Application of a constrained-transfer-matrix method to metastability in the $d$$=$$2$ Ising ferromagnet

TL;DR

This paper uses a transfer-matrix method to numerically analyze metastability in the 2D Ising ferromagnet, confirming droplet-model predictions and exploring finite-size effects and droplet shapes.

Contribution

It introduces a transfer-matrix approach to compute complex constrained free energies, extending the droplet-model analysis into the metastable phase.

Findings

Metastable free energy matches droplet-model predictions.

Different finite-size scaling in weak and intermediate fields.

Critical droplet surface energy follows Wulff construction.

Abstract

Applying a numerical transfer-matrix formalism, we obtain complex-valued constrained free energies for the two-dimensional square-lattice nearest-neighbor Ising ferromagnet below its critical temperature and in an external magnetic field. In particular, we study the imaginary part of the constrained free-energy branch that corresponds to the metastable phase. Although droplets are not introduced explicitly, the metastable free energy is obtained in excellent agreement with field-theoretical droplet-model predictions. The finite-size scaling properties are different in the weak-field and intermediate-field regimes, and we identify the corresponding different critical-droplet shapes. For intermediate fields, we show that the surface free energy of the critical droplet is given by a Wulff construction with the equilibrium surface tension. We also find a prefactor exponent in complete agreement with the field-theoretical droplet model. Our results extend the region of validity for known results of this field-theoretical droplet model, and they indicate that this transfer-matrix approach provides a nonperturbative numerical continuation of the equilibrium free energy into the metastable phase.