Correlations around an interface

Abstract

We compute one-loop correlation functions for the fluctuations of an interface using a ${\ensuremath{\varphi}}^{4}$ field theory model in $d=2$ and $d=3$ dimensions. We obtain them from Feynman diagrams drawn with a propagator that is the inverse of the Hamiltonian of a P\"oschl-Teller problem. We derive an expression for the propagator in terms of elementary functions, show that it corresponds to the usual spectral sum, and use it to calculate quantities such as the surface tension and interface profile in two and three spatial dimensions. The three-dimensional quantities are rederived in a simple, unified manner, whereas those in two dimensions extend the existing literature, and are applicable to thin films. In addition, we compute the one-loop self-energy, which may be extracted from experiment or from Monte Carlo simulations. Our results may be applied in various scenarios, which include fluctuations around topological defects in cosmology, supersymmetric domain walls, $Z(N)$ bubbles in QCD, domain walls in magnetic systems, interfaces separating Bose-Einstein condensates, and interfaces in binary liquid mixtures.