Pseudopotential of an interaction with a power-law decay in two-dimensional systems

Abstract

We analytically derive the general pseudopotential operator of an arbitrary isotropic interaction for particles confined in two-dimensional (2D) systems using the frame work developed by Huang and Yang for three-dimensional scattering. We also analytically derive the low-energy dependence of the scattering phase shift for an arbitrary interaction with a power-law decaying tail, ${V}_{2\text{D}}(\ensuremath{\rho})\ensuremath{\propto}{\ensuremath{\rho}}^{\ensuremath{-}\ensuremath{\alpha}}$ (for $\ensuremath{\alpha}>2$). We apply our results to the 2D dipolar gases $(\ensuremath{\alpha}=3)$ as an example, calculating the momentum and dipole moment dependences of the pseudopotential for both $s$- and $p$-wave scattering channels if the two scattering particles are in the same 2D layer. Results for the $s$-wave scattering between particles in two different (parallel) layers are also investigated. Our results can be directly applied to the systems of dipolar atoms and/or polar molecules in a general 2D geometry.