Semiclassical calculation of the nucleation rate for first order phase transitions in the 2-dimensional phi⁴-model beyond the thin wall approximation

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.