Appearance of branched motifs in the spectra of \(BC_N\) type Polychronakos spin chains

• Published in: arXiv.org 2019-09
• Main Authors: Basu-Mallick, Bireswar, Sinha, Madhurima
• Format: Article
• Language: English
• Subjects: Binary digits; Chain branching; Chains; Deformation; Digits; Energy levels; Fermions; Functions (mathematics); Mathematical analysis; Multivariate analysis; Operators (mathematics); Partitions; Partitions (mathematics); Polynomials; Spectra; Supersymmetry
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1909.12125

As is well known, energy levels appearing in the highly degenerate spectra of the \(A_{N-1}\) type of Haldane-Shastry and Polychronakos spin chains can be classified through the motifs, which are characterized by some sequences of the binary digits like `0' and `1'. In a similar way, at present we classify all energy levels appearing in the spectra of the \(BC_N\) type of Polychronakos spin chains with Hamiltonians containing supersymmetric analogue of polarized spin reversal operators. To this end, we show that the \(BC_N\) type of multivariate super Rogers-Szeg\"o (SRS) polynomials, which at a certain limit reduce to the partition functions of the later type of Polychronakos spin chains, satisfy some recursion relation involving a \(q\)-deformation of the elementary supersymmetric polynomials. Subsequently, we use a Jacobi-Trudi like formula to define the corresponding \(q\)-deformed super Schur polynomials and derive a novel expression for the \(BC_N\) type of multivariate SRS polynomials as suitable linear combinations of the \(q\)-deformed super Schur polynomials. Such an expression for SRS polynomials leads to a complete classification of all energy levels appearing in the spectra of the \(BC_N\) type of Polychronakos spin chains through the `branched' motifs, which are characterized by some sequences of integers of the form \((\delta_1, \delta_2,..., \delta_{N-1}|l)\), where \(\delta_i \in \{ 0,1 \}\) and \( l \in \{ 0,1,...,N \}\). Finally, we derive an extended boson-fermion duality relation among the restricted super Schur polynomials and show that the partition functions of the \(BC_N\) type of Polychronakos spin chains also exhibit similar type of duality relation.