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

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, $V_{2\text{D}}\left(\rho \right)\propto {\rho }^{-\alpha }$ (for $\alpha >2$). We apply our results to the 2D dipolar gases $\left(\alpha =3\right)$ 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.

Figure 1

(Color online) [(a) and (b)] Calculated $s$-wave scattering length and amplitude, $a_3^{\left(0\right)}$ and ${\mathcal{P}}_0^{\left(0\right)}\left(k\right)$, as a function of ${\Delta }_3$ for case A. The inset shows results for ${\left({\tilde{P}}_0^{\left(0\right)}\right)}^{-1}$. Here we set $W=0.1\mu \text{m}$ (see the text). (c) shows the calculated ${\mathcal{P}}_1^{\left(0\right)}\left(k\right)$ for case B. Results for different values of incident wave vector, $k$, are also shown together.

Figure 2

(Color online) Results for case C: (a) is the value of $a_3^{\left(1\right)}/d$ as a function of ${\Delta }_3/d$. In the inset: the upper one shows ${\left({\tilde{P}}_0^{\left(1\right)}\right)}^{-1}$ for zero energy scattering, and the lower one shows the bound-state energy in unit of recoil energy, $E_R\sim {\hslash }^2{\rho }^2/2M{d}^2$. (b) shows the value of ${\mathcal{P}}_0^{\left(1\right)}\left(k\right)$ for different incident wave vectors, $k$. Inset: the magnified plot for the first resonance.