Clusters of bound particles in the derivative δ-function Bose gas

Abstract

In this paper we discuss a novel procedure for constructing clusters of bound particles in the case of a quantum integrable derivative δ-function Bose gas in one dimension. It is shown 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. We also establish a connection between these 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 associated with the derivative δ-function Bose gas and allows us to study various properties of these clusters like their size and their stability under the variation of the coupling constant.

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

“…Using the above mentioned procedure, we have found that [43] clusters of bound particles associated with fractions of type III become unstable and cease to exist for any small change of the coupling constant. On the other hand, for the case of fractions of type I, clusters of bound particles transmute to a localized bound state containing only one cluster if the value of φ is slightly changed towards the direction of the nearest fraction n/N.…”

“…Through a lengthy analysis which uses the properties (3.5a,b) and (3.6) of a Farey sequence, we have shown that fractions of types I and III belonging to F ′ N −1 satisfy the relation (3.1b), while type II and type IV fractions do not satisfy (3.1b) [43]. Hence, clusters of bound particles are formed for the case of derivative δ-function Bose gas, if and only if the corresponding coupling constant φ/π = a/b ∈ F ′ N −1 is a fraction of type I or type III.…”

“…In analogy with the case of the δ-function Bose gas, one may think that clusters of bound particles can only be constructed for the case of derivative δ-function Bose gas by properly assigning the corresponding quasi-momenta on several concentric circles or circular 'strings' in the complex momentum plane. However, we have recently found that, for the Hamiltonian (1.1) with some special values of the coupling constant η, clusters of bound particles can be constructed in a much simpler way by assigning the corresponding quasi-momenta as equidistant points on a single circle in the complex momentum plane [43]. The purpose of the present article is mainly to review the key results of Ref.…”

“…The purpose of the present article is mainly to review the key results of Ref. [43] and make some additional comment. The arrangement of this article is as follows.…”