Coordinate space representation of a one-dimensional odd-parity pseudopotential

We propose a discrete-space representation of a one-dimensional zero-range odd-parity pseudopotential. The proposed representation is validated by applying it to the analytically solvable case of two fermions in a harmonic trap and successfully recovering the exact energy spectrum and eigenfunctions. Furthermore, we use the square-well and modified Pöschl–Teller potentials as finite-range representations of the odd-parity interaction and study their convergence to the contact interaction when the range tends to zero. Finally, we perform natural orbital analysis and compute the eigenvalues of the one-body density matrix for different particle numbers, examining their dependence on the one-dimensional scattering length and identifying distinct physical regimes.

💡 Research Summary

This paper addresses the longstanding challenge of representing p‑wave interactions in one‑dimensional (1D) ultracold fermionic systems. In 1D, the usual notion of p‑wave scattering, which relies on angular momentum, collapses; instead, the correct low‑energy scattering physics is captured by an odd‑parity (antisymmetric) zero‑range pseudopotential. Conventional δ‑function contact potentials fail to enforce the required boundary condition while preserving antisymmetry, because a δ‑potential only acts when two particles occupy the same point, which never occurs for an antisymmetric wavefunction. The authors therefore construct a proper representation based on the derivative of the δ‑function, −g_F δ′(x_ij) ∂̂_ij, where g_F is related to the 1D scattering length a_1D via g_F = 2ℏ²a_1D/m.

The core contribution is an explicit discrete‑space formulation of this odd‑parity interaction that is exact in the zero‑range limit. The derivation proceeds by exploiting the analytically known bound‑state solution for two spin‑polarized fermions in free space with the odd‑parity contact interaction. The relative‑coordinate wavefunction is ψ_rel(x)=sgn(x) exp(−|x|/a_1D), with binding energy E_rel = −4ℏ⁶/(m³g_F²). By inserting this exact solution into the stationary Schrödinger equation and using the standard second‑difference approximation for the Laplacian on a uniform grid (spacing Δx), the authors solve for the on‑site potential V that reproduces the same wavefunction. The resulting potential is non‑zero only at the two nearest‑neighbour points x=±Δx, with magnitude

V(±Δx)= (ℏ²/mΔx²)