Discretized vs. continuous models of p-wave interacting fermions in 1D

TL;DR

This paper establishes optimal mappings between continuous and lattice models for 1D Bose- and Fermi-gases with local interactions, enabling accurate numerical analysis of their ground states and correlations.

Contribution

It introduces a general, optimal mapping framework for 1D Bose- and Fermi-gases, connecting continuous models to lattice models like Bose-Hubbard and XXZ.

Findings

Mapping errors minimized for given discretization

Ground state properties computed numerically

Convergence verified in known limits

Abstract

We present a general mapping between continuous and lattice models of Bose- and Fermi-gases in one dimension, interacting via local two-body interactions. For s-wave interacting bosons we arrive at the Bose-Hubbard model in the weakly interacting, low density regime. The dual problem of p-wave interacting fermions is mapped to the spin-1/2 XXZ model close to the critical point in the highly polarized regime. The mappings are shown to be optimal in the sense that they produce the least error possible for a given discretization length. As an application we examine the ground state of a interacting Fermi gas in a harmonic trap, calculating numerically real-space and momentum-space distributions as well as two-particle correlations. In the analytically known limits the convergence of the results of the lattice model to the continuous one is shown.