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

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.