Super Rogers-Szeg\"o polynomials associated with $BC_N$ type of Polychronakos spin chains

TL;DR

This paper derives new multivariate super Rogers-Szeg"o polynomials linked to $BC_N$ type Polychronakos spin chains, expanding the mathematical framework for analyzing these integrable models with supersymmetric features.

Contribution

It introduces novel multivariate super Rogers-Szeg"o polynomials for $BC_N$ chains, constructs their generating functions, and establishes recursion relations for their partition functions.

Findings

Derived partition functions for $BC_N$ Polychronakos chains with supersymmetry.

Constructed generating functions for the new super Rogers-Szeg"o polynomials.

Established recursion relations for chain partition functions.

Abstract

As is well known, multivariate Rogers-Szeg\"o polynomials are closely connected with the partition functions of the $A_{N-1}$ type of Polychronakos spin chains having long-range interactions. Applying the `freezing trick', here we derive the partition functions for a class of $BC_N$ type of Polychronakos spin chains containing supersymmetric analogues of polarized spin reversal operators and subsequently use those partition functions to obtain novel multivariate super Rogers-Szeg\"o (SRS) polynomials depending on four types of variables. We construct the generating functions for such SRS polynomials and show that these polynomials can be written as some bilinear combinations of the $A_{N-1}$ type of SRS polynomials. We also use the above mentioned generating functions to derive a set of recursion relations for the partition functions of the $BC_N$ type of Polychronakos spin chains involving different numbers of lattice sites and internal degrees of freedom.