Pseudo-potential of a power-law decaying interaction in two-dimensional systems

TL;DR

This paper derives a general pseudo-potential operator for isotropic interactions in 2D systems with power-law decay, providing analytical expressions for scattering phase-shifts and applying them to dipolar gases.

Contribution

It introduces a new analytical framework for the pseudo-potential of arbitrary isotropic 2D interactions with power-law tails, extending Huang and Yang's 3D approach.

Findings

Derived the pseudo-potential operator for 2D isotropic interactions.

Analyzed low-energy scattering phase-shifts for power-law potentials.

Applied results to 2D dipolar gases, detailing s- and p-wave scattering behaviors.

Abstract

We analytically derive the general pseudo-potential operator of an arbitrary isotropic interaction for particles confined in two-dimensional (2D) systems, using the frame work developed by Huang and Yang for 3D scattering. We also analytically derive the low energy dependence of the scattering phase-shift for an arbitrary interaction with a power-law decaying tail, $V_{\rm 2D}(\rho)\propto \rho^{-\alpha}$ (for $\alpha>2$). We apply our results to the 2D dipolar gases ($\alpha=3$) as an example, calculating the momentum and dipole moment dependence of the pseudo-potential 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.