Clusters of bound particles in a quantum integrable many-body system and number theory

TL;DR

This paper constructs and classifies clusters of bound particles in a quantum integrable Bose gas, revealing a novel connection between the system's coupling constants and Farey sequences in number theory, and analyzing their properties.

Contribution

It introduces a method to form bound particle clusters at specific coupling constants linked to Farey sequences, providing a new classification framework for these clusters.

Findings

Clusters form at special coupling constants related to Farey fractions.

The size and stability of clusters depend on the coupling constant.

A classification scheme for clusters based on number theory is developed.

Abstract

We construct clusters of bound particles for a quantum integrable derivative delta-function Bose gas in one dimension. It is found that clusters of bound particles can be constructed for this Bose gas for some special values of the coupling constant, by taking the quasi-momenta associated with the corresponding Bethe state to be equidistant points on a single circle in the complex momentum plane. Interestingly, there exists a connection between the above mentioned special values of the coupling constant and some fractions belonging to the Farey sequences in number theory. This connection leads to a classification of the clusters of bound particles for the derivative delta-function Bose gas and the determination of various properties of these clusters like their size and their stability under a variation of the coupling constant.