study develops a nonperturbative transfer-matrix approach to characterize metastability and critical clusters in the 2D Ising model under nonzero field, extending previous methods.

Findings

Agreement of imaginary free energies with droplet model

Validation of cluster free energy via Wulff construction

Consistent metastable lifetime estimates with Monte Carlo simulations

Abstract

We apply a generalized numerical transfer-matrix method to the 2-d Ising ferro- magnet in a nonzero field to obtain complex constrained free energies. Below $T_c$ certain eigenstates of the transfer matrix are identified as representing a metastable phase. The imaginary parts of the metastable constrained free energies are found to agree with a field-theoretic droplet model for a wide range of fields, allowing us to numerically estimate the average free-energy cost of a critical cluster. We find excellent agreement with the equilibrium cluster free energy obtained by a Wulff construction with the exact, aniso- tropic zero-field surface tension, and we present strong evidence for Gold- stone modes on the critical cluster surface. Our results are also fully con- sistent with average metastable lifetimes from previous Monte-Carlo simula- tions. The study indicates that our constrained-transfer-matrix technique pro- vides a nonperturbative numerical method to obtain an analytic continuation of the free energy around the essential singularity at the first-order transition.