  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. Traditionally, this process is described by the semiclassical
nucleation theory developed by Langer and, in the context of quantum field
theory, by Callan and Coleman. They have shown that 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. It is not possible to find explicit expressions for the constants
$\mathcal{A}$ and $\mathcal{H}_{c}$ in the general case of a finite difference
$\eta$ between the energies of the stable and metastable vacua. For small
$\eta$, the constant $\mathcal{A}$ can be determined within the leading
approximation in $\eta$, which is an extension of the ``thin wall
approximation''. We have calculated the leading approximation of the prefactor
for the case of a model with a real-valued order parameter field in two
dimensions.

Statistical Mechanics