Ground state of one-dimensional bosons with delta interaction: link to the BCS model

TL;DR

This paper establishes a connection between the ground state of one-dimensional bosons with delta interactions and the BCS model, showing that Bethe roots relate to Laguerre polynomial zeros and deriving explicit energy results.

Contribution

It reveals a novel link between weakly interacting bosons and the strong coupling BCS pairing model within integrable 1D systems.

Findings

Bethe roots satisfy Richardson equations in the weak interaction limit.

Bethe roots are zeros of Laguerre polynomials.

Explicit expressions for ground state energy and excitations.

Abstract

The Bethe roots describing the ground state energy of the integrable 1D model of interacting bosons with weakly repulsive two-body delta interactions are seen to satisfy the set of Richardson equations appearing in the strong coupling limit of an integrable BCS pairing model. The BCS model describes boson-boson interactions with zero centre of mass momentum of pairs. It follows that the Bethe roots of the weakly interacting boson model are given by the zeros of Laguerre polynomials. The ground state energy and the lowest excitation are obtained explicitly via the Bethe roots. A direct link has thus been established, in the context of integrable 1D models, between bosons interacting via weakly repulsive two-body delta-interactions and strongly interacting Cooper pairs of bosons.