Exact Analysis of Delta-Function Attractive Fermions and Repulsive Bosons in One-Dimension

TL;DR

This paper provides an exact analytical solution to the Gaudin integral equation for one-dimensional delta-function attractive fermions, deriving asymptotic expansions and connecting to repulsive boson solutions, enhancing understanding of 1D quantum gases.

Contribution

It offers a power series solution to the Gaudin integral equation and links attractive fermion and repulsive boson models in one dimension.

Findings

Analytical asymptotic expansions for strong and weak coupling regimes.

Ground state energy expressed in terms of a dimensionless parameter gamma.

Explicit connection between attractive fermion and repulsive boson solutions.

Abstract

The Gaudin integral equation for the ground state of a one-dimensional delta-function attractive spin-1/2 fermions is solved in the form of power series. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities such as the ground state energy are expressed in terms of a single dimensionless parameter gamma =c/D, where c is the coupling constant and D is the number density. The results agree with those obtained from the perturbation calculations, which include the one in the classical electrostatics originally by Kirchhoff. In the strong coupling limit, the connection to the solutions of the Lieb-Liniger integral equation for the ground state of a one-dimensional delta-function repulsive bose gas is shown explicitly.