The classical nucleation rate in two dimensions

TL;DR

This paper calculates the nucleation rate in two-dimensional systems with a real order parameter, extending the thin wall approximation to determine the prefactor and critical bubble energy in first order phase transitions.

Contribution

It provides an explicit calculation of the nucleation rate's prefactor and critical bubble energy in 2D models using an extended thin wall approximation for small energy differences.

Findings

Derived explicit expressions for nucleation rate components in 2D.

Extended thin wall approximation to include prefactor calculation.

Applicable to systems with small energy differences between vacua.

Abstract

In many systems in condensed matter physics and quantum field theory, first order phase transitions are initiated by the nucleation of bubbles of the stable phase. In homogeneous nucleation theory the nucleation rate $\Gamma$ can be written in the form of the Arrhenius law: $\Gamma=\mathcal{A} e^{-\mathcal{H}_{c}}$. Here $\mathcal{H}_{c}$ is the energy of the critical bubble, and the prefactor $\mathcal{A}$ can be expressed in terms of the determinant of the operator of fluctuations near the critical bubble state. In general it is not possible to find explicit expressions for $\mathcal{A}$ and $\mathcal{H}_{c}$. If the difference $\eta$ between the energies of the stable and metastable vacua is small, the constant $\mathcal{A}$ can be determined within the leading approximation in $\eta$, which is an extension of the ``thin wall approximation''. We have done this calculation for the case of a model with a real-valued order parameter in two dimensions.