One-dimensional multicomponent fermions with delta function interaction in strong and weak coupling limits: Two-component Fermi gas

TL;DR

This paper analytically solves the Fredholm equations for one-dimensional two-component fermions with delta interactions in various coupling regimes, deriving ground state energy expansions and revealing an intriguing identity between repulsive and attractive cases.

Contribution

It provides the first asymptotic expansions of the ground state energy for polarized two-component fermions in different interaction regimes and proves the equivalence of weakly repulsive and attractive Fredholm equations.

Findings

Derived asymptotic expansions for ground state energy in four regimes.

Proved the equivalence of Fredholm equations for weakly repulsive and attractive interactions.

Identified that energy expansions are identical in the weak coupling region, despite potential non-analyticity.

Abstract

The Fredholm equations for one-dimensional two-component Fermions with repulsive and with attractive delta-function interactions are solved by an asymptotic expansion for A) strong repulsion, B) weak repulsion, C) weak attraction and D) strong attraction. Consequently, we obtain the first few terms of the expansion of ground state energy for the Fermi gas with polarization for these regimes. We also prove that the two sets of the Fredhom equations for weakly repulsive and attractive interactions are identical as long as the integration boundaries match each other between the two sides. Thus the asymptotic expansions of the energies of the repulsive and attractive Fermions are identical to all orders in this region. The identity of the asymptotic expansions may not mean that the energy analytically connects.