Exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions

TL;DR

This paper derives exact solutions for a one-dimensional mixture of spinor bosons and fermions with delta interactions, revealing supersymmetry, ground state polarization, and bound state formation depending on interaction type.

Contribution

It introduces new Bethe ansatz equations for the system and analyzes its supersymmetric properties and ground state characteristics.

Findings

Wave function exhibits $SU(2|2)$ supersymmetry.

Ground state is partially polarized with fermions in a spin singlet.

Attractive interactions lead to bound states and boson condensation.

Abstract

The exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions with $\delta$-function interactions are studied. Some new sets of Bethe ansatz equations are obtained by using the graded nest quantum inverse scattering method. Many interesting features appear in the system. For example, the wave function has the $SU(2|2)$ supersymmetry. It is also found that the ground state of the system is partial polarized, where the fermions form a spin singlet state and the bosons are totally polarized. From the solution of Bethe ansatz equations, it is shown that all the momentum, spin and isospin rapidities at the ground state are real if the interactions between the particles are repulsive; while the fermions form two-particle bounded states and the bosons form one large bound state, which means the bosons condensed at the zero momentum point, if the interactions are attractive. The charge, spin and isospin excitations are discussed in detail. The thermodynamic Bethe ansatz equations are also derived and their solutions at some special cases are obtained analytically.